Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation
NASA Astrophysics Data System (ADS)
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
The LEM exponential integrator for advection-diffusion-reaction equations
NASA Astrophysics Data System (ADS)
Caliari, Marco; Vianello, Marco; Bergamaschi, Luca
2007-12-01
We implement a second-order exponential integrator for semidiscretized advection-diffusion-reaction equations, obtained by coupling exponential-like Euler and Midpoint integrators, and computing the relevant matrix exponentials by polynomial interpolation at Leja points. Numerical tests on 2D models discretized in space by finite differences or finite elements, show that the Leja-Euler-Midpoint (LEM) exponential integrator can be up to 5 times faster than a classical second-order implicit solver.
An operator splitting algorithm for the three-dimensional advection-diffusion equation
NASA Astrophysics Data System (ADS)
Khan, Liaqat Ali; Liu, Philip L.-F.
1998-09-01
Operator splitting algorithms are frequently used for solving the advection-diffusion equation, especially to deal with advection dominated transport problems. In this paper an operator splitting algorithm for the three-dimensional advection-diffusion equation is presented. The algorithm represents a second-order-accurate adaptation of the Holly and Preissmann scheme for three-dimensional problems. The governing equation is split into an advection equation and a diffusion equation, and they are solved by a backward method of characteristics and a finite element method, respectively. The Hermite interpolation function is used for interpolation of concentration in the advection step. The spatial gradients of concentration in the Hermite interpolation are obtained by solving equations for concentration gradients in the advection step. To make the composite algorithm efficient, only three equations for first-order concentration derivatives are solved in the diffusion step of computation. The higher-order spatial concentration gradients, necessary to advance the solution in a computational cycle, are obtained by numerical differentiations based on the available information. The simulation characteristics and accuracy of the proposed algorithm are demonstrated by several advection dominated transport problems.
Solving the advection-diffusion equations in biological contexts using the cellular Potts model
NASA Astrophysics Data System (ADS)
Dan, Debasis; Mueller, Chris; Chen, Kun; Glazier, James A.
2005-10-01
The cellular Potts model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approximate the Laplacian. Directed spin flips in the CPM handle the advective movement of the fluid particles. A constraint on relative velocities in the fluid explicitly accounts for fluid viscosity. We use the CPM to solve various diffusion examples including multiple instantaneous sources, continuous sources, moving sources, and different boundary geometries and conditions to validate our approximation against analytical and established numerical solutions. We also verify the CPM results for Poiseuille flow and Taylor-Aris dispersion.
Lin, Guang; Tartakovsky, Alexandre M.
2010-04-01
In this study, we solve the three-dimensional stochastic Darcy's equation and stochastic advection-diffusion-dispersion equation using a probabilistic collocation method (PCM) on sparse grids. Karhunen-Lo\\`{e}ve (KL) decomposition is employed to represent the three-dimensional log hydraulic conductivity $Y=\\ln K_s$. The numerical examples which demonstrate the convergence of PCM are presented. It appears that the faster convergence rate in the variance can be obtained by using the Jacobi-chaos representing the truncated Gaussian distributions than using the Hermite-chaos for the Gaussian distribution. The effect of dispersion coefficient on the mean and standard deviation of the hydraulic head and solute concentration is investigated. Additionally, we also study how the statistical properties of the hydraulic head and solute concentration vary while using different types of random distributions and different standard deviations of random hydraulic conductivity.
An enriched finite element method to fractional advection-diffusion equation
NASA Astrophysics Data System (ADS)
Luan, Shengzhi; Lian, Yanping; Ying, Yuping; Tang, Shaoqiang; Wagner, Gregory J.; Liu, Wing Kam
2017-03-01
In this paper, an enriched finite element method with fractional basis [ 1,x^{α }] for spatial fractional partial differential equations is proposed to obtain more stable and accurate numerical solutions. For pure fractional diffusion equation without advection, the enriched Galerkin finite element method formulation is demonstrated to simulate the exact solution successfully without any numerical oscillation, which is advantageous compared to the traditional Galerkin finite element method with integer basis [ 1,x] . For fractional advection-diffusion equation, the oscillatory behavior becomes complex due to the introduction of the advection term which can be characterized by a fractional element Peclet number. For the purpose of addressing the more complex numerical oscillation, an enriched Petrov-Galerkin finite element method is developed by using a dimensionless fractional stabilization parameter, which is formulated through a minimization of the residual of the nodal solution. The effectiveness and accuracy of the enriched finite element method are demonstrated by a series of numerical examples of fractional diffusion equation and fractional advection-diffusion equation, including both one-dimensional and two-dimensional, steady-state and time-dependent cases.
NASA Astrophysics Data System (ADS)
Paparella, F.; Oliveri, F.
2009-04-01
The interplay of advection, reaction and diffusion terms in ADR equations is a rather difficult one to be modeled numerically. The kind of spurious oscillations that is usually harmless for non-reacting scalars is often amplified without bounds by reaction terms. Furthermore, in most biogeochimical applications, such as mesoscale or global-scale plankton modeling, the diffusive fluxes may be smaller than the numerical ones. Inspired by the particle-mesh methods used by cosmologists, we propose to discretize on a grid only the diffusive term of the equation, and solve the advection-reaction terms as ordinary differential equations along the characteristic lines. Diffusion happens by letting the concentration field carried by each particle to relax towards the diffusive field known on the grid, without redistributing the particles. This method, in the limit of vanishing diffusivity and for a fixed mesh size, recovers the advection-reaction solution with no numerical diffusion. We show some example numerical solutions of the ADR equations stemming from a simple predator-prey model.
New Solution of Diffusion-Advection Equation for Cosmic-Ray Transport Using Ultradistributions
NASA Astrophysics Data System (ADS)
Rocca, M. C.; Plastino, A. R.; Plastino, A.; Ferri, G. L.; de Paoli, A.
2015-11-01
In this paper we exactly solve the diffusion-advection equation (DAE) for cosmic-ray transport. For such a purpose we use the Theory of Ultradistributions of J. Sebastiao e Silva, to give a general solution for the DAE. From the ensuing solution, we obtain several approximations as limiting cases of various situations of physical and astrophysical interest. One of them involves Solar cosmic-rays' diffusion.
Quantification of numerical diffusivity due to TVD schemes in the advection equation
NASA Astrophysics Data System (ADS)
Bidadi, Shreyas; Rani, Sarma L.
2014-03-01
In this study, the numerical diffusivity νnum inherent to the Roe-MUSCL scheme has been quantified for the scalar advection equation. The Roe-MUSCL scheme employed is a combination of: (1) the standard extension of the original Roe's formulation to the advection equation, and (2) van Leer's Monotone Upwind Scheme for Conservation Laws (MUSCL) technique that applies a linear variable reconstruction in a cell along with a scaled limiter function. An explicit expression is derived for the numerical diffusivity in terms of the limiter function, the distance between the cell centers on either side of a face, and the face-normal velocity. The numerical diffusivity formulation shows that a scaled limiter function is more appropriate for MUSCL in order to consistently recover the central-differenced flux at the maximum value of the limiter. The significance of the scaling factor is revealed when the Roe-MUSCL scheme, originally developed for 1-D scenarios, is applied to 2-D scalar advection problems. It is seen that without the scaling factor, the MUSCL scheme may not necessarily be monotonic in multi-dimensional scenarios. Numerical diffusivities of the minmod, superbee, van Leer and Barth-Jesperson TVD limiters were quantified for four problems: 1-D advection of a step function profile, and 2-D advection of step, sinusoidal, and double-step profiles. For all the cases, it is shown that the superbee scheme provides the lowest numerical diffusivity that is also most confined to the vicinity of the discontinuity. The minmod scheme is the most diffusive, as well as active in regions away from high gradients. As expected, the grid resolution study demonstrates that the magnitude and the spatial extent of the numerical diffusivity decrease with increasing resolution.
Comparison of Nonlinear and Linear Stabilization Schemes for Advection-Diffusion Equations
NASA Astrophysics Data System (ADS)
Grove, R. R.; Heister, T.
2015-12-01
Accurately solving advection-diffusion equations that appear in the finite element discretization of a mantle convection simulation is an important computational issue to the computational geoscience community. This is because it allows for users studying mantle convection to create reliable simulations for something as small and simple as a 2D simulation on their personal laptop to something as complex as a massively parallel 3D simulation on their university supercomputer. Standard finite element discretizations of advection-diffusion equations introduce unphysical oscillations around steep gradients. Therefore, stabilization must be added to the discrete formulation to obtain correct solutions. Using the open source scientific library ASPECT, the SUPG and Entropy Viscosity schemes are compared using stationary and non-stationary test equations. Differences in maximum overshoot and undershoot, smear, and convergence orders are compared to see if improvements can be made to the existing numerical method existing in ASPECT.
Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.
Sánchez-Garduño, Faustino; Pérez-Velázquez, Judith
This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h'(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1) h'(u) is constant k, (2) h'(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.
Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations
Sánchez-Garduño, Faustino
2016-01-01
This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h′(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1) h′(u) is constant k, (2) h′(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE. PMID:27689131
Pointwise interactions of finite element modeling of advection-diffusion equations
Yeh, G.T.
1984-07-01
Pointwise iteration techniques including successive under-relaxation (SUR), Gauss-Seidel (G-S), and successive over-relaxation (SOR) schemes, are applied to advection-diffusion equations to derive the matrix equation with finite element methods. These schemes are tested using two simple examples for which analytical solutions are available so that numerical results can be checked to ensure code consistency. Numerical experiments indicate that the iteration schemes, if convergent, produce almost identical solutions as those obtained by the direct elimination scheme. For diffusion dominant transport, all three iteration schemes generate convergent computations. However, for advection-diffusion equally dominant or advection dominant transport, only SUR and G-S schemes yield convergent calculations, the SOR scheme leads to divergent computations. Pointwise iteration schemes offer substantial savings in central process unit (CPU) memory over the direct elimination scheme, even for the small, two-dimensional verification example, without complicating the programming efforts and, in the meantime, keeps the CPU time comparable. A realistic, hypothetical problem is used to demonstrate the applicability and versatility of pointwise iterations and direct elimination schemes. The saving in CPU memory using the pointwise iterations is more than tenfold that using the direct elimination solution for this hypothetical problem. The saving in CPU time is even better, more than 40 fold.
Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes
NASA Astrophysics Data System (ADS)
Montecinos, Gino I.; Toro, Eleuterio F.
2014-10-01
Following Cattaneo's original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advection-diffusion-reaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a locally implicit version of the ADER method to solve these stiff systems to high accuracy. The new ingredient of the numerical methodology is a locally implicit solution of the generalised Riemann problem. We illustrate the formulations and the resulting numerical approach by solving the compressible Navier-Stokes equations.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2016-04-01
Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods were combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (2013) [19] to efficiently solve stiff nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection-diffusion-reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.
Multigrid techniques for the solution of the passive scalar advection-diffusion equation
NASA Technical Reports Server (NTRS)
Phillips, R. E.; Schmidt, F. W.
1985-01-01
The solution of elliptic passive scalar advection-diffusion equations is required in the analysis of many turbulent flow and convective heat transfer problems. The accuracy of the solution may be affected by the presence of regions containing large gradients of the dependent variables. The multigrid concept of local grid refinement is a method for improving the accuracy of the calculations in these problems. In combination with the multilevel acceleration techniques, an accurate and efficient computational procedure is developed. In addition, a robust implementation of the QUICK finite-difference scheme is described. Calculations of a test problem are presented to quantitatively demonstrate the advantages of the multilevel-multigrid method.
Preconditioned iterative methods for space-time fractional advection-diffusion equations
NASA Astrophysics Data System (ADS)
Zhao, Zhi; Jin, Xiao-Qing; Lin, Matthew M.
2016-08-01
In this paper, we propose practical numerical methods for solving a class of initial-boundary value problems of space-time fractional advection-diffusion equations. First, we propose an implicit method based on two-sided Grünwald formulae and discuss its stability and consistency. Then, we develop the preconditioned generalized minimal residual (preconditioned GMRES) method and preconditioned conjugate gradient normal residual (preconditioned CGNR) method with easily constructed preconditioners. Importantly, because resulting systems are Toeplitz-like, fast Fourier transform can be applied to significantly reduce the computational cost. We perform numerical experiments to demonstrate the efficiency of our preconditioners, even in cases with variable coefficients.
Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations
NASA Astrophysics Data System (ADS)
Busto, S.; Toro, E. F.; Vázquez-Cendón, M. E.
2016-12-01
We construct, analyze and assess various schemes of second order of accuracy in space and time for model advection-diffusion-reaction differential equations. The constructed schemes are meant to be of practical use in solving industrial problems and are derived following two related approaches, namely ADER and MUSCL-Hancock. Detailed analysis of linear stability and local truncation error are carried out. In addition, the schemes are implemented and assessed for various test problems. Empirical convergence rate studies confirm the theoretically expected accuracy in both space and time.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2013-11-01
Implicit integration factor (IIF) methods are originally a class of efficient “exactly linear part” time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection-diffusion-reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.
Barth, Andrea Lang, Annika
2012-12-15
In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, cadlag, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L{sup 2} and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler-Maruyama approximation. Finally, simulations complete the paper.
Application of a Particle Method to the Advection-Diffusion-Reaction Equation
NASA Astrophysics Data System (ADS)
Paster, A.; Bolster, D.; Benson, D. A.
2012-12-01
A reaction between two chemical species can only happen if molecules collide and react. Thus, the mixing of a system can become a limiting factor in the onset of reaction. Solving for reaction rate in a well-mixed system is typically a straightforward task. However, when incomplete mixing kicks in, obtaining a solution becomes more challenging. Since reaction can only happen in regions where both reactants co-exist, the incomplete mixing may slow down the reaction rate, when compared to a well-mixed system. The effect of incomplete mixing upon reaction is a highly important aspect of various processes in natural and engineered systems, ranging from mineral precipitation in geological formations to groundwater remediation in aquifers. We study a relatively simple system with a bi-molecular irreversible kinetic reaction A+B → Ø where the underlying transport of reactants is governed by an advection-diffusion equation, and the initial concentrations are given in terms of an average and a perturbation. Such a system does not have an analytical solution to date, even for the zero advection case. We model the system by a Monte Carlo particle tracking method, where particles represent some reactant mass. In this method, diffusion is modeled by a random walk of the particles, and reaction is modeled by annihilation of particles. The probability of the annihilation is proportional to the reaction rate constant and the probability density associated with particle co-location. We study the numerical method in depth, characterizing typical numerical errors and time step restrictions. In particular, we show that the numerical method converges to the advection-diffusion-reaction equation at the limit Δt →0. We also rigorously derive the relationship between the initial number of particles in the system and the initial concentrations perturbations represented by that number. We then use the particle simulations of zero-advection system to demonstrate the well
Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport
Litvinenko, Yuri E.; Effenberger, Frederic
2014-12-01
Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.
Analytical Solutions of a Fractional Diffusion-advection Equation for Solar Cosmic-Ray Transport
NASA Astrophysics Data System (ADS)
Litvinenko, Yuri E.; Effenberger, Frederic
2014-12-01
Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.
Kordilla, Jannes; Pan, Wenxiao; Tartakovsky, Alexandre
2014-12-14
We propose a novel smoothed particle hydrodynamics (SPH) discretization of the fully coupled Landau-Lifshitz-Navier-Stokes (LLNS) and stochastic advection-diffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and the self-diffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations is found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for coupled LLNS and advection-diffusion equations, we simulate the interface between two miscible fluids. We study formation of the so-called "giant fluctuations" of the front between light and heavy fluids with and without gravity, where the light fluid lies on the top of the heavy fluid. We find that the power spectra of the simulated concentration field are in good agreement with the experiments and analytical solutions. In the absence of gravity, the power spectra decay as the power -4 of the wavenumber-except for small wavenumbers that diverge from this power law behavior due to the effect of finite domain size. Gravity suppresses the fluctuations, resulting in much weaker dependence of the power spectra on the wavenumber. Finally, the model is used to study the effect of thermal fluctuation on the Rayleigh-Taylor instability, an unstable dynamics of the front between a heavy fluid overlaying a light fluid. The front dynamics is shown to agree well with the analytical solutions.
Kordilla, Jannes; Pan, Wenxiao Tartakovsky, Alexandre
2014-12-14
We propose a novel smoothed particle hydrodynamics (SPH) discretization of the fully coupled Landau-Lifshitz-Navier-Stokes (LLNS) and stochastic advection-diffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and the self-diffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations is found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for coupled LLNS and advection-diffusion equations, we simulate the interface between two miscible fluids. We study formation of the so-called “giant fluctuations” of the front between light and heavy fluids with and without gravity, where the light fluid lies on the top of the heavy fluid. We find that the power spectra of the simulated concentration field are in good agreement with the experiments and analytical solutions. In the absence of gravity, the power spectra decay as the power −4 of the wavenumber—except for small wavenumbers that diverge from this power law behavior due to the effect of finite domain size. Gravity suppresses the fluctuations, resulting in much weaker dependence of the power spectra on the wavenumber. Finally, the model is used to study the effect of thermal fluctuation on the Rayleigh-Taylor instability, an unstable dynamics of the front between a heavy fluid overlaying a light fluid. The front dynamics is shown to agree well with the analytical solutions.
Kordilla, Jannes; Pan, Wenxiao; Tartakovsky, Alexandre M.
2014-12-14
We propose a novel Smoothed Particle Hydrodynamics (SPH) discretization of the fully-coupled Landau-Lifshitz-Navier-Stokes (LLNS) and advection-diffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and self-diffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations are found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for the coupled LLNS and advection-diffusion equations, we simulate the interface between two miscible fluids. We study the formation of the so-called giant fluctuations of the front between light and heavy fluids with and without gravity, where the light fluid lays on the top of the heavy fluid. We find that the power spectra of the simulated concentration field is in good agreement with the experiments and analytical solutions. In the absence of gravity the the power spectra decays as the power -4 of the wave number except for small wave numbers which diverge from this power law behavior due to the effect of finite domain size. Gravity suppresses the fluctuations resulting in the much weaker dependence of the power spectra on the wave number. Finally the model is used to study the effect of thermal fluctuation on the Rayleigh-Taylor instability, an unstable dynamics of the front between a heavy fluid overlying a light fluid. The front dynamics is shown to agree well with the analytical solutions.
Preconditioned time-difference methods for advection-diffusion-reaction equations
Aro, C.; Rodrigue, G.; Wolitzer, D.
1994-12-31
Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions placed on stepsize due to stability. Stability bounds for explicit time differencing methods on advection-diffusion-reaction problems are generally quite severe and implicit methods are used instead. The linear systems arising from these implicit methods are large and sparse so that iterative methods must be used to solve them. In this paper the authors develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods.
NASA Astrophysics Data System (ADS)
Costa, C. P.; Vilhena, M. T.; Moreira, D. M.; Tirabassi, T.
We present a three-dimensional solution of the steady-state advection-diffusion equation considering a vertically inhomogeneous planetary boundary layer (PBL). We reach this goal applying the generalized integral transform technique (GITT), a hybrid method that had solved a wide class of direct and inverse problems mainly in the area of heat transfer and fluid mechanics. The transformed problem is solved by the advection-diffusion multilayer model (ADMM) method, a semi-analytical solution based on a discretization of the PBL in sub-layers where the advection-diffusion equation is solved by the Laplace transform technique. Numerical simulations are presented and the performances of the solution are compared against field experiments data.
Barajas-Solano, David A.; Tartakovsky, A. M.
2016-10-13
We present a hybrid scheme for the coupling of macro and microscale continuum models for reactive contaminant transport in fractured and porous media. The transport model considered is the advection-dispersion equation, subject to linear heterogeneous reactive boundary conditions. The Multiscale Finite Volume method (MsFV) is employed to define an approximation to the microscale concentration field defined in terms of macroscopic or \\emph{global} degrees of freedom, together with local interpolator and corrector functions capturing microscopic spatial variability. The macroscopic mass balance relations for the MsFV global degrees of freedom are coupled with the macroscopic model, resulting in a global problem for the simultaneous time-stepping of all macroscopic degrees of freedom throughout the domain. In order to perform the hybrid coupling, the micro and macroscale models are applied over overlapping subdomains of the simulation domain, with the overlap denoted as the handshake subdomain $\\Omega^{hs}$, over which continuity of concentration and transport fluxes between models is enforced. Continuity of concentration is enforced by posing a restriction relation between models over $\\Omega^{hs}$. Continuity of fluxes is enforced by prolongating the macroscopic model fluxes across the boundary of $\\Omega^{hs}$ to microscopic resolution. The microscopic interpolator and corrector functions are solutions to local microscopic advection-diffusion problems decoupled from the global degrees of freedom and from each other by virtue of the MsFV decoupling ansatz. The error introduced by the decoupling ansatz is reduced iteratively by the preconditioned GMRES algorithm, with the hybrid MsFV operator serving as the preconditioner.
NASA Astrophysics Data System (ADS)
Sanskrityayn, Abhishek; Kumar, Naveen
2016-12-01
Some analytical solutions of one-dimensional advection-diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green's function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant's mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.
NASA Astrophysics Data System (ADS)
Bhrawy, A. H.; Baleanu, D.
2013-10-01
An efficient Legendre-Gauss-Lobatto collocation (L-GL-C) method is applied to solve the space-fractional advection diffusion equation with nonhomogeneous initial-boundary conditions. The Legendre-Gauss-Lobatto points are used as collocation nodes for spatial fractional derivatives as well as the Caputo fractional derivative. This approach is reducing the problem to the solution of a system of ordinary differential equations in time which can be solved by using any standard numerical techniques. The proposed numerical solutions when compared with the exact solutions reveal that the obtained solution produces highly accurate results. The results show that the proposed method has high accuracy and is efficient for solving the space-fractional advection diffusion equation.
NASA Astrophysics Data System (ADS)
Lowry, Thomas; Li, Shu-Guang
2005-02-01
Difficulty in solving the transient advection-diffusion equation (ADE) stems from the relationship between the advection derivatives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space-time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solution methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection-diffusion equation. Water Resour Res 38 (7), 10.1029/2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian-Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods.
1992-01-01
formulations for the advection- diffusion equation with time-dependent domains 0. Pironneau Universite Paris 6. Analyse Numerique . T 55-65/5. 4 place...solution of the advection- dissipation equation, Correspondence to: Dr. 0. Pironneau, Universit6 Paris 6, Analyse Numerique . T 55-65/5, 4 place...for Partial Differential Equations (Cambridize Univ. Press. Cambridge. 1989). 1121 TE. Tezduvar. .I. Behr and J. Liou. A new strategy for finite
AN EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD FOR THE ADVECTION-DIFFUSION EQUATION
Many numerical methods use characteristic analysis to accommodate the advective component of transport. Such characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. A generalization of characteri...
NASA Astrophysics Data System (ADS)
Vukadinovic, J.; Dedits, E.; Poje, A. C.; Schäfer, T.
2015-08-01
We consider the two-dimensional advection-diffusion equation (ADE) on a bounded domain subject to Dirichlet or von Neumann boundary conditions involving a Liouville integrable Hamiltonian. Transformation to action-angle coordinates permits averaging in time and angle, resulting in an equation that allows for separation of variables. The Fourier transform in the angle coordinate transforms the equation into an effective diffusive equation and a countable family of non-self-adjoint Schrödinger equations. For the corresponding Liouville-Sturm problem, we apply complex-plane WKB methods to study the spectrum in the semi-classical limit for vanishing diffusivity. The spectral limit graph is found to consist of analytic curves (branches) related to Stokes graphs forming a tree-structure. Eigenvalues in the neighborhood of branches emanating from the imaginary axis are subject to various sublinear power laws with respect to diffusivity, leading to convection-enhanced rates of dissipation of the corresponding modes. The solution of the ADE converges in the limit of vanishing diffusivity to the solution of the effective diffusion equation on convective time scales that are sublinear with respect to the diffusive time scales.
Erratum: A Comparison of Closures for Stochastic Advection-Diffusion Equations
Jarman, Kenneth D.; Tartakovsky, Alexandre M.
2015-01-01
This note corrects an error in the authors' article [SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 319 347] in which the cited work [Neuman, Water Resour. Res., 29(3) (1993), pp. 633 645] was incorrectly represented and attributed. Concentration covariance equations presented in our article as new were in fact previously derived in the latter work. In the original abstract, the phrase " . . .we propose a closed-form approximation to two-point covariance as a measure of uncertainty. . ." should be replaced by the phrase " . . .we study a closed-form approximation to two-point covariance, previously derived in [Neuman 1993], as a measure of uncertainty." The primary results in our article--the analytical and numerical comparison of existing closure methods for specific example problems are not changed by this correction.
NASA Astrophysics Data System (ADS)
Gharehbaghi, Amin
2016-10-01
In this paper, a numerical solution of one-dimensional time-dependent advection-diffusion equation with variable coefficients in semi-infinite domain is presented by using the differential quadrature method. Both the explicit and implicit approaches are provided. Totally, two solute dispersion problems are employed to simulate various conditions. The inhomogeneity of the domain is supplied by the spatially dependent flow. The problem domains are modeled with Chebyshev-Gauss-Lobatto grid points. In order to examine the accuracy and the efficiency of the suggested explicit and implicit approaches, analytical solutions, which are presented in the literature, are employed. In addition, the results of the above-mentioned method are compared with outcomes of the finite difference method. The results show that both of the explicit and implicit forms of the differential quadrature method are efficient, robust and reliable. But between these two forms, numerical predictions of implicit form are more accurate than explicit form.
Diffusion and Advection using Cellular Potts Model
NASA Astrophysics Data System (ADS)
Dan, Debasis; Glazier, James
2005-03-01
The Cellular Potts Model (CPM) is a robust cell level methodology for simulation of biological tissues and morphogenesis. Standard diffusion solvers in the CPM use finite difference methods on the underlying CPM lattice. These methods have difficulty in simulating local advection in the ECM due to physiology and morphogenesis. To circumvent the problem of instabilities we simulate advection-diffusion within the framework of CPM using off-lattice finite-difference methods. We define a set of generalised fluid "cells" or particles which separate advection and diffusion from the lattice. Diffusion occurs between neighboring fluid cells by local averaging rules which approximate the Laplacian. CPM movement of the cells by spin flips handles the advection. The extension allows the CPM to model viscosity explicitly by including a relative velocity constraint on the fluid. The extended CPM correctly reproduces flow profiles of viscous fluids in cylindrical tube, during Stokes flow across a sphere and in flow in concentric cylindrical shells. We illustrate various conditions for diffusion including multiple instantaneous sources, continuous sources, moving sources and different boundary geometries and conditions to validate our approximation by comparing with analytical and established numerical solutions.
NASA Astrophysics Data System (ADS)
Mudunuru, M. K.; Nakshatrala, K. B.
2016-01-01
We present a robust computational framework for advective-diffusive-reactive systems that satisfies maximum principles, the non-negative constraint, and element-wise species balance property. The proposed methodology is valid on general computational grids, can handle heterogeneous anisotropic media, and provides accurate numerical solutions even for very high Péclet numbers. The significant contribution of this paper is to incorporate advection (which makes the spatial part of the differential operator non-self-adjoint) into the non-negative computational framework, and overcome numerical challenges associated with advection. We employ low-order mixed finite element formulations based on least-squares formalism, and enforce explicit constraints on the discrete problem to meet the desired properties. The resulting constrained discrete problem belongs to convex quadratic programming for which a unique solution exists. Maximum principles and the non-negative constraint give rise to bound constraints while element-wise species balance gives rise to equality constraints. The resulting convex quadratic programming problems are solved using an interior-point algorithm. Several numerical results pertaining to advection-dominated problems are presented to illustrate the robustness, convergence, and the overall performance of the proposed computational framework.
Parker, Jack C; Kim, Ungtae
2015-11-01
The mono-continuum advection-dispersion equation (mADE) is commonly regarded as unsuitable for application to media that exhibit rapid breakthrough and extended tailing associated with diffusion between high and low permeability regions. This paper demonstrates that the mADE can be successfully used to model such conditions if certain issues are addressed. First, since hydrodynamic dispersion, unlike molecular diffusion, cannot occur upstream of the contaminant source, models must be formulated to prevent "back-dispersion." Second, large variations in aquifer permeability will result in differences between volume-weighted average concentration (resident concentration) and flow-weighted average concentration (flux concentration). Water samples taken from wells may be regarded as flux concentrations, while soil samples may be analyzed to determine resident concentrations. While the mADE is usually derived in terms of resident concentration, it is known that a mADE of the same mathematical form may be written in terms of flux concentration. However, when solving the latter, the mathematical transformation of a flux boundary condition applied to the resident mADE becomes a concentration type boundary condition for the flux mADE. Initial conditions must also be consistent with the form of the mADE that is to be solved. Thus, careful attention must be given to the type of concentration data that is available, whether resident or flux concentrations are to be simulated, and to boundary and initial conditions. We present 3-D analytical solutions for resident and flux concentrations, discuss methods of solving numerical models to obtain resident and flux concentrations, and compare results for hypothetical problems. We also present an upscaling method for computing "effective" dispersivities and other mADE model parameters in terms of physically meaningful parameters in a diffusion-limited mobile-immobile model. Application of the latter to previously published studies of
Space-fractional advection-diffusion and reflective boundary condition.
Krepysheva, Natalia; Di Pietro, Liliana; Néel, Marie-Christine
2006-02-01
Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRWs) with transition probability densities showing space- and/or time-diverging moments were developed to account for anomalous behaviors. A broad class of CTRWs was shown to correspond, on the macroscopic scale, to advection-diffusion equations involving derivatives of noninteger order. In particular, CTRWs with Lévy distribution of jumps and finite mean waiting time lead to a space-fractional equation that accounts for superdiffusion and involves a nonlocal integral-differential operator. Within this framework, we analyze the evolution of particles performing symmetric Lévy flights with respect to a fluid moving at uniform speed . The particles are restricted to a semi-infinite domain limited by a reflective barrier. We show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator. Thus, the macroscopic space-fractional advection-diffusion equation obtained is different from that in an infinite medium.
Analytical solution for the advection-dispersion transport equation in layered media
Technology Transfer Automated Retrieval System (TEKTRAN)
The advection-dispersion transport equation with first-order decay was solved analytically for multi-layered media using the classic integral transform technique (CITT). The solution procedure used an associated non-self-adjoint advection-diffusion eigenvalue problem that had the same form and coef...
Advection, diffusion, and delivery over a network.
Heaton, Luke L M; López, Eduardo; Maini, Philip K; Fricker, Mark D; Jones, Nick S
2012-08-01
Many biological, geophysical, and technological systems involve the transport of a resource over a network. In this paper, we present an efficient method for calculating the exact quantity of the resource in each part of an arbitrary network, where the resource is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. The key conceptual step is to partition the resource into material that does or does not reach a node over a given time step. As an example application, we consider resource allocation within fungal networks, and analyze the spatial distribution of the resource that emerges as such networks grow over time. Fungal growth involves the expansion of fluid filled vessels, and such growth necessarily involves the movement of fluid. We develop a model of delivery in growing fungal networks, and find good empirical agreement between our model and experimental data gathered using radio-labeled tracers. Our results lead us to suggest that in foraging fungi, growth-induced mass flow is sufficient to account for long-distance transport, if the system is well insulated. We conclude that active transport mechanisms may only be required at the very end of the transport pathway, near the growing tips.
Overcoming diffusion-limited processes using enhanced advective fields
Rasmussen, T.C.
1995-12-31
Many subsurface cleanup activities focus on the remediation of organic contaminants using induced advective fields. Subsurface heterogeneities cause most advective transport to occur in more permeable zones, with transport from the lower permeability units being limited by diffusion to the higher permeable units. While diffusion rates can be enhanced using thermal sources, many of the treatment strategies, including pump and treat, vapor extraction and bioremediation, are limited by mass exchange rates between the higher and lower permeability sand and clay mixtures. Instead of relying on the enhancement of diffusion rates, it is proposed that remediation strategies should focus on the enhancement of induced advective transport rates through the lower permeability units. Injection-extraction strategies using crosshole and huff-and-puff methods are presented for maximizing advective transport through lower permeability units. Optimization of the design can incorporate diffusion-enhancement technologies, bionourishment, capillary confinement in the unsaturated zone, and DNAPL slurping.
First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems
NASA Technical Reports Server (NTRS)
Mazaheri, Alireza; Nishikawa, Hiroaki
2014-01-01
A time-dependent extension of the first-order hyperbolic system method for advection-diffusion problems is introduced. Diffusive/viscous terms are written and discretized as a hyperbolic system, which recovers the original equation in the steady state. The resulting scheme offers advantages over traditional schemes: a dramatic simplification in the discretization, high-order accuracy in the solution gradients, and orders-of-magnitude convergence acceleration. The hyperbolic advection-diffusion system is discretized by the second-order upwind residual-distribution scheme in a unified manner, and the system of implicit-residual-equations is solved by Newton's method over every physical time step. The numerical results are presented for linear and nonlinear advection-diffusion problems, demonstrating solutions and gradients produced to the same order of accuracy, with rapid convergence over each physical time step, typically less than five Newton iterations.
Population persistence under advection-diffusion in river networks.
Ramirez, Jorge M
2012-11-01
An integro-differential equation on a tree graph is used to model the time evolution and spatial distribution of a population of organisms in a river network. Individual organisms become mobile at a constant rate, and disperse according to an advection-diffusion process with coefficients that are constant on the edges of the graph. Appropriate boundary conditions are imposed at the outlet and upstream nodes of the river network. The local rates of population growth/decay and that by which the organisms become mobile, are assumed constant in time and space. Imminent extinction of the population is understood as the situation whereby the zero solution to the integro-differential equation is stable. Lower and upper bounds for the eigenvalues of the dispersion operator, and related Sturm-Liouville problems are found. The analysis yields sufficient conditions for imminent extinction and/or persistence in terms of the values of water velocity, channel length, cross-sectional area and diffusivity throughout the river network.
Advective-diffusive contaminant migration in unsaturated sand and gravel
Rowe, R.K.; Badv, K.
1996-12-01
A method is presented for estimating the diffusion coefficients for chloride and sodium in unsaturated coarse sand and fine gravel based on parameters obtained from saturated diffusion tests conducted for similar material. The method is tested by comparing the observed and predicted diffusion profiles through unsaturated soil. The method is shown to work well for predicting the advective-diffusive migration of chloride and sodium through a two-layer soil system consisting of a compacted clayey silt underlain by an unsaturated fine gravel. Over the range of conditions examined, it is concluded that existing solute transport theory along with the proposed procedure for estimating the unsaturated diffusion coefficients can adequately predict chloride and sodium diffusion through both unsaturated coarse sand and fine gravel as well as predict advective-diffusive transport through a compacted clayey layer and underlying unsaturated fine gravel.
A Quasi-Conservative Adaptive Semi-Lagrangian Advection-Diffusion Scheme
NASA Astrophysics Data System (ADS)
Behrens, Joern
2014-05-01
Many processes in atmospheric or oceanic tracer transport are conveniently represented by advection-diffusion type equations. Depending on the magnitudes of both components, the mathematical representation and consequently the discretization is a non-trivial problem. We will focus on advection-dominated situations and will introduce a semi-Lagrangian scheme with adaptive mesh refinement for high local resolution. This scheme is well suited for pollutant transport from point sources, or transport processes featuring fine filamentation with corresponding local concentration maxima. In order to achieve stability, accuracy and conservation, we combine an adaptive mesh refinement quasi-conservative semi-Lagrangian scheme, based on an integral formulation of the underlying advective conservation law (Behrens, 2006), with an advection diffusion scheme as described by Spiegelman and Katz (2006). The resulting scheme proves to be conservative and stable, while maintaining high computational efficiency and accuracy.
Critical time scales for advection-diffusion-reaction processes
NASA Astrophysics Data System (ADS)
Ellery, Adam J.; Simpson, Matthew J.; McCue, Scott W.; Baker, Ruth E.
2012-04-01
The concept of local accumulation time (LAT) was introduced by Berezhkovskii and co-workers to give a finite measure of the time required for the transient solution of a reaction-diffusion equation to approach the steady-state solution [A. M. Berezhkovskii, C. Sample, and S. Y. Shvartsman, Biophys. J.BIOJAU0006-349510.1016/j.bpj.2010.07.045 99, L59 (2010); A. M. Berezhkovskii, C. Sample, and S. Y. Shvartsman, Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.83.051906 83, 051906 (2011)]. Such a measure is referred to as a critical time. Here, we show that LAT is, in fact, identical to the concept of mean action time (MAT) that was first introduced by McNabb [A. McNabb and G. C. Wake, IMA J. Appl. Math.IJAMDM0272-496010.1093/imamat/47.2.193 47, 193 (1991)]. Although McNabb's initial argument was motivated by considering the mean particle lifetime (MPLT) for a linear death process, he applied the ideas to study diffusion. We extend the work of these authors by deriving expressions for the MAT for a general one-dimensional linear advection-diffusion-reaction problem. Using a combination of continuum and discrete approaches, we show that MAT and MPLT are equivalent for certain uniform-to-uniform transitions; these results provide a practical interpretation for MAT by directly linking the stochastic microscopic processes to a meaningful macroscopic time scale. We find that for more general transitions, the equivalence between MAT and MPLT does not hold. Unlike other critical time definitions, we show that it is possible to evaluate the MAT without solving the underlying partial differential equation (pde). This makes MAT a simple and attractive quantity for practical situations. Finally, our work explores the accuracy of certain approximations derived using MAT, showing that useful approximations for nonlinear kinetic processes can be obtained, again without treating the governing pde directly.
Technology Transfer Automated Retrieval System (TEKTRAN)
Analytical solutions of the advection-dispersion solute transport equation remain useful for a large number of applications in science and engineering. In this paper we extend the Duhamel theorem, originally established for diffusion type problems, to the case of advective-dispersive transport subj...
Optimal Stretching in Advection-Reaction-Diffusion Systems
NASA Astrophysics Data System (ADS)
Nevins, Thomas D.; Kelley, Douglas H.
2016-10-01
We investigate growth of the excitable Belousov-Zhabotinsky reaction in chaotic, time-varying flows. In slow flows, reacted regions tend to lie near vortex edges, whereas fast flows restrict reacted regions to vortex cores. We show that reacted regions travel toward vortex centers faster as flow speed increases, but nonreactive scalars do not. For either slow or fast flows, reaction is promoted by the same optimal range of the local advective stretching, but stronger stretching causes reaction blowout and can hinder reaction from spreading. We hypothesize that optimal stretching and blowout occur in many advection-diffusion-reaction systems, perhaps creating ecological niches for phytoplankton in the ocean.
Optimal Stretching in Advection-Reaction-Diffusion Systems.
Nevins, Thomas D; Kelley, Douglas H
2016-10-14
We investigate growth of the excitable Belousov-Zhabotinsky reaction in chaotic, time-varying flows. In slow flows, reacted regions tend to lie near vortex edges, whereas fast flows restrict reacted regions to vortex cores. We show that reacted regions travel toward vortex centers faster as flow speed increases, but nonreactive scalars do not. For either slow or fast flows, reaction is promoted by the same optimal range of the local advective stretching, but stronger stretching causes reaction blowout and can hinder reaction from spreading. We hypothesize that optimal stretching and blowout occur in many advection-diffusion-reaction systems, perhaps creating ecological niches for phytoplankton in the ocean.
Update on Advection-Diffusion Purge Flow Model
NASA Technical Reports Server (NTRS)
Brieda, Lubos
2015-01-01
Gaseous purge is commonly used in sensitive spacecraft optical or electronic instruments to prevent infiltration of contaminants and/or water vapor. Typically, purge is sized using simplistic zero-dimensional models that do not take into account instrument geometry, surface effects, and the dependence of diffusive flux on the concentration gradient. For this reason, an axisymmetric computational fluid dynamics (CFD) simulation was recently developed to model contaminant infiltration and removal by purge. The solver uses a combined Navier-Stokes and Advection-Diffusion approach. In this talk, we report on updates in the model, namely inclusion of a particulate transport model.
A global spectral element model for poisson equations and advective flow over a sphere
NASA Astrophysics Data System (ADS)
Mei, Huan; Wang, Faming; Zeng, Zhong; Qiu, Zhouhua; Yin, Linmao; Li, Liang
2016-03-01
A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. Highprecision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.
Horizontal advection, diffusion and plankton spectra at the sea surface.
NASA Astrophysics Data System (ADS)
Bracco, A.; Clayton, S.; Pasquero, C.
2009-04-01
Plankton patchiness is ubiquitous in the oceans, and various physical and biological processes have been proposed as its generating mechanisms. However, a coherent statement on the problem is missing, due to both a small number of suitable observations and to an incomplete understanding of the properties of reactive tracers in turbulent media. Abraham (1998) suggested that horizontal advection may be the dominant process behind the observed distributions of phytoplankton and zooplankton, acting to mix tracers with longer reaction times (Rt) down to smaller scales. Conversely, Mahadevan and Campbell (2002) attributed the relative distributions of sea surface temperature and phytoplankton to small scale upwelling, where tracers with longer Rt are able to homogenize more than those with shorter reaction times. Neither of the above mechanisms can explain simultaneously the (relative) spectral slopes of temperature, phytoplankton and zooplankton. Here, with a simple advection model and a large suite of numerical experiments, we concentrate on some of the physical processes influencing the relative distributions of tracers at the ocean surface, and we investigate: 1) the impact of the spatial scale of tracer supply; 2) the role played by coherent eddies on the distribution of tracers with different Rt; 3) the role of diffusion (so far neglected). We show that diffusion determines the distribution of temperature, regardless of the nature of the forcing. We also find that coherent structures together with differential diffusion of tracers with different Rt impact the tracer distributions. This may help in understanding the highly variable nature of observed plankton spectra.
Dependence of advection-diffusion-reaction on flow coherent structures
NASA Astrophysics Data System (ADS)
Tang, Wenbo; Luna, Christopher
2013-10-01
A study on an advection-diffusion-reaction system is presented. Variability of the reaction process in such a system triggered by a highly localized source is quantified. It is found, for geophysically motivated parameter regimes, that the difference in bulk concentration subject to realizations of different source locations is highly correlated with the local flow topology of the source. Such flow topologies can be highlighted by Lagrangian coherent structures. Reaction is relatively enhanced in regions of strong stretching, and relatively suppressed in regions where vortices are present. In any case, the presence of a divergence-free background flow helps speed up the reaction process, especially when the flow is time-dependent. Probability density of various quantities characterizing the reaction processes is also obtained. This reveals the inherent complexity of the reaction-diffusion process subject to nonlinear background stirring.
Oxygen Advection and Diffusion in a Three Dimensional Vascular Anatomical Network
Fang, Qianqian; Sakadžić, Sava; Ruvinskaya, Lana; Devor, Anna; Dale, Anders M.; Boas, David A.
2008-01-01
There is an increasing need for quantitative and computationally affordable models for analyzing tissue metabolism and hemodynamics in microvascular networks. In this work, we develop a hybrid model to solve for the time-varying oxygen advection-diffusion equation in the vessels and tissue. To obtain a three-dimensional temporal evolution of tissue oxygen concentration for realistic complex vessel networks, we used a graph-based advection model combined with a finite-element based diffusion model and an implicit time-advancing scheme. We validated this algorithm for both static and dynamic conditions. We also applied it to a complex vascular network obtained from a rodent somatosensory cortex. Qualitative agreement was found with in-vivo experiments. PMID:18958033
The role of advection and diffusion in waste disposal by sea urchin embryos
NASA Astrophysics Data System (ADS)
Clark, Aaron; Licata, Nicholas
2014-03-01
We determine the first passage probability for the absorption of waste molecules released from the microvilli of sea urchin embryos. We calculate a perturbative solution of the advection-diffusion equation for a linear shear profile similar to the fluid environment which the embryos inhabit. Rapid rotation of the embryo results in a concentration boundary layer of comparable thickness to the length of the microvilli. A comparison of the results to the regime of diffusion limited transport indicates that fluid flow is advantageous for efficient waste disposal.
Computation of traveling wave fronts for a nonlinear diffusion-advection model.
Mansour, M B A
2009-01-01
This paper utilizes a nonlinear reaction-diffusion-advection model for describing the spatiotemporal evolution of bacterial growth. The traveling wave solutions of the corresponding system of partial differential equations are analyzed. Using two methods, we then find such solutions numerically. One of the methods involves the traveling wave equations and solving an initial-value problem, which leads to accurate computations of the wave profiles and speeds. The second method is to construct time-dependent solutions by solving an initial-moving boundary-value problem for the PDE system, showing another approximation for such wave solutions.
NASA Astrophysics Data System (ADS)
Nefedov, Nikolay
2017-02-01
This is an extended variant of the paper presented at MURPHYS-HSFS 2016 conference in Barcelona. We discuss further development of the asymptotic method of differential inequalities to investigate existence and stability of sharp internal layers (fronts) for nonlinear singularly perturbed periodic parabolic problems and initial boundary value problems with blow-up of fronts for reaction-diffusion-advection equations. In particular, we consider periodic solutions with internal layer in the case of balanced reaction. For the initial boundary value problems we prove the existence of fronts and give their asymptotic approximation including the new case of blowing-up fronts. This case we illustrate by the generalised Burgers equation.
Boso, F; Broyda, S V; Tartakovsky, D M
2014-06-08
We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddy-diffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.
How Hydrate Saturation Anomalies are Diffusively Constructed and Advectively Smoothed
NASA Astrophysics Data System (ADS)
Rempel, A. W.; Irizarry, J. T.; VanderBeek, B. P.; Handwerger, A. L.
2015-12-01
The physical processes that control the bulk characteristics of hydrate reservoirs are captured reasonably well by long-established model formulations that are rooted in laboratory-verified phase equilibrium parameterizations and field-based estimates of in situ conditions. More detailed assessments of hydrate distribution, especially involving the occurrence of high-saturation hydrate anomalies have been more difficult to obtain. Spatial variations in sediment properties are of central importance for modifying the phase behavior and promoting focussed fluid flow. However, quantitative predictions of hydrate anomaly development cannot be made rigorously without also addressing the changes in phase behavior and mechanical balances that accompany changes in hydrate saturation level. We demonstrate how pore-scale geometrical controls on hydrate phase stability can be parameterized for incorporation in simulations of hydrate anomaly development along dipping coarse-grained layers embedded in a more fine-grained background that is less amenable to fluid transport. Model simulations demonstrate how hydrate anomaly growth along coarse-layer boundaries is promoted by diffusive gas transport from the adjacent fine-grained matrix, while advective transport favors more distributed growth within the coarse-grained material and so effectively limits the difference between saturation peaks and background levels. Further analysis demonstrates how sediment contacts are unloaded once hydrate saturation reaches sufficient levels to form a load-bearing skeleton that can evolve to produce segregated nodules and lenses. Decomposition of such growth forms poses a significant geohazard that is expected to be particularly sensitive to perturbations induced by gas extraction. The figure illustrates the predicted evolution of hydrate saturation Sh in a coarse-grained dipping layer showing how prominent bounding hydrate anomalies (spikes) supplied by diffusive gas transport at early times
Fractional chemotaxis diffusion equations.
Langlands, T A M; Henry, B I
2010-05-01
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.
NASA Astrophysics Data System (ADS)
Monobe, Harunori; Wu, Chang-Hong
2016-12-01
In this paper, we investigate a reaction-diffusion-advection equation with a free boundary which models the spreading of an invasive species in one-dimensional heterogeneous environments. We assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal, and the spreading mechanism of species is determined by a Stefan-type condition. Investigating the sign of the principal eigenvalue of the associated linearized eigenvalue problem, under certain conditions we obtain the sharp criteria for spreading and vanishing via system parameters. Also, we establish the long-time behavior of the solution and the asymptotic spreading speed. Finally, some biological implications are discussed.
Analyzing critical propagation in a reaction-diffusion-advection model using unstable slow waves.
Kneer, Frederike; Obermayer, Klaus; Dahlem, Markus A
2015-02-01
The effect of advection on the propagation and in particular on the critical minimal speed of traveling waves in a reaction-diffusion model is studied. Previous theoretical studies estimated this effect on the velocity of stable fast waves and predicted the existence of a critical advection strength below which propagating waves are not supported anymore. In this paper, an analytical expression for the advection-velocity relation of the unstable slow wave is derived. In addition, the critical advection strength is calculated taking into account the unstable slow wave solution. We also analyze a two-variable reaction-diffusion-advection model numerically in a wide parameter range. Due to the new control parameter (advection) we can find stable wave propagation in the otherwise non-excitable parameter regime, if the advection strength exceeds a critical value. Comparing theoretical predictions to numerical results, we find that they are in good agreement. Theory provides an explanation for the observed behaviour.
Nonlocal electrical diffusion equation
NASA Astrophysics Data System (ADS)
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.
A new Remesh-Lagrange technique for advecting temperature that minimizes numerical diffusion
NASA Astrophysics Data System (ADS)
Hasenclever, J.; Phipps Morgan, J.; Shi, C.
2007-12-01
The proper treatment of heat-advection is a generally underappreciated problem within CFD, yet particularly critical for calculating physically sound erosion in plume-lithosphere interactions and temperature sensitive melting processes. Typically, Eulerian (fixed-mesh) codes have been preferred to solve for fluid flow and they are almost essential for finite-difference-based algorithms. Unfortunately, the Eulerian approach introduces numerical artifacts into the solution of the advection-diffusion heat transport problem that can only be suppressed by adding 'too-diffusive' artificial diffusion to the equations, as for example in the Smolarkiewicz formulation for heat advection. We have developed a 'Remesh-Lagrange' method using a partly deforming finite element mesh and find it to be significantly more accurate than our previous methods. In several test scenarios we show the large improvement in accuracy that can be obtained by using a Lagrangian approach for 10-30 time steps (depending upon the distortion of the finite elements in the deformed Lagrangian mesh) and then regridding to the initial mesh. When an element becomes too distorted the nodes connected to it become fixed and we switch from Lagrange to a Semi-Lagrange formulation for these nodes. Instead of the standard 'linear backward' Semi-Lagrange we are also experimenting with a more accurate interpolation scheme for an unstructured mesh that additionally includes the nodal derivatives of the temperature field when calculating the value at the Semi-Lagrange traceback point. The same bicubic interpolation method for an unstructured grid is used to remesh the 'too-distorted' Lagrange grid back to the initial undistorted mesh. We compare the Remesh-Lagrange technique against the following Eulerian methods in a series of 2-D numerical experiments advecting stripes and Gaussian peaks in steady circulating flow: linear back-interpolation Semi-Lagrange method; bicubic back-interpolation Semi-Lagrange method
von Kameke, A; Huhn, F; Muñuzuri, A P; Pérez-Muñuzuri, V
2013-02-22
In the absence of advection, reaction-diffusion systems are able to organize into spatiotemporal patterns, in particular spiral and target waves. Whenever advection is present that can be parametrized in terms of effective or turbulent diffusion D(*), these patterns should be attainable on a much greater, boosted length scale. However, so far, experimental evidence of these boosted patterns in a turbulent flow was lacking. Here, we report the first experimental observation of boosted target and spiral patterns in an excitable chemical reaction in a quasi-two-dimensional turbulent flow. The wave patterns observed are ~50 times larger than in the case of molecular diffusion only. We vary the turbulent diffusion coefficient D(*) of the flow and find that the fundamental Fisher-Kolmogorov-Petrovsky-Piskunov equation, v(f) proportional sqrt[D(*)], for the asymptotic speed of a reactive wave remains valid. However, not all measures of the boosted wave scale with D(*) as expected from molecular diffusion, since the wave fronts turn out to be highly filamentous.
2014-04-01
downstream boundary (when needed) is obtained by extrapolation, taking into account the hyperbolic character of the equation . By separating the...for Developing Reduced Order Models of Reaction-Advection Equations 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT...advection scalar equation is used as a representative equation to investigate the overall approach. Both linear and nonlinear model equations are
A high-order discontinuous Galerkin method for unsteady advection-diffusion problems
NASA Astrophysics Data System (ADS)
Borker, Raunak; Farhat, Charbel; Tezaur, Radek
2017-03-01
A high-order discontinuous Galerkin method with Lagrange multipliers is presented for the solution of unsteady advection-diffusion problems in the high Péclet number regime. It operates directly on the second-order form of the governing equation and does not require any stabilization. Its spatial basis functions are chosen among the free-space solutions of the homogeneous form of the partial differential equation obtained after time-discretization. It also features Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. This leads to a system of differential-algebraic equations which are time-integrated by an implicit family of schemes. The numerical stability of these schemes and the well-posedness of the overall discretization method are supported by a theoretical analysis. The performance of this method is demonstrated for various high Péclet number constant-coefficient model flow problems.
NASA Astrophysics Data System (ADS)
Gusti, T. P.; Hertanti, D. R.; Bahsan, E.; Soeryantono, H.
2013-12-01
Particle-based numerical methods, such as Smoothed Particle Hydrodynamics (SPH), may be able to simulate some hydrodynamic and morphodynamic behaviors better than grid-based numerical methods. This study simulates hydrodynamics in meanders and advection and turbulent diffusion in straight river channels using Microsoft Excel and Visual Basic. The simulators generate three-dimensional data for hydrodynamics and one-dimensional data for advection-turbulent diffusion. Fluid at rest, sloshing, and helical flow are simulated in the river meanders. Spill loading and step loading are done to simulate concentration patterns associated with advection-turbulent diffusion. Results indicate that helical flow is formed due to disturbance in morphology and particle velocity in the stream and the number of particles does not have a significant effect on the pattern of advection-turbulent diffusion concentration.
Perko, Janez; Patel, Ravi A
2014-05-01
The paper presents an approach that extends the flexibility of the standard lattice Boltzmann single relaxation time scheme in terms of spatial variation of dissipative terms (e.g., diffusion coefficient) and stability for high Péclet mass transfer problems. Spatial variability of diffusion coefficient in SRT is typically accommodated through the variation of relaxation time during the collision step. This method is effective but cannot deal with large diffusion coefficient variations, which can span over several orders of magnitude in some natural systems. The approach explores an alternative way of dealing with large diffusion coefficient variations in advection-diffusion transport systems by introducing so-called diffusion velocity. The diffusion velocity is essentially an additional convective term that replaces variations in diffusion coefficients vis-à-vis a chosen reference diffusion coefficient which defines the simulation time step. Special attention is paid to the main idea behind the diffusion velocity formulation and its implementation into the lattice Boltzmann framework. Finally, the performance, stability, and accuracy of the diffusion velocity formulation are discussed via several advection-diffusion transport benchmark examples. These examples demonstrate improved stability and flexibility of the proposed scheme with marginal consequences on the numerical performance.
An exact peak capturing and essentially oscillation-free (EPCOF) algorithm, consisting of advection-dispersion decoupling, backward method of characteristics, forward node tracking, and adaptive local grid refinement, is developed to solve transport equations. This algorithm repr...
Spectral Theory of Advective Diffusion in the Ocean
2013-09-19
to study this enhancement of sea ice thermal conductivity and better understand temperature data collected during a 2007 Antarctic expedition. 15...conductivity and better understand temperature data collected during a 2007 Antarctic expedition. Activities and Findings: 1. Advection-enhanced...critically on the properties of this Hilbert space. More specifically, it is only on a special subset of this space that the random operator is Hermitian
A general mechanism for “inexact” phase differences in reaction-diffusion-advection systems
NASA Astrophysics Data System (ADS)
Satnoianu, Razvan A.; Menzinger, Michael
2002-11-01
‘Inexact’ phase differences that may take any value in the range [0, π], between the chemical morphogens diffusing in an embryo, have been proposed [M.A. Russell, Dev. Biol. 108 (1985) 269] to improve the positional information theory [L. Wolpert, J. Theor. Biol. 25 (1969) 1] by encoding this information with higher resolution than that provided by other mechanisms. Reaction-diffusion systems, including Turing systems, show only ‘exact’ phase differences 0 and/or π. We demonstrate here that inexact phase differences arise naturally in reactive flows described by reaction-diffusion-advection equations and illustrate them by the stationary waves in open flows (flow- and diffusion-distributed structures FDS [R.A. Satnoianu, M. Menzinger, Phys. Rev. E 62 (2000) 113; R.A. Satnoianu, P.K. Maini, M. Menzinger, Physica D 160 (2001) 79] and travelling waves in differential-induced flow systems (DIFI) [A.B. Rovinsky, M. Menzinger, Phys. Rev. Lett. 70 (1993) 778; R.A. Satnoianu, J.H. Merkin, S.K. Scott, Physica D 124 (1998) 345]. The ability of cells in a developing organism to read phase differences in addition to morphogen concentrations would endow them with a robust mechanism for producing segmentation patterns that is richer, shows higher spatial resolution and is more stable than Turing's and Wolpert's positional information mechanisms.
Moments of action provide insight into critical times for advection-diffusion-reaction processes.
Ellery, Adam J; Simpson, Matthew J; McCue, Scott W; Baker, Ruth E
2012-09-01
Berezhkovskii and co-workers introduced the concept of local accumulation time as a finite measure of the time required for the transient solution of a reaction-diffusion equation to effectively reach steady state [Biophys J. 99, L59 (2010); Phys. Rev. E 83, 051906 (2011)]. Berezhkovskii's approach is a particular application of the concept of mean action time (MAT) that was introduced previously by McNabb [IMA J. Appl. Math. 47, 193 (1991)]. Here, we generalize these previous results by presenting a framework to calculate the MAT, as well as the higher moments, which we call the moments of action. The second moment is the variance of action time, the third moment is related to the skew of action time, and so on. We consider a general transition from some initial condition to an associated steady state for a one-dimensional linear advection-diffusion-reaction partial differential equation (PDE). Our results indicate that it is possible to solve for the moments of action exactly without requiring the transient solution of the PDE. We present specific examples that highlight potential weaknesses of previous studies that have considered the MAT alone without considering higher moments. Finally, we also provide a meaningful interpretation of the moments of action by presenting simulation results from a discrete random-walk model together with some analysis of the particle lifetime distribution. This work shows that the moments of action are identical to the moments of the particle lifetime distribution for certain transitions.
NASA Technical Reports Server (NTRS)
Wang, Xiao-Yen; Chow, Chuen-Yen; Chang, Sin-Chung
1999-01-01
Test problems are used to examine the performance of several one-dimensional numerical schemes based on the space-time conservation and solution element (CE/SE) method. Investigated in this paper are the CE/SE schemes constructed previously for solving the linear unsteady advection-diffusion equation and the schemes derived here for solving the nonlinear viscous and inviscid Burgers equations. In comparison with the numerical solutions obtained using several traditional finite-difference schemes with similar accuracy, the CE/SE solutions display much lower numerical dissipation and dispersion errors.
Advective-diffusive motion on large scales from small-scale dynamics with an internal symmetry
NASA Astrophysics Data System (ADS)
Marino, Raffaele; Aurell, Erik
2016-06-01
We consider coupled diffusions in n -dimensional space and on a compact manifold and the resulting effective advective-diffusive motion on large scales in space. The effective drift (advection) and effective diffusion are determined as a solvability conditions in a multiscale analysis. As an example, we consider coupled diffusions in three-dimensional space and on the group manifold SO(3) of proper rotations, generalizing results obtained by H. Brenner [J. Colloid Interface Sci. 80, 548 (1981), 10.1016/0021-9797(81)90214-9]. We show in detail how the analysis can be conveniently carried out using local charts and invariance arguments. As a further example, we consider coupled diffusions in two-dimensional complex space and on the group manifold SU(2). We show that although the local operators may be the same as for SO(3), due to the global nature of the solvability conditions the resulting diffusion will differ and generally be more isotropic.
NASA Astrophysics Data System (ADS)
Lueptow, Richard M.; Schlick, Conor P.; Umbanhowar, Paul B.; Ottino, Julio M.
2013-11-01
We investigate chaotic advection and diffusion in competitive autocatalytic reactions. To study this subject, we use a computationally efficient method for solving advection-reaction-diffusion equations for periodic flows using a mapping method with operator splitting. In competitive autocatalytic reactions, there are two species, B and C, which both react autocatalytically with species A (A +B -->2B and A +C -->2C). If there is initially a small amount of spatially localized B and C and a large amount of A, all three species will be advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that the small scale interactions associated with the chaotic velocity field, specifically the local finite-time Lyapunov exponents (FTLEs), can accurately predict the final average concentrations of B and C after the reaction is complete. The species, B or C, that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If species B and C start in regions having similar FTLEs, their average concentrations at the end of the reaction will also be similar. Funded by NSF Grant CMMI-1000469.
Magnetic flux and heat losses by diffusive, advective, and Nernst effects in MagLIF-like plasma
Velikovich, A. L. Giuliani, J. L.; Zalesak, S. T.
2014-12-15
The MagLIF approach to inertial confinement fusion involves subsonic/isobaric compression and heating of a DT plasma with frozen-in magnetic flux by a heavy cylindrical liner. The losses of heat and magnetic flux from the plasma to the liner are thereby determined by plasma advection and gradient-driven transport processes, such as thermal conductivity, magnetic field diffusion and thermomagnetic effects. Theoretical analysis based on obtaining exact self-similar solutions of the classical collisional Braginskii's plasma transport equations in one dimension demonstrates that the heat loss from the hot plasma to the cold liner is dominated by the transverse heat conduction and advection, and the corresponding loss of magnetic flux is dominated by advection and the Nernst effect. For a large electron Hall parameter ω{sub e}τ{sub e} effective diffusion coefficients determining the losses of heat and magnetic flux are both shown to decrease with ω{sub e}τ{sub e} as does the Bohm diffusion coefficient, which is commonly associated with low collisionality and two-dimensional transport. This family of exact solutions can be used for verification of codes that model the MagLIF plasma dynamics.
Velikovich, A. L.; Giuliani, J. L.; Zalesak, S. T.
2015-04-15
The magnetized liner inertial fusion (MagLIF) approach to inertial confinement fusion [Slutz et al., Phys. Plasmas 17, 056303 (2010); Cuneo et al., IEEE Trans. Plasma Sci. 40, 3222 (2012)] involves subsonic/isobaric compression and heating of a deuterium-tritium plasma with frozen-in magnetic flux by a heavy cylindrical liner. The losses of heat and magnetic flux from the plasma to the liner are thereby determined by plasma advection and gradient-driven transport processes, such as thermal conductivity, magnetic field diffusion, and thermomagnetic effects. Theoretical analysis based on obtaining exact self-similar solutions of the classical collisional Braginskii's plasma transport equations in one dimension demonstrates that the heat loss from the hot compressed magnetized plasma to the cold liner is dominated by transverse heat conduction and advection, and the corresponding loss of magnetic flux is dominated by advection and the Nernst effect. For a large electron Hall parameter (ω{sub e}τ{sub e}≫1), the effective diffusion coefficients determining the losses of heat and magnetic flux to the liner wall are both shown to decrease with ω{sub e}τ{sub e} as does the Bohm diffusion coefficient cT/(16eB), which is commonly associated with low collisionality and two-dimensional transport. We demonstrate how this family of exact solutions can be used for verification of codes that model the MagLIF plasma dynamics.
NASA Astrophysics Data System (ADS)
Velikovich, A. L.; Giuliani, J. L.; Zalesak, S. T.
2015-04-01
The magnetized liner inertial fusion (MagLIF) approach to inertial confinement fusion [Slutz et al., Phys. Plasmas 17, 056303 (2010); Cuneo et al., IEEE Trans. Plasma Sci. 40, 3222 (2012)] involves subsonic/isobaric compression and heating of a deuterium-tritium plasma with frozen-in magnetic flux by a heavy cylindrical liner. The losses of heat and magnetic flux from the plasma to the liner are thereby determined by plasma advection and gradient-driven transport processes, such as thermal conductivity, magnetic field diffusion, and thermomagnetic effects. Theoretical analysis based on obtaining exact self-similar solutions of the classical collisional Braginskii's plasma transport equations in one dimension demonstrates that the heat loss from the hot compressed magnetized plasma to the cold liner is dominated by transverse heat conduction and advection, and the corresponding loss of magnetic flux is dominated by advection and the Nernst effect. For a large electron Hall parameter ( ωeτe≫1 ), the effective diffusion coefficients determining the losses of heat and magnetic flux to the liner wall are both shown to decrease with ωeτe as does the Bohm diffusion coefficient c T /(16 e B ) , which is commonly associated with low collisionality and two-dimensional transport. We demonstrate how this family of exact solutions can be used for verification of codes that model the MagLIF plasma dynamics.
High-Order Residual-Distribution Hyperbolic Advection-Diffusion Schemes: 3rd-, 4th-, and 6th-Order
NASA Technical Reports Server (NTRS)
Mazaheri, Alireza R.; Nishikawa, Hiroaki
2014-01-01
In this paper, spatially high-order Residual-Distribution (RD) schemes using the first-order hyperbolic system method are proposed for general time-dependent advection-diffusion problems. The corresponding second-order time-dependent hyperbolic advection- diffusion scheme was first introduced in [NASA/TM-2014-218175, 2014], where rapid convergences over each physical time step, with typically less than five Newton iterations, were shown. In that method, the time-dependent hyperbolic advection-diffusion system (linear and nonlinear) was discretized by the second-order upwind RD scheme in a unified manner, and the system of implicit-residual-equations was solved efficiently by Newton's method over every physical time step. In this paper, two techniques for the source term discretization are proposed; 1) reformulation of the source terms with their divergence forms, and 2) correction to the trapezoidal rule for the source term discretization. Third-, fourth, and sixth-order RD schemes are then proposed with the above techniques that, relative to the second-order RD scheme, only cost the evaluation of either the first derivative or both the first and the second derivatives of the source terms. A special fourth-order RD scheme is also proposed that is even less computationally expensive than the third-order RD schemes. The second-order Jacobian formulation was used for all the proposed high-order schemes. The numerical results are then presented for both steady and time-dependent linear and nonlinear advection-diffusion problems. It is shown that these newly developed high-order RD schemes are remarkably efficient and capable of producing the solutions and the gradients to the same order of accuracy of the proposed RD schemes with rapid convergence over each physical time step, typically less than ten Newton iterations.
A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species.
Peng, Rui; Zhao, Xiao-Qiang
2016-02-01
In this article, we are concerned with a nonlocal reaction-diffusion-advection model which describes the evolution of a single phytoplankton species in a eutrophic vertical water column where the species relies solely on light for its metabolism. The new feature of our modeling equation lies in that the incident light intensity and the death rate are assumed to be time periodic with a common period. We first establish a threshold type result on the global dynamics of this model in terms of the basic reproduction number R0. Then we derive various characterizations of R0 with respect to the vertical turbulent diffusion rate, the sinking or buoyant rate and the water column depth, respectively, which in turn give rather precise conditions to determine whether the phytoplankton persist or become extinct. Our theoretical results not only extend the existing ones for the time-independent case, but also reveal new interesting effects of the modeling parameters and the time-periodic heterogeneous environment on persistence and extinction of the phytoplankton species, and thereby suggest important implications for phytoplankton growth control.
NASA Astrophysics Data System (ADS)
Mudunuru, M. K.; Shabouei, M.; Nakshatrala, K.
2015-12-01
Advection-diffusion-reaction (ADR) equations appear in various areas of life sciences, hydrogeological systems, and contaminant transport. Obtaining stable and accurate numerical solutions can be challenging as the underlying equations are coupled, nonlinear, and non-self-adjoint. Currently, there is neither a robust computational framework available nor a reliable commercial package known that can handle various complex situations. Herein, the objective of this poster presentation is to present a novel locally conservative non-negative finite element formulation that preserves the underlying physical and mathematical properties of a general linear transient anisotropic ADR equation. In continuous setting, governing equations for ADR systems possess various important properties. In general, all these properties are not inherited during finite difference, finite volume, and finite element discretizations. The objective of this poster presentation is two fold: First, we analyze whether the existing numerical formulations (such as SUPG and GLS) and commercial packages provide physically meaningful values for the concentration of the chemical species for various realistic benchmark problems. Furthermore, we also quantify the errors incurred in satisfying the local and global species balance for two popular chemical kinetics schemes: CDIMA (chlorine dioxide-iodine-malonic acid) and BZ (Belousov--Zhabotinsky). Based on these numerical simulations, we show that SUPG and GLS produce unphysical values for concentration of chemical species due to the violation of the non-negative constraint, contain spurious node-to-node oscillations, and have large errors in local and global species balance. Second, we proposed a novel finite element formulation to overcome the above difficulties. The proposed locally conservative non-negative computational framework based on low-order least-squares finite elements is able to preserve these underlying physical and mathematical properties
Moments of action provide insight into critical times for advection-diffusion-reaction processes
NASA Astrophysics Data System (ADS)
Ellery, Adam J.; Simpson, Matthew J.; McCue, Scott W.; Baker, Ruth E.
2012-09-01
Berezhkovskii and co-workers introduced the concept of local accumulation time as a finite measure of the time required for the transient solution of a reaction-diffusion equation to effectively reach steady state [Biophys J.BIOJAU0006-349510.1016/j.bpj.2010.07.045 99, L59 (2010); Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.83.051906 83, 051906 (2011)]. Berezhkovskii's approach is a particular application of the concept of mean action time (MAT) that was introduced previously by McNabb [IMA J. Appl. Math.IJAMDM0272-496010.1093/imamat/47.2.193 47, 193 (1991)]. Here, we generalize these previous results by presenting a framework to calculate the MAT, as well as the higher moments, which we call the moments of action. The second moment is the variance of action time, the third moment is related to the skew of action time, and so on. We consider a general transition from some initial condition to an associated steady state for a one-dimensional linear advection-diffusion-reaction partial differential equation (PDE). Our results indicate that it is possible to solve for the moments of action exactly without requiring the transient solution of the PDE. We present specific examples that highlight potential weaknesses of previous studies that have considered the MAT alone without considering higher moments. Finally, we also provide a meaningful interpretation of the moments of action by presenting simulation results from a discrete random-walk model together with some analysis of the particle lifetime distribution. This work shows that the moments of action are identical to the moments of the particle lifetime distribution for certain transitions.
NASA Astrophysics Data System (ADS)
Chauhan, R. P.; Kumar, Amit
The present work is aimed that out of diffusive and advective transport which is dominant process for indoor radon entry under normal room conditions. For this purpose the radon diffusion coefficient and permeability of concrete were measured by specially designed experimental set up. The radon diffusion coefficient of concrete was measured by continuous radon monitor. The measured value was (3.78 ± 0.39)×10-8 m2/s and found independent of the radon gas concentration in source chamber. The radon permeability of concrete varied between 1.85×10-17 to 1.36×10-15 m2 for the bulk pressure difference fewer than 20 Pa to 73.3 kPa. From the measured diffusion coefficient and absolute permeability, the radon flux from the concrete surface having concentrations gradient 12-40 kBq/m3 and typical floor thickness 0.1 m was calculated by the application of Fick and Darcy laws. Using the measured flux attributable to diffusive and advective transport, the indoor radon concentration for a typical Indian model room having dimension (5×6×7) m3 was calculated under average room ventilation (0.63 h-1). The results showed that the contribution of diffusive transport through intact concrete is dominant over the advective transport, as expected from the low values of concrete permeability.
Technology Transfer Automated Retrieval System (TEKTRAN)
Analytical solutions of the advection-dispersion equation and related models are indispensable for predicting or analyzing contaminant transport processes in streams and rivers, as well as in other surface water bodies. Many useful analytical solutions originated in disciplines other than surface-w...
A validated 2-D diffusion-advection model for prediction of drift from ground boom sprayers
NASA Astrophysics Data System (ADS)
Baetens, K.; Ho, Q. T.; Nuyttens, D.; De Schampheleire, M.; Melese Endalew, A.; Hertog, M. L. A. T. M.; Nicolaï, B.; Ramon, H.; Verboven, P.
Correct field drift prediction is a key element in environmental risk assessment of spraying applications. A reduced order drift prediction model based on the diffusion-advection equation is presented. It allows fast assessment of the drift potential of specific ground boom applications under specific environmental wind conditions that obey the logarithmic wind profile. The model was calibrated based on simulations with a validated Computational Fluid Dynamics (CFD) model. Validation of both models against 38 carefully conducted field experiments is successfully performed for distances up to 20 m from the field edge, for spraying on flat pasture land. The reduced order model succeeded in correct drift predictions for different nozzle types, wind velocities, boom heights and spray pressures. It used 4 parameters representing the physical aspects of the drift cloud; the height of the cloud at the field edge, the mass flux crossing the field edge, the settling velocity of the droplets and the turbulence. For the parameter set and range considered, it is demonstrated for the first time that the effect of the droplet diameter distribution of the different nozzle types on the amount of deposition spray drift can be evaluated by a single parameter, i.e., the volume fraction of droplets with a diameter smaller than 191 μm. The reduced order model can be solved more than 4 orders of magnitude faster than the comprehensive CFD model.
Noise Prevents Infinite Stretching of the Passive Field in a Stochastic Vector Advection Equation
NASA Astrophysics Data System (ADS)
Flandoli, Franco; Maurelli, Mario; Neklyudov, Mikhail
2014-09-01
A linear stochastic vector advection equation is considered; the equation may model a passive magnetic field in a random fluid. When the driving velocity field is rough but deterministic, in particular just Hölder continuous and bounded, one can construct examples of infinite stretching of the passive field, arising from smooth initial conditions. The purpose of the paper is to prove that infinite stretching is prevented if the driving velocity field contains in addition a white noise component.
Karniadakis, George Em
2014-03-11
The main objective of this project is to develop new computational tools for uncertainty quantifica- tion (UQ) of systems governed by stochastic partial differential equations (SPDEs) with applications to advection-diffusion-reaction systems. We pursue two complementary approaches: (1) generalized polynomial chaos and its extensions and (2) a new theory on deriving PDF equations for systems subject to color noise. The focus of the current work is on high-dimensional systems involving tens or hundreds of uncertain parameters.
A novel method for analytically solving a radial advection-dispersion equation
NASA Astrophysics Data System (ADS)
Lai, Keng-Hsin; Liu, Chen-Wuing; Liang, Ching-Ping; Chen, Jui-Sheng; Sie, Bing-Ruei
2016-11-01
An analytical solution for solute transport in a radial flow field has a variety of practical applications in the study of the transport in push-pull/divergent/convergent flow tracer tests, aquifer remediation by pumping and aquifer storage and recovery. However, an analytical solution for radial advective-dispersive transport has been proven very difficult to develop and relatively few in subsurface hydrology have made efforts to do so, because variable coefficients in the governing partial differential equations. Most of the solutions for radial advective-dispersive transport presented in the literature have generally been solved semi-analytically with the final concentration values being obtained with the help of a numerical Laplace inversion. This study presents a novel solution strategy for analytically solving the radial advective-dispersive transport problem. A Laplace transform with respect to the time variable and a generalized integral transform technique with respect to the spatial variable are first performed to convert the transient governing partial differential equations into an algebraic equation. Subsequently, the algebraic equation is solved using simple algebraic manipulations, easily yielding the solution in the transformed domain. The solution in the original domain is ultimately obtained by successive applications of the Laplace and corresponding generalized integral transform inversions. A convergent flow tracer test is used to demonstrate the robustness of the proposed method for deriving an exact analytical solution to the radial advective-dispersive transport problem. The developed analytical solution is verified against a semi-analytical solution taken from the literature. The results show perfect agreement between our exact analytical solution and the semi-analytical solution. The solution method presented in this study can be applied to create more comprehensive analytical models for a great variety of radial advective
NASA Astrophysics Data System (ADS)
Yochelis, Arik; Bar-On, Tomer; Gov, Nir S.
2016-04-01
Unconventional myosins belong to a class of molecular motors that walk processively inside cellular protrusions towards the tips, on top of actin filament. Surprisingly, in addition, they also form retrograde moving self-organized aggregates. The qualitative properties of these aggregates are recapitulated by a mass conserving reaction-diffusion-advection model and admit two distinct families of modes: traveling waves and pulse trains. Unlike the traveling waves that are generated by a linear instability, pulses are nonlinear structures that propagate on top of linearly stable uniform backgrounds. Asymptotic analysis of isolated pulses via a simplified reaction-diffusion-advection variant on large periodic domains, allows to draw qualitative trends for pulse properties, such as the amplitude, width, and propagation speed. The results agree well with numerical integrations and are related to available empirical observations.
Multiphase Advection and Radiation Diffusion with Material Interfaces on Unstructured Meshes
Anninos, P
2002-10-03
A collection of numerical methods are presented for the advection or remapping of material properties on unstructured and staggered polyhedral meshes in arbitrary Lagrange-Eulerian calculations. The methods include several new procedures to track and capture sharp interface boundaries, and to partition radiation energy into multi-material thermal states. The latter is useful for extending and applying consistently single material radiation diffusion solvers to multi-material problems.
An improved lattice Boltzmann method for simulating advective-diffusive processes in fluids
NASA Astrophysics Data System (ADS)
Aursjø, Olav; Jettestuen, Espen; Vinningland, Jan Ludvig; Hiorth, Aksel
2017-03-01
Lattice Boltzmann methods are widely used to simulate advective-diffusive processes in fluids. Lattice Bhatnagar-Gross-Krook methods presented in the literature mostly just exhibit first order spatial accuracy and introduce errors proportional to the velocity squared. Formulations proposed to alleviate this have only been partly successful and are valid only in certain specific situations. We present and demonstrate here a formulation that produces no such second order errors. This formulation suggests that a subtle, but important, adjustment is all it takes to improve the accuracy of the method. The key to the improved accuracy of this new model is the non-standard definition of the concentration that relates to the distribution function describing the advection-diffusion in lattice Boltzmann. The main advantage of the algorithm comes to view when simulating situations where fluid density variations appear. The present formulation of the advection-diffusion algorithm will, by taking into account these fluid density variations, drastically reduce the errors produced compared to the standard formulations. We also show how a source term is included in this new formulation without it losing its second order spatial accuracy.
İbiş, Birol
2014-01-01
This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. PMID:24578662
Siebert, Julien; Alonso, Sergio; Bär, Markus; Schöll, Eckehard
2014-05-01
A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.
Least-Squares Spectral Method for the solution of a fractional advection-dispersion equation
NASA Astrophysics Data System (ADS)
Carella, Alfredo Raúl; Dorao, Carlos Alberto
2013-01-01
Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection-dispersion equations using Caputo or Riemann-Liouville fractional derivatives. A Gauss-Lobatto-Jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition. Exponential convergence rate of the operator is verified when increasing the order of the approximation. Solutions are calculated for fractional-time and fractional-space differential equations. Comparisons with finite difference schemes are included. A significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate.
Assessment of nitrate transport parameters using the advection-diffusion cell.
Aljazzar, Taiseer; Al-Qinna, Mohammed
2016-11-01
This study aimed to better understand nitrate transport in the soil system in a part of the state of North Rhine-Westphalia, in Germany, and to aid in the development of groundwater protection plans. An advection-diffusion (AD) cell was used in a miscible displacement experiment setup to characterize nitrate transport in 12 different soil samples from the study area. The three nitrate sorption isotherms were tested to define the exact nitrate interaction with the soil matrix. Soils varied in their properties which in its turn explain the variations in nitrate transport rates. Soil texture and organic matter content showed to have the most important effect on nitrate recovery and retardation. The miscible displacement experiment indicated a decrease in retardation by increasing sand fraction, and an increase in retardation by increasing soil organic matter content. Soil samples with high sand fractions (up to 94 %) exhibited low nitrate sorption capacity of less than 10 %, while soils with high organic matter content showed higher sorption of about 30 %. Based on parameterization for nitrate transport equation, the pore water velocity for both sandy and loamy soils were significantly different (P < 0.001). Pore water velocity in sandy soil (about 4 × 10(-3) m/s) was about 100 to 1000 larger than in loamy soils (8.7 × 10(-5) m/s). On the other hand, the reduction in nitrate transport in soils associated with high organic matter was due to fine pore pathways clogged by fine organic colloids. It is expected that the existing micro-phobicity increased the nitrate recovery from 9 to 32 % resulting in maximum diffusion rates of about 3.5 × 10(-5) m/s(2) in sandy soils (sample number CS-04) and about 1.4 × 10(-7) m/s(2) in silt loam soils (sample number FS-02).
Bioturbation, advection, and diffusion of a conserved tracer in a laboratory flume
NASA Astrophysics Data System (ADS)
Work, P. A.; Moore, P. R.; Reible, D. D.
2002-06-01
Laboratory experiments indicating the relative influences of advection, diffusion, and bioturbation on transport of NaCl tracer between a stream and streambed are described. Data were collected in a recirculating flume housing a box filled with test sediments. Peclet numbers ranged from 0 to 1.5. Sediment components included a medium sand (d50 = 0.31 mm), kaolinite, and topsoil. Lumbriculus variegatus were introduced as bioturbators. Conductivity probes were employed to document the flux of the tracer solution out of the bed. Measurements are compared to one-dimensional effective diffusion models assuming one or two horizontal sediment layers. These simple models provide a good indication of tracer half-life in the bed if a suitable effective diffusion coefficient is chosen but underpredict initial flux and overpredict flux at long times. Organism activity was limited to the upper reaches of the sediment test box but eventually exerts a secondary influence on flux from deeper regions.
Choi, Jee-Won; Tillman, Fred D; Smith, James A
2002-07-15
It was hypothesized that atmospheric pressure changes can induce gas flow in the unsaturated zone to such an extent that the advective flux of organic vapors in unsaturated-zone soil gas can be significant relative to the gas-phase diffusion flux of these organic vapors. To test this hypothesis, a series of field measurements and computer simulations were conducted to simulate and compare diffusion and advection fluxes at a trichloroethene-contaminated field site at Picatinny Arsenal in north-central New Jersey. Moisture content temperature, and soil-gas pressure were measured at multiple depths (including at land surface) and times for three distinct sampling events in August 1996, October 1996, and August 1998. Gas pressures in the unsaturated zone changed significantly over time and followed changes measured in the atmosphere. Gas permeability of the unsaturated zone was estimated using data from a variety of sources, including laboratory gas permeability measurements made on intact soil cores from the site, a field air pump test, and calibration of a gas-flow model to the transient, one-dimensional gas pressure data. The final gas-flow model reproduced small pressure gradients as observed in the field during the three distinct sampling events. The velocities calculated from the gas-flow model were used in transient, one-dimensional transport simulations to quantify advective and diffusive fluxes of TCE vapor from the subsurface to the atmosphere as a function of time for each sampling event. Effective diffusion coefficients used for these simulations were determined from independent laboratory measurements made on intact soil cores collected from the field site. For two of the three sampling events (August 1996 and August 1998), the TCE gas-phase diffusion flux at land surface was significantly greater than the advection flux over the entire sampling period. For the second sampling event (October 1996), the advection flux was frequently larger than the
NASA Astrophysics Data System (ADS)
Pérez Guerrero, J. S.; Skaggs, T. H.
2010-08-01
SummaryMathematical models describing contaminant transport in heterogeneous porous media are often formulated as an advection-dispersion transport equation with distance-dependent transport coefficients. In this work, a general analytical solution is presented for the linear, one-dimensional advection-dispersion equation with distance-dependent coefficients. An integrating factor is employed to obtain a transport equation that has a self-adjoint differential operator, and a solution is found using the generalized integral transform technique (GITT). It is demonstrated that an analytical expression for the integrating factor exists for several transport equation formulations of practical importance in groundwater transport modeling. Unlike nearly all solutions available in the literature, the current solution is developed for a finite spatial domain. As an illustration, solutions for the particular case of a linearly increasing dispersivity are developed in detail and results are compared with solutions from the literature. Among other applications, the current analytical solution will be particularly useful for testing or benchmarking numerical transport codes because of the incorporation of a finite spatial domain.
A tracer-based inversion method for diagnosing eddy-induced diffusivity and advection
NASA Astrophysics Data System (ADS)
Bachman, S. D.; Fox-Kemper, B.; Bryan, F. O.
2015-02-01
A diagnosis method is presented which inverts a set of tracer flux statistics into an eddy-induced transport intended to apply for all tracers. The underlying assumption is that a linear flux-gradient relationship describes eddy-induced tracer transport, but a full tensor coefficient is assumed rather than a scalar coefficient which allows for down-gradient and skew transports. Thus, Lagrangian advection and anisotropic diffusion not necessarily aligned with the tracer gradient can be diagnosed. In this method, multiple passive tracers are initialized in an eddy-resolving flow simulation. Their spatially-averaged gradients form a matrix, where the gradient of each tracer is assumed to satisfy an identical flux-gradient relationship. The resulting linear system, which is overdetermined when using more than three tracers, is then solved to obtain an eddy transport tensor R which describes the eddy advection (antisymmetric part of R) and potentially anisotropic diffusion (symmetric part of R) in terms of coarse-grained variables. The mathematical basis for this inversion method is presented here, along with practical guidelines for its implementation. We present recommendations for initialization of the passive tracers, maintaining the required misalignment of the tracer gradients, correcting for nonconservative effects, and quantifying the error in the diagnosed transport tensor. A method is proposed to find unique, tracer-independent, distinct rotational and divergent Lagrangian transport operators, but the results indicate that these operators are not meaningfully relatable to tracer-independent eddy advection or diffusion. With the optimal method of diagnosis, the diagnosed transport tensor is capable of predicting the fluxes of other tracers that are withheld from the diagnosis, including even active tracers such as buoyancy, such that relative errors of 14% or less are found.
Lichtner, P.C.; Helgeson, H.C.
1986-06-20
A general formulation of multi-phase fluid flow coupled to chemical reactions was developed based on a continuum description of porous media. A preliminary version of the computer code MCCTM was constructed which implemented the general equations for a single phase fluid. The computer code MCCTM incorporates mass transport by advection-diffusion/dispersion in a one-dimensional porous medium coupled to reversible and irreversible, homogeneous and heterogeneous chemical reactions. These reactions include aqueous complexing, oxidation/reduction reactions, ion exchange, and hydrolysis reactions of stoichiometric minerals. The code MCCTM uses a fully implicit finite difference algorithm. The code was tested against analytical calculations. Applications of the code included investigation of the propagation of sharp chemical reaction fronts, metasomatic alteration of microcline at elevated temperatures and pressures, and ion-exchange in a porous column. Finally numerical calculations describing fluid flow in crystalline rock in the presence of a temperature gradient were compared with experimental results for quartzite.
Space-fractional advection-dispersion equations by the Kansa method
NASA Astrophysics Data System (ADS)
Pang, Guofei; Chen, Wen; Fu, Zhuojia
2015-07-01
The paper makes the first attempt at applying the Kansa method, a radial basis function meshless collocation method, to the space-fractional advection-dispersion equations, which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer-order derivative. However, because of non-local property of integro-differential operator of space-fractional derivative, numerical solution of these novel models is very challenging and little has been reported in literature. It is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non-local property. Thus, the global methods appear to have certain advantages in numerical simulation of these non-local models because of their high accuracy and smaller size resultant matrix equation. Compared with the finite difference method, popular in the solution of fractional equations, the Kansa method is a recent meshless global technique and is promising for high-dimensional irregular domain problems. In this study, the resultant matrix of the Kansa method is accurately calculated by the Gauss-Jacobi quadrature rule. Numerical results show that the Kansa method is highly accurate and computationally efficient for space-fractional advection-dispersion problems.
Variational Integration for Ideal MHD with Built-in Advection Equations
Zhou, Yao; Qin, Hong; Burby, J. W.; Bhattacharjee, A.
2014-08-05
Newcomb's Lagrangian for ideal MHD in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
Variational integration for ideal magnetohydrodynamics with built-in advection equations
Zhou, Yao; Burby, J. W.; Bhattacharjee, A.; Qin, Hong
2014-10-15
Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
NASA Astrophysics Data System (ADS)
Butler, N. L.; Hunt, J. R.; Tompkins, M. R.
2011-12-01
Hyporheic exchange can locally mitigate thermal stress caused by high water temperatures by upwelling water cooler than ambient stream temperatures and thus providing thermal refuge for critical cold water organisms like salmonids. Ten hyporheic exchange locations were identified by dye tracer experiments along a 16 km stretch of Deer Creek near Vina, California. Four months of continuous temperature measurements were made in the late summer of 2005 at each downwelling and upwelling location and revealed upwelled temperatures that were lagged in time and damped in amplitude. Upwelling hyporheic temperatures that could provide thermal refuge were observed in seven of the ten temperature records. This data was modeled by an analytical one-dimensional advection-diffusion equation solution using subsurface water velocity and the hydrodynamic dispersivity fitting parameters. At each location variations in upwelling temperature are explained by changing subsurface water velocities and flow pathways. The lag time in hyporheic heat flow ranged from a few hours to 44 hours over distances of 15 to 76 meters. The daily stream temperature variation was on the order of 10°C, which was reduced to 1 to 8°C in the upwelling hyporheic flow. At four locations, there was evidence that changes in stream flow produced changes in the amplitude and phase of the upwelling hyporheic water temperature by altering both the subsurface water velocity and hydrodynamic dispersivity. At two locations, additional cold water refuge was created by decreases in surface water flow because it reduced the estimated subsurface water velocity increasing the lag time between the peak surface water and subsurface water temperatures. Increases in surface water flow increased the dispersivity at three locations providing more cold water refuge by reducing the amplitude of the upwelling hyporheic temperature. Such changes alter thermal refuge for salmonids placing a new emphasis on managing surface water
An advection-diffusion model for cross-field runaway electron transport in perturbed magnetic fields
NASA Astrophysics Data System (ADS)
Särkimäki, Konsta; Hirvijoki, Eero; Decker, Joan; Varje, Jari; Kurki-Suonio, Taina
2016-12-01
Disruption-generated runaway electrons (RE) present an outstanding issue for ITER. The predictive computational studies of RE generation rely on orbit-averaged computations and, as such, they lack the effects from the magnetic field stochasticity. Since stochasticity is naturally present in post-disruption plasma, and externally induced stochastization offers a prominent mechanism to mitigate RE avalanche, we present an advection-diffusion model that can be used to couple an orbit-following code to an orbit-averaged tool in order to capture the cross-field transport and to overcome the latter’s limitation. The transport coefficients are evaluated via a Monte Carlo method. We show that the diffusion coefficient differs significantly from the well-known Rechester-Rosenbluth result. We also demonstrate the importance of including the advection: it has a two-fold role both in modelling transport barriers created by magnetic islands and in amplifying losses in regions where the islands are not present.
NASA Astrophysics Data System (ADS)
You, Kehua; Zhan, Hongbin
2013-02-01
Diffusive flux is traditionally treated as the dominant mechanism of gas transport in unsaturated zones under natural conditions, and advective flux is usually neglected. However, some researchers have found that pressure-driven and density-driven advective flux may also be significant under certain conditions. This article compares the diffusive, pressure-driven and density-driven advective fluxes of gas phase volatile organic compound (VOCs) in unsaturated zones under various natural conditions. The presence of a less or more permeable layer at ground surface in a layered unsaturated zone is investigated for its impact on the net contribution of advective and diffusive fluxes. Results show although the transient advective flux can be greater than the diffusive flux, under most of the field conditions the net contribution of the advective flux is one to three orders of magnitude less than the diffusive flux, and the influence of the density-driven flux is undetectable. The advective flux contributes comparably with the diffusive flux only when the gas-filled porosity is less than 0.05. The presence of a less permeable layer at ground surface slightly increases the total flux in the underlying layer, while the presence of a more permeable layer at ground surface significantly increases the total flux in it. When the magnitude of water table fluctuation is less than 1 cm, and the period is greater than 0.5 day, the fluctuation of the water table can be simulated by fixing the water table position and setting a fluctuating moving velocity at the water table.
Richon, Patrick; Perrier, Frédéric; Koirala, Bharat Prasad; Girault, Frédéric; Bhattarai, Mukunda; Sapkota, Soma Nath
2011-02-01
Temporal variation of radon-222 concentration was studied at the Syabru-Bensi hot springs, located on the Main Central Thrust zone in Central Nepal. This site is characterized by several carbon dioxide discharges having maximum fluxes larger than 10 kg m(-2) d(-1). Radon concentration was monitored with autonomous Barasol™ probes between January 2008 and November 2009 in two small natural cavities with high CO(2) concentration and at six locations in the soil: four points having a high flux, and two background reference points. At the reference points, dominated by radon diffusion, radon concentration was stable from January to May, with mean values of 22 ± 6.9 and 37 ± 5.5 kBq m(-3), but was affected by a large increase, of about a factor of 2 and 1.6, respectively, during the monsoon season from June to September. At the points dominated by CO(2) advection, by contrast, radon concentration showed higher mean values 39.0 ± 2.6 to 78 ± 1.4 kBq m(-3), remarkably stable throughout the year with small long-term variation, including a possible modulation of period around 6 months. A significant difference between the diffusion dominated reference points and the advection-dominated points also emerged when studying the diurnal S(1) and semi-diurnal S(2) periodic components. At the advection-dominated points, radon concentration did not exhibit S(1) or S(2) components. At the reference points, however, the S(2) component, associated with barometric tide, could be identified during the dry season, but only when the probe was installed at shallow depth. The S(1) component, associated with thermal and possibly barometric diurnal forcing, was systematically observed, especially during monsoon season. The remarkable short-term and long-term temporal stability of the radon concentration at the advection-dominated points, which suggests a strong pressure source at depth, may be an important asset to detect possible temporal variations associated with the
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems
NASA Astrophysics Data System (ADS)
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-07-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic dissipative particle dynamics (DPD) framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between tDPD particles, and the advection is implicitly considered by the movements of these Lagrangian particles. An analytical formula is proposed to relate the tDPD parameters to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the conventional DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers.
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems
Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-01-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic dissipative particle dynamics (DPD) framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between tDPD particles, and the advection is implicitly considered by the movements of these Lagrangian particles. An analytical formula is proposed to relate the tDPD parameters to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the conventional DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers. PMID:26156459
Xu, Bruce S; Lollar, Barbara Sherwood; Passeport, Elodie; Sleep, Brent E
2016-04-15
Aqueous phase diffusion-related isotope fractionation (DRIF) for carbon isotopes was investigated for common groundwater contaminants in systems in which transport could be considered to be one-dimensional. This paper focuses not only on theoretically observable DRIF effects in these systems but introduces the important concept of constraining "observable" DRIF based on constraints imposed by the scale of measurements in the field, and on standard limits of detection and analytical uncertainty. Specifically, constraints for the detection of DRIF were determined in terms of the diffusive fractionation factor, the initial concentration of contaminants (C0), the method detection limit (MDL) for isotopic analysis, the transport time, and the ratio of the longitudinal mechanical dispersion coefficient to effective molecular diffusion coefficient (Dmech/Deff). The results allow a determination of field conditions under which DRIF may be an important factor in the use of stable carbon isotope measurements for evaluation of contaminant transport and transformation for one-dimensional advective-dispersive transport. This study demonstrates that for diffusion-dominated transport of BTEX, MTBE, and chlorinated ethenes, DRIF effects are only detectable for the smaller molar mass compounds such as vinyl chloride for C0/MDL ratios of 50 or higher. Much larger C0/MDL ratios, corresponding to higher source concentrations or lower detection limits, are necessary for DRIF to be detectable for the higher molar mass compounds. The distance over which DRIF is observable for VC is small (less than 1m) for a relatively young diffusive plume (<100years), and DRIF will not easily be detected by using the conventional sampling approach with "typical" well spacing (at least several meters). With contaminant transport by advection, mechanical dispersion, and molecular diffusion this study suggests that in field sites where Dmech/Deff is larger than 10, DRIF effects will likely not be
NASA Astrophysics Data System (ADS)
Siero, E.; Doelman, A.; Eppinga, M. B.; Rademacher, J. D. M.; Rietkerk, M.; Siteur, K.
2015-03-01
For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns.
ON A NONHOMOGENEOUS RANDOM DIFFUSION EQUATION.
CORRELATION TECHNIQUES, FUNCTIONS(MATHEMATICS)), (*STOCHASTIC PROCESSES, EQUATIONS), STATISTICAL FUNCTIONS, PROBABILITY, HILBERT SPACE, GREEN’S FUNCTION, SERIES(MATHEMATICS), ALLOYS, DIFFUSION , INTEGRALS
Transport dissipative particle dynamics model for mesoscopic advection- diffusion-reaction problems
Zhen, Li; Yazdani, Alireza; Tartakovsky, Alexandre M.; Karniadakis, George E.
2015-07-07
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic DPD framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between particles, and an analytical formula is proposed to relate the mesoscopic concentration friction to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers.
Effect of advective flow in fractures and matrix diffusion on natural gas production
Karra, Satish; Makedonska, Nataliia; Viswanathan, Hari S.; Painter, Scott L.; Hyman, Jeffrey D.
2015-10-12
Although hydraulic fracturing has been used for natural gas production for the past couple of decades, there are significant uncertainties about the underlying mechanisms behind the production curves that are seen in the field. A discrete fracture network based reservoir-scale work flow is used to identify the relative effect of flow of gas in fractures and matrix diffusion on the production curve. With realistic three dimensional representations of fracture network geometry and aperture variability, simulated production decline curves qualitatively resemble observed production decline curves. The high initial peak of the production curve is controlled by advective fracture flow of free gas within the network and is sensitive to the fracture aperture variability. Matrix diffusion does not significantly affect the production decline curve in the first few years, but contributes to production after approximately 10 years. These results suggest that the initial flushing of gas-filled background fractures combined with highly heterogeneous flow paths to the production well are sufficient to explain observed initial production decline. Lastly, these results also suggest that matrix diffusion may support reduced production over longer time frames.
Effect of advective flow in fractures and matrix diffusion on natural gas production
Karra, Satish; Makedonska, Nataliia; Viswanathan, Hari S.; ...
2015-10-12
Although hydraulic fracturing has been used for natural gas production for the past couple of decades, there are significant uncertainties about the underlying mechanisms behind the production curves that are seen in the field. A discrete fracture network based reservoir-scale work flow is used to identify the relative effect of flow of gas in fractures and matrix diffusion on the production curve. With realistic three dimensional representations of fracture network geometry and aperture variability, simulated production decline curves qualitatively resemble observed production decline curves. The high initial peak of the production curve is controlled by advective fracture flow of freemore » gas within the network and is sensitive to the fracture aperture variability. Matrix diffusion does not significantly affect the production decline curve in the first few years, but contributes to production after approximately 10 years. These results suggest that the initial flushing of gas-filled background fractures combined with highly heterogeneous flow paths to the production well are sufficient to explain observed initial production decline. Lastly, these results also suggest that matrix diffusion may support reduced production over longer time frames.« less
NASA Astrophysics Data System (ADS)
Mazaheri, Alireza; Nishikawa, Hiroaki
2016-09-01
We propose arbitrary high-order discontinuous Galerkin (DG) schemes that are designed based on a first-order hyperbolic advection-diffusion formulation of the target governing equations. We present, in details, the efficient construction of the proposed high-order schemes (called DG-H), and show that these schemes have the same number of global degrees-of-freedom as comparable conventional high-order DG schemes, produce the same or higher order of accuracy solutions and solution gradients, are exact for exact polynomial functions, and do not need a second-derivative diffusion operator. We demonstrate that the constructed high-order schemes give excellent quality solution and solution gradients on irregular triangular elements. We also construct a Weighted Essentially Non-Oscillatory (WENO) limiter for the proposed DG-H schemes and apply it to discontinuous problems. We also make some accuracy comparisons with conventional DG and interior penalty schemes. A relative qualitative cost analysis is also reported, which indicates that the high-order schemes produce orders of magnitude more accurate results than the low-order schemes for a given CPU time. Furthermore, we show that the proposed DG-H schemes are nearly as efficient as the DG and Interior-Penalty (IP) schemes as these schemes produce results that are relatively at the same error level for approximately a similar CPU time.
Solution of the advection-dispersion equation: Continuous load of finite duration
Runkel, R.L.
1996-01-01
Field studies of solute fate and transport in streams and rivers often involve an. experimental release of solutes at an upstream boundary for a finite period of time. A review of several standard references on surface-water-quality modeling indicates that the analytical solution to the constant-parameter advection-dispersion equation for this type of boundary condition has been generally overlooked. Here an exact analytical solution that considers a continuous load of unite duration is compared to an approximate analytical solution presented elsewhere. Results indicate that the exact analytical solution should be used for verification of numerical solutions and other solute-transport problems wherein a high level of accuracy is required. ?? ASCE.
Barbour, S Lee; Hendry, M Jim; Wassenaar, Leonard I
2012-04-01
Solute transport in clay-rich aquitards is characterized as molecular diffusion- or advection-dominated based on the Péclet number (P(e)). However, few field-based measurements of the coefficient of molecular diffusion (D(e)) exist, and none with a range of advection- or diffusion-dominated conditions in the same aquitard. In this long-term field experiment, standing water in a recovering well was spiked with deuterium ((2)H), then water-level recovery and δ(2)H values were monitored as the well returned to static conditions over 1054 days. After a second (2)H spike, water levels and δ(2)H values were monitored to day 1644 while under near static conditions. Modeling of the second spike was used to define the D(e) of (2)H as (3-4)× 10(-10)m(2)s(-1) for an accessible porosity of 0.31. Reservoir concentrations from the initial spike were modeled to define the transition from advection- to diffusion-dominated transport. This occurred after 200 days, consistent with a transition in P(e) from <1 to >1 when the length term is taken as the radial extent of the tracer plume (normalized concentration <0.05). This study verifies plume extent as the characteristic length term in the calculation of P(e) and demonstrates the transition from advection- to diffusion-dominated transport as the value of P(e) decreases below unity.
Numerical applications of the advective-diffusive codes for the inner magnetosphere
NASA Astrophysics Data System (ADS)
Aseev, N. A.; Shprits, Y. Y.; Drozdov, A. Y.; Kellerman, A. C.
2016-11-01
In this study we present analytical solutions for convection and diffusion equations. We gather here the analytical solutions for the one-dimensional convection equation, the two-dimensional convection problem, and the one- and two-dimensional diffusion equations. Using obtained analytical solutions, we test the four-dimensional Versatile Electron Radiation Belt code (the VERB-4D code), which solves the modified Fokker-Planck equation with additional convection terms. The ninth-order upwind numerical scheme for the one-dimensional convection equation shows much more accurate results than the results obtained with the third-order scheme. The universal limiter eliminates unphysical oscillations generated by high-order linear upwind schemes. Decrease in the space step leads to convergence of a numerical solution of the two-dimensional diffusion equation with mixed terms to the analytical solution. We compare the results of the third- and ninth-order schemes applied to magnetospheric convection modeling. The results show significant differences in electron fluxes near geostationary orbit when different numerical schemes are used.
Space shuttle exhaust plumes in the lower thermosphere: Advective transport and diffusive spreading
NASA Astrophysics Data System (ADS)
Stevens, Michael H.; Lossow, Stefan; Siskind, David E.; Meier, R. R.; Randall, Cora E.; Russell, James M.; Urban, Jo; Murtagh, Donal
2014-02-01
The space shuttle main engine plume deposited between 100 and 115 km altitude is a valuable tracer for global-scale dynamical processes. Several studies have shown that this plume can reach the Arctic or Antarctic to form bursts of polar mesospheric clouds (PMCs) within a few days. The rapid transport of the shuttle plume is currently not reproduced by general circulation models and is not well understood. To help delineate the issues, we present the complete satellite datasets of shuttle plume observations by the Sounding of the Atmosphere using Broadband Emission Radiometry instrument and the Sub-Millimeter Radiometer instrument. From 2002 to 2011 these two instruments observed 27 shuttle plumes in over 600 limb scans of water vapor emission, from which we derive both advective meridional transport and diffusive spreading. Each plume is deposited at virtually the same place off the United States east coast so our results are relevant to northern mid-latitudes. We find that the advective transport for the first 6-18 h following deposition depends on the local time (LT) of launch: shuttle plumes deposited later in the day (~13-22 LT) typically move south whereas they otherwise typically move north. For these younger plumes rapid transport is most favorable for launches at 6 and 18 LT, when the displacement is 10° in latitude corresponding to an average wind speed of 30 m/s. For plumes between 18 and 30 h old some show average sustained meridional speeds of 30 m/s. For plumes between 30 and 54 h old the observations suggest a seasonal dependence to the meridional transport, peaking near the beginning of year at 24 m/s. The diffusive spreading of the plume superimposed on the transport is on average 23 m/s in 24 h. The plume observations show large variations in both meridional transport and diffusive spreading so that accurate modeling requires knowledge of the winds specific to each case. The combination of transport and spreading from the STS-118 plume in August
NASA Astrophysics Data System (ADS)
Ginzburg, Irina
2017-01-01
The effect of the heterogeneity in the soil structure or the nonuniformity of the velocity field on the modeled resident time distribution (RTD) and breakthrough curves is quantified by their moments. While the first moment provides the effective velocity, the second moment is related to the longitudinal dispersion coefficient (kT) in the developed Taylor regime; the third and fourth moments are characterized by their normalized values skewness (Sk) and kurtosis (Ku), respectively. The purpose of this investigation is to examine the role of the truncation corrections of the numerical scheme in kT, Sk, and Ku because of their interference with the second moment, in the form of the numerical dispersion, and in the higher-order moments, by their definition. Our symbolic procedure is based on the recently proposed extended method of moments (EMM). Originally, the EMM restores any-order physical moments of the RTD or averaged distributions assuming that the solute concentration obeys the advection-diffusion equation in multidimensional steady-state velocity field, in streamwise-periodic heterogeneous structure. In our work, the EMM is generalized to the fourth-order-accurate apparent mass-conservation equation in two- and three-dimensional duct flows. The method looks for the solution of the transport equation as the product of a long harmonic wave and a spatially periodic oscillating component; the moments of the given numerical scheme are derived from a chain of the steady-state fourth-order equations at a single cell. This mathematical technique is exemplified for the truncation terms of the two-relaxation-time lattice Boltzmann scheme, using plug and parabolic flow in straight channel and cylindrical capillary with the d2Q9 and d3Q15 discrete velocity sets as simple but illustrative examples. The derived symbolic dependencies can be readily extended for advection by another, Newtonian or non-Newtonian, flow profile in any-shape open-tabular conduits. It is
Ginzburg, Irina
2017-01-01
The effect of the heterogeneity in the soil structure or the nonuniformity of the velocity field on the modeled resident time distribution (RTD) and breakthrough curves is quantified by their moments. While the first moment provides the effective velocity, the second moment is related to the longitudinal dispersion coefficient (k_{T}) in the developed Taylor regime; the third and fourth moments are characterized by their normalized values skewness (Sk) and kurtosis (Ku), respectively. The purpose of this investigation is to examine the role of the truncation corrections of the numerical scheme in k_{T}, Sk, and Ku because of their interference with the second moment, in the form of the numerical dispersion, and in the higher-order moments, by their definition. Our symbolic procedure is based on the recently proposed extended method of moments (EMM). Originally, the EMM restores any-order physical moments of the RTD or averaged distributions assuming that the solute concentration obeys the advection-diffusion equation in multidimensional steady-state velocity field, in streamwise-periodic heterogeneous structure. In our work, the EMM is generalized to the fourth-order-accurate apparent mass-conservation equation in two- and three-dimensional duct flows. The method looks for the solution of the transport equation as the product of a long harmonic wave and a spatially periodic oscillating component; the moments of the given numerical scheme are derived from a chain of the steady-state fourth-order equations at a single cell. This mathematical technique is exemplified for the truncation terms of the two-relaxation-time lattice Boltzmann scheme, using plug and parabolic flow in straight channel and cylindrical capillary with the d2Q9 and d3Q15 discrete velocity sets as simple but illustrative examples. The derived symbolic dependencies can be readily extended for advection by another, Newtonian or non-Newtonian, flow profile in any-shape open-tabular conduits. It is
Lichtenberg, Mads; Nørregaard, Rasmus Dyrmose; Kühl, Michael
2017-03-01
The role of hyaline hairs on the thallus of brown algae in the genus Fucus is long debated and several functions have been proposed. We used a novel motorized set-up for two-dimensional and three-dimensional mapping with O2 microsensors to investigate the spatial heterogeneity of the diffusive boundary layer (DBL) and O2 flux around single and multiple tufts of hyaline hairs on the thallus of Fucus vesiculosus. Flow was a major determinant of DBL thickness, where higher flow decreased DBL thickness and increased O2 flux between the algal thallus and the surrounding seawater. However, the topography of the DBL varied and did not directly follow the contour of the underlying thallus. Areas around single tufts of hyaline hairs exhibited a more complex mass-transfer boundary layer, showing both increased and decreased thickness when compared with areas over smooth thallus surfaces. Over thallus areas with several hyaline hair tufts, the overall effect was an apparent increase in the boundary layer thickness. We also found indications for advective O2 transport driven by pressure gradients or vortex shedding downstream from dense tufts of hyaline hairs that could alleviate local mass-transfer resistances. Mass-transfer dynamics around hyaline hair tufts are thus more complex than hitherto assumed and may have important implications for algal physiology and plant-microbe interactions.
Correlation Networks from Flows. The Case of Forced and Time-Dependent Advection-Diffusion Dynamics.
Tupikina, Liubov; Molkenthin, Nora; López, Cristóbal; Hernández-García, Emilio; Marwan, Norbert; Kurths, Jürgen
2016-01-01
Complex network theory provides an elegant and powerful framework to statistically investigate different types of systems such as society, brain or the structure of local and long-range dynamical interrelationships in the climate system. Network links in climate networks typically imply information, mass or energy exchange. However, the specific connection between oceanic or atmospheric flows and the climate network's structure is still unclear. We propose a theoretical approach for verifying relations between the correlation matrix and the climate network measures, generalizing previous studies and overcoming the restriction to stationary flows. Our methods are developed for correlations of a scalar quantity (temperature, for example) which satisfies an advection-diffusion dynamics in the presence of forcing and dissipation. Our approach reveals that correlation networks are not sensitive to steady sources and sinks and the profound impact of the signal decay rate on the network topology. We illustrate our results with calculations of degree and clustering for a meandering flow resembling a geophysical ocean jet.
A Time-Adaptive Integrator Based on Radau Methods for Advection Diffusion Reaction PDEs
NASA Astrophysics Data System (ADS)
Gonzalez-Pinto, S.; Perez-Rodriguez, S.
2009-09-01
The numerical integration of time-dependent PDEs, especially of Advection Diffusion Reaction type, for two and three spatial variables (in short, 2D and 3D problems) in the MoL framework is considered. The spatial discretization is made by using Finite Differences and the time integration is carried out by means of the L-stable, third order formula known as the two stage Radau IIA method. The main point for the solution of the large dimensional ODEs is not to solve the stage values of the Radau method until convergence (because the convergence is very slow on the stiff components), but only giving a very few iterations and take as advancing solution the latter stage value computed. The iterations are carried out by using the Approximate Matrix Factorization (AMF) coupled to a Newton-type iteration (SNI) as indicated in [5], which turns out in an acceptably cheap iteration, like Alternating Directions Methods (ADI) of Peaceman and Rachford (1955). Some stability results for the whole process (AMF)-(SNI) and a local error estimate for an adaptive time-integration are also given. Numerical results on two standard PDEs are presented and some conclusions about our method and other well-known solvers are drawn.
Correlation Networks from Flows. The Case of Forced and Time-Dependent Advection-Diffusion Dynamics
Tupikina, Liubov; Molkenthin, Nora; López, Cristóbal; Hernández-García, Emilio; Marwan, Norbert; Kurths, Jürgen
2016-01-01
Complex network theory provides an elegant and powerful framework to statistically investigate different types of systems such as society, brain or the structure of local and long-range dynamical interrelationships in the climate system. Network links in climate networks typically imply information, mass or energy exchange. However, the specific connection between oceanic or atmospheric flows and the climate network’s structure is still unclear. We propose a theoretical approach for verifying relations between the correlation matrix and the climate network measures, generalizing previous studies and overcoming the restriction to stationary flows. Our methods are developed for correlations of a scalar quantity (temperature, for example) which satisfies an advection-diffusion dynamics in the presence of forcing and dissipation. Our approach reveals that correlation networks are not sensitive to steady sources and sinks and the profound impact of the signal decay rate on the network topology. We illustrate our results with calculations of degree and clustering for a meandering flow resembling a geophysical ocean jet. PMID:27128846
2012-06-19
Squares [13] for advection- diffusion with a reaction term, or the Unusual Stabilized Finite Element Method (USFEM) [14, 15] are a few examples. In...the underlying numerical scheme [21]. However, Godunov’s theorem [22] implies that the latter property may be violated in the prox - imity of...capturing finite element for- mulations for nonlinear convection-diffusion- reaction equations, Com- put. Methods Appl. Mech. and Engrg. 59 (1986) 307–325
NASA Astrophysics Data System (ADS)
Kemner, K. M.; Boyanov, M.; Flynn, T. M.; O'Loughlin, E. J.; Antonopoulos, D. A.; Kelly, S.; Skinner, K.; Mishra, B.; Brooks, S. C.; Watson, D. B.; Wu, W. M.
2015-12-01
FeIII- and SO42--reducing microorganisms and the mineral phases they produce have profound implications for many processes in aquatic and terrestrial systems. In addition, many of these microbially-catalysed geochemical transformations are highly dependent upon introduction of reactants via advective and diffusive hydrological transport. We have characterized microbial communities from a set of static microcosms to test the effect of ethanol diffusion and sulfate concentration on UVI-contaminated sediment. The spatial distribution, valence states, and speciation of both U and Fe were monitored in situ throughout the experiment by synchrotron x-ray absorption spectroscopy, in parallel with solution measurements of pH and the concentrations of sulfate, ethanol, and organic acids. After reaction initiation, a ~1-cm thick layer of sediment near the sediment-water (S-W) interface became visibly dark. Fe XANES spectra of the layer were consistent with the formation of FeS. Over the 4 year duration of the experiment, U LIII-edge XANES indicated reduction of U, first in the dark layer and then throughout the sediment. Next, the microcosms were disassembled and samples were taken from the overlying water and different sediment regions. We extracted DNA and characterized the microbial community by sequencing 16S rRNA gene amplicons with the Illumina MiSeq platform and found that the community evolved from its originally homogeneous composition, becoming significantly spatially heterogeneous. We have also developed an x-ray accessible column to probe elemental transformations as they occur along the flow path in a porous medium with the purpose of refining reactive transport models (RTMs) that describe coupled physical and biogeochemical processes in environmental systems. The elemental distribution dynamics and the RTMs of the redox driven processes within them will be presented.
Diffusive instabilities in hyperbolic reaction-diffusion equations.
Zemskov, Evgeny P; Horsthemke, Werner
2016-03-01
We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.
Trotter products and reaction-diffusion equations
NASA Astrophysics Data System (ADS)
Popescu, Emil
2010-01-01
In this paper, we study a class of generalized diffusion-reaction equations of the form , where A is a pseudodifferential operator which generates a Feller semigroup. Using the Trotter product formula we give a corresponding discrete time integro-difference equation for numerical solutions.
Shapiro, A.M.; Renken, R.A.; Harvey, R.W.; Zygnerski, M.R.; Metge, D.W.
2008-01-01
A tracer experiment, using a nonreactive tracer, was conducted as part of an investigation of the potential for chemical and pathogen migration to public supply wells that draw groundwater from the highly transmissive karst limestone of the Biscayne aquifer in southeastern Florida. The tracer was injected into the formation over approximately 1 h, and its recovery was monitored at a pumping well approximately 100 m from the injection well. The first detection of the tracer occurred after approximately 5 h, and the peak concentration occurred at about 8 h after the injection. The tracer was still detected in the production well more than 6 days after injection, and only 42% of the tracer mass was recovered. It is hypothesized that a combination of chemical diffusion and slow advection resulted in significant retention of the tracer in the formation, despite the high transmissivity of the karst limestone. The tail of the breakthrough curve exhibited a straight-line behavior with a slope of -2 on a log-log plot of concentration versus time. The -2 slope is hypothesized to be a function of slow advection, where the velocities of flow paths are hypothesized to range over several orders of magnitude. The flow paths having the slowest velocities result in a response similar to chemical diffusion. Chemical diffusion, due to chemical gradients, is still ongoing during the declining limb of the breakthrough curve, but this process is dwarfed by the magnitude of the mass flux by slow advection.
Sabelnikov, V A; Petrova, N N; Lipatnikov, A N
2016-10-01
Within the framework of the Maxwell-Cattaneo relaxation model extended to reaction-diffusion systems with nonlinear advection, travelling wave (TW) solutions are analytically investigated by studying a normalized reaction-telegraph equation in the case of the reaction and advection terms described by quadratic functions. The problem involves two governing parameters: (i) a ratio φ^{2} of the relaxation time in the Maxwell-Cattaneo model to the characteristic time scale of the reaction term, and (ii) the normalized magnitude N of the advection term. By linearizing the equation at the leading edge of the TW, (i) necessary conditions for the existence of TW solutions that are smooth in the entire interval of -∞<ζ<∞ are obtained, (ii) the smooth TW speed is shown to be less than the maximal speed φ^{-1} of the propagation of a substance, (iii) the lowest TW speed as a function of φ and N is determined. If the necessary condition of N>φ-φ^{-1} does not hold, e.g., if the magnitude N of the nonlinear advection is insufficiently high in the case of φ^{2}>1, then, the studied equation admits piecewise smooth TW solutions with sharp leading fronts that propagate at the maximal speed φ^{-1}, with the substance concentration or its spatial derivative jumping at the front. An increase in N can make the solution smooth in the entire spatial domain. Moreover, an explicit TW solution to the considered equation is found provided that N>φ. Subsequently, by invoking a principle of the maximal decay rate of TW solution at its leading edge, relevant TW solutions are selected in a domain of (φ,N) that admits the smooth TWs. Application of this principle to the studied problem yields transition from pulled (propagation speed is controlled by the TW leading edge) to pushed (propagation speed is controlled by the entire TW structure) TW solutions at N=N_{cr}=sqrt[1+φ^{2}], with the pulled (pushed) TW being relevant at smaller (larger) N. An increase in the normalized
Inertial-diffusive range for a passive scalar advected by a white-in-time velocity field
NASA Astrophysics Data System (ADS)
Frisch, U.; Wirth, A.
1996-09-01
It is shown analytically and by Monte Carlo simulations that a passive scalar with finite diffusivity, advected by a white-in-time velocity field with a power law spectrum propto k-1-ξ (0 < ξ < 2), has an inertial-diffusive range with a spectrum propto k-3-ξ. This is the analog of the Batchelor-Howells-Townsend (J. Fluid Mech., 5 (1959) 134) phenomenological derivation of the k-17/3 law for low-Schmidt-number passive-scalar dynamics in ordinary turbulence.
NASA Astrophysics Data System (ADS)
Park, A. J.; Chan, M. A.
2006-12-01
Abundant iron oxide concretions occurring in Navajo Sandstone of southern Utah and those discovered at Meridiani Planum, Mars share many common observable physical traits such as their spheriodal shapes, occurrence, and distribution patterns in sediments. Terrestrial concretions are products of interaction between oxygen-rich aquifer water and basin-derived reducing (iron-rich) water. Water-rock interaction simulations show that diffusion of oxygen and iron supplied by slow-moving water is a reasonable mechanism for producing observed concretion patterns. In short, southern Utah iron oxide concretions are results of Liesegang-type diffusive infiltration reactions in sediments. We propose that the formation of blueberry hematite concretions in Mars sediments followed a similar diagenetic mechanism where iron was derived from the alteration of volcanic substrate and oxygen was provided by the early Martian atmosphere. Although the terrestrial analog differs in the original host rock composition, both the terrestrial and Mars iron-oxide precipitation mechanisms utilize iron and oxygen interactions in sedimentary host rock with diffusive infiltration of solutes from two opposite sources. For the terrestrial model, slow advection of iron-rich water is an important factor that allowed pervasive and in places massive precipitation of iron-oxide concretions. In Mars, evaporative flux of water at the top of the sediment column may have produced a slow advective mass-transfer mechanism that provided a steady source and the right quantity of iron. The similarities of the terrestrial and Martian systems are demonstrated using a water-rock interaction simulator Sym.8, initially in one-dimensional systems. Boundary conditions such as oxygen content of water, partial pressure of oxygen, and supply rate of iron were varied. The results demonstrate the importance of slow advection of water and diffusive processes for producing diagenetic iron oxide concretions.
Transformed Fourier and Fick equations for the control of heat and mass diffusion
Guenneau, S.; Petiteau, D.; Zerrad, M.; Amra, C.; Puvirajesinghe, T.
2015-05-15
We review recent advances in the control of diffusion processes in thermodynamics and life sciences through geometric transforms in the Fourier and Fick equations, which govern heat and mass diffusion, respectively. We propose to further encompass transport properties in the transformed equations, whereby the temperature is governed by a three-dimensional, time-dependent, anisotropic heterogeneous convection-diffusion equation, which is a parabolic partial differential equation combining the diffusion equation and the advection equation. We perform two dimensional finite element computations for cloaks, concentrators and rotators of a complex shape in the transient regime. We precise that in contrast to invisibility cloaks for waves, the temperature (or mass concentration) inside a diffusion cloak crucially depends upon time, its distance from the source, and the diffusivity of the invisibility region. However, heat (or mass) diffusion outside cloaks, concentrators and rotators is unaffected by their presence, whatever their shape or position. Finally, we propose simplified designs of layered cylindrical and spherical diffusion cloaks that might foster experimental efforts in thermal and biochemical metamaterials.
Kurikami, Hiroshi; Malins, Alex; Takeishi, Minoru; Saito, Kimiaki; Iijima, Kazuki
2017-02-17
Radiocesium is an important environmental contaminant in fallout from nuclear reactor accidents and atomic weapons testing. A modified Diffusion-Sorption-Fixation (mDSF) model, based on the advection-dispersion equation, is proposed to describe the vertical migration of radiocesium in soils following fallout. The model introduces kinetics for the reversible binding of radiocesium. We test the model by comparing its results to depth profiles measured in Fukushima Prefecture, Japan, since 2011. The results from the mDSF model are a better fit to the measurement data (as quantified by R(2)) than results from a simple diffusion model and the original DSF model. The introduction of reversible sorption kinetics means that the exponential-shape depth distribution can be reproduced immediately following fallout. The initial relaxation mass depth of the distribution is determined by the diffusion length, which depends on the distribution coefficient, sorption rate and dispersion coefficient. The mDSF model captures the long tails of the radiocesium distribution at large depths, which are caused by different rates for kinetic sorption and desorption. The mDSF model indicates that depth distributions displaying a peak in activity below the surface are possible for soils with high organic matter content at the surface. The mDSF equations thus offers a physical basis for various types of radiocesium depth profiles observed in contaminated environments.
Kile, D.E.; Eberl, D.D.
2003-01-01
Crystal growth experiments were conducted using potassium alum and calcite crystals in aqueous solution under both non-stirred and stirred conditions to elucidate the mechanism for size-dependent (proportionate) and size-independent (constant) crystal growth. Growth by these two laws can be distinguished from each other because the relative size difference among crystals is maintained during proportionate growth, leading to a constant crystal size variance (??2) for a crystal size distribution (CSD) as the mean size increases. The absolute size difference among crystals is maintained during constant growth, resulting in a decrease in size variance. Results of these experiments show that for centimeter-sized alum crystals, proportionate growth occurs in stirred systems, whereas constant growth occurs in non-stirred systems. Accordingly, the mechanism for proportionate growth is hypothesized to be related to the supply of reactants to the crystal surface by advection, whereas constant growth is related to supply by diffusion. Paradoxically, micrometer-sized calcite crystals showed proportionate growth both in stirred and in non-stirred systems. Such growth presumably results from the effects of convection and Brownian motion, which promote an advective environment and hence proportionate growth for minute crystals in non-stirred systems, thereby indicating the importance of solution velocity relative to crystal size. Calcite crystals grown in gels, where fluid motion was minimized, showed evidence for constant, diffusion-controlled growth. Additional investigations of CSDs of naturally occurring crystals indicate that proportionate growth is by far the most common growth law, thereby suggesting that advection, rather than diffusion, is the dominant process for supplying reactants to crystal surfaces.
Wang, Chi-Jen
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Wang, Wei; Shu, Chi-Wang; Yee, H.C.; Sjögreen, Björn
2012-01-01
A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.
Fast permutation preconditioning for fractional diffusion equations.
Wang, Sheng-Feng; Huang, Ting-Zhu; Gu, Xian-Ming; Luo, Wei-Hua
2016-01-01
In this paper, an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from [Formula: see text] to [Formula: see text], where N is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.
Shifted Feedback Suppression of Turbulent Behavior in Advection-Diffusion Systems
Evain, C.; Szwaj, C.; Bielawski, S.; Couprie, M.-E.; Hosaka, M.; Mochihashi, A.; Katoh, M.
2009-04-03
In spatiotemporal systems with advection, suppression of noise-sustained structures involves questions that are outside of the framework of deterministic dynamical systems control (such as Ott-Grebogi-Yorke-type methods). Here we propose and test an alternate strategy where a nonlocal additive feedback is applied, with the objective to create a new deterministic solution that becomes robust to noise. As a remarkable fact - though the needed parameter perturbations required have essentially a finite size - they turn out to be extraordinarily small in principle: 10{sup -8} in the free-electron laser experiment presented here.
Fast parareal iterations for fractional diffusion equations
NASA Astrophysics Data System (ADS)
Wu, Shu-Lin; Zhou, Tao
2017-01-01
Numerical methods for fractional PDEs is a hot topic recently. This work is concerned with the parareal algorithm for system of ODEs u‧ (t) + Au (t) = f that arising from semi-discretizations of time-dependent fractional diffusion equations with nonsymmetric Riemann-Liouville fractional derivatives. The spatial semi-discretization of this kind of fractional derivatives often results in a coefficient matrix A with spectrum σ (A)
NASA Astrophysics Data System (ADS)
Wheaton, Daniel D.; Singha, Kamini
2010-09-01
Multiple types of physical heterogeneity have been suggested to explain anomalous solute transport behavior, yet determining exactly what controls transport at a given site is difficult from concentration histories alone. Differences in timing between co-located fluid and bulk apparent electrical conductivity data have previously been used to estimate solute mass transfer rates between mobile and less-mobile domains; here, we consider if this behavior can arise from other types of heterogeneity. Numerical models are used to investigate the electrical signatures associated with large-scale hydraulic conductivity heterogeneity and small-scale dual-domain mass transfer, and address issues regarding the scale of the geophysical measurement. We examine the transport behavior of solutes with and without dual-domain mass transfer, in: 1) a homogeneous medium, 2) a discretely fractured medium, and 3) a hydraulic conductivity field generated with sequential Gaussian simulation. We use the finite-element code COMSOL Multiphysics to construct two-dimensional cross-sectional models and solve the coupled flow, transport, and electrical conduction equations. Our results show that both large-scale heterogeneity and subscale heterogeneity described by dual-domain mass transfer produce a measurable hysteresis between fluid and bulk apparent electrical conductivity, indicating a lag between electrical conductivity changes in the mobile and less-mobile domains of an aquifer, or mass transfer processes, at some scale. The shape and magnitude of the observed hysteresis is controlled by the spatial distribution of hydraulic heterogeneity, mass transfer rate between domains, and the ratio of mobile to immobile porosity. Because the rate of mass transfer is related to the inverse square of a diffusion length scale, our results suggest that the shape of the hysteresis curve is indicative of the length scale over which mass transfer is occurring. We also demonstrate that the difference in
Belucz, Bernadett; Forgács-Dajka, Emese; Dikpati, Mausumi E-mail: dikpati@ucar.edu
2015-06-20
Babcock–Leighton type-solar dynamo models with single-celled meridional circulation are successful in reproducing many solar cycle features. Recent observations and theoretical models of meridional circulation do not indicate a single-celled flow pattern. We examine the role of complex multi-cellular circulation patterns in a Babcock–Leighton solar dynamo in advection- and diffusion-dominated regimes. We show from simulations that the presence of a weak, second, high-latitude reverse cell speeds up the cycle and slightly enhances the poleward branch in the butterfly diagram, whereas the presence of a second cell in depth reverses the tilt of the butterfly wing to an antisolar type. A butterfly diagram constructed from the middle of convection zone yields a solar-like pattern, but this may be difficult to realize in the Sun because of magnetic buoyancy effects. Each of the above cases behaves similarly in higher and lower magnetic diffusivity regimes. However, our dynamo with a meridional circulation containing four cells in latitude behaves distinctly differently in the two regimes, producing solar-like butterfly diagrams with fast cycles in the higher diffusivity regime, and complex branches in butterfly diagrams in the lower diffusivity regime. We also find that dynamo solutions for a four-celled pattern, two in radius and two in latitude, prefer to quickly relax to quadrupolar parity if the bottom flow speed is strong enough, of similar order of magnitude as the surface flow speed.
Tartakovsky, Daniel
2013-08-30
We developed new CDF and PDF methods for solving non-linear stochastic hyperbolic equations that does not rely on linearization approximations and allows for rigorous formulation of the boundary conditions.
Healy, R.W.; Russell, T.F.
1993-01-01
Test results demonstrate that the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) outperforms standard finite-difference methods for solute transport problems that are dominated by advection. FVELLAM systematically conserves mass globally with all types of boundary conditions. Integrated finite differences, instead of finite elements, are used to approximate the governing equation. This approach, in conjunction with a forward tracking scheme, greatly facilitates mass conservation. The mass storage integral is numerically evaluated at the current time level, and quadrature points are then tracked forward in time to the next level. Forward tracking permits straightforward treatment of inflow boundaries, thus avoiding the inherent problem in backtracking of characteristic lines intersecting inflow boundaries. FVELLAM extends previous results by obtaining mass conservation locally on Lagrangian space-time elements. -from Authors
On the supercritically diffusive magnetogeostrophic equations
NASA Astrophysics Data System (ADS)
Friedlander, Susan; Rusin, Walter; Vicol, Vlad
2012-11-01
We address the well-posedness theory for the magneto-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-Δ)γ with 0 < γ < 1, we discover that for γ > 1/2 the equations are locally well-posed, while for γ < 1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γ goes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of γ ∈ (0, 1). In fact, these solutions are global in time when γ ∈ [1/2, 1).
A semi-discrete finite element method for a class of time-fractional diffusion equations.
Sun, HongGuang; Chen, Wen; Sze, K Y
2013-05-13
As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection-diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.
NASA Astrophysics Data System (ADS)
Möller, Johannes; Narayanan, Theyencheri
In colloidal suspensions internal or external fields can induce directed motions of particles in addition to Brownian diffusion. Here, gradients in temperature or chemical potential, shear flow as well as gravity can act as an external field. Examples for internal motions can be found in synthetic self-propelling particles and microorganisms, generally coined as active matter. We present multi-speckle X-ray photon correlation spectroscopy measurements in the Ultra-Small-Angle scattering range which probes an expanded length scale comparable to DLS and optical microscopy. To demonstrate the advanced capabilities, we show measurements probing the motions within a settling suspension of sub-micron sized silica particles. A global fitting procedure has been applied to separate the diffusive and advective contributions to the particle dynamics. With this, macroscopic parameters such as the sedimentation velocity can be probed on a microscopic level in highly opaque and concentrated systems, which are in general difficult to access for optical investigations. This procedure may prove its value for investigating various kinds of non-equilibrium systems.
A dynamical equation for the distribution of a scalar advected by turbulence
NASA Astrophysics Data System (ADS)
Venaille, Antoine; Sommeria, Joel
2007-02-01
A phenomenological model for the dissipation of scalar fluctuations due to the straining by the fluid motion is proposed in this Brief Communication. An explicit equation is obtained for the time evolution of the probability distribution function of a coarse-grained scalar concentration. The model relies on a self-convolution process. We first present this model in the Batchelor regime and then extend empirically our result to the turbulent case. This approach is finally compared with other models.
2011-06-16
introduce this anti-diffusive WENO scheme for Eq . (11). 5 Let xi, i = 1, . . . , N be a uniform (for simplicity) mesh of the computational domain, with mesh...example is the model problem of [23]. Consider Eq . (4) with f(u) = u, the source term given by Eq . ( 5 ), and the initial condition: u(x, 0) = { 1, x ≤ 0.3...should be always zero. However, if µ in the source term Eq . ( 5 ) is very large, the numerical errors of u in the transition region can result in large
Nonlocal diffusion second order partial differential equations
NASA Astrophysics Data System (ADS)
Benedetti, I.; Loi, N. V.; Malaguti, L.; Taddei, V.
2017-02-01
The paper deals with a second order integro-partial differential equation in Rn with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.
Embry, Irucka; Roland, Victor; Agbaje, Oluropo; ...
2013-01-01
A new residence-time distribution (RTD) function has been developed and applied to quantitative dye studies as an alternative to the traditional advection-dispersion equation (AdDE). The new method is based on a jointly combined four-parameter gamma probability density function (PDF). The gamma residence-time distribution (RTD) function and its first and second moments are derived from the individual two-parameter gamma distributions of randomly distributed variables, tracer travel distance, and linear velocity, which are based on their relationship with time. The gamma RTD function was used on a steady-state, nonideal system modeled as a plug-flow reactor (PFR) in the laboratory to validate themore » effectiveness of the model. The normalized forms of the gamma RTD and the advection-dispersion equation RTD were compared with the normalized tracer RTD. The normalized gamma RTD had a lower mean-absolute deviation (MAD) (0.16) than the normalized form of the advection-dispersion equation (0.26) when compared to the normalized tracer RTD. The gamma RTD function is tied back to the actual physical site due to its randomly distributed variables. The results validate using the gamma RTD as a suitable alternative to the advection-dispersion equation for quantitative tracer studies of non-ideal flow systems.« less
Wang, Lei; Zhao, Cunlu; Wijnperlé, Daniel; Duits, Michel H G; Mugele, Frieder
2016-05-01
Establishing and maintaining concentration gradients that are stable in space and time is critical for applications that require screening the adsorption behavior of organic or inorganic species onto solid surfaces for wide ranges of fluid compositions. In this work, we present a design of a simple and compact microfluidic device based on steady-state diffusion of the analyte, between two control channels where liquid is pumped through. The device generates a near-linear distribution of concentrations. We demonstrate this via experiments with dye solutions and comparison to finite-element numerical simulations. In a subsequent step, the device is combined with total internal reflection ellipsometry to study the adsorption of (cat)ions on silica surfaces from CsCl solutions at variable pH. Such a combined setup permits a fast determination of an adsorption isotherm. The measured optical thickness is compared to calculations from a triple layer model for the ion distribution, where surface complexation reactions of the silica are taken into account. Our results show a clear enhancement of the ion adsorption with increasing pH, which can be well described with reasonable values for the equilibrium constants of the surface reactions.
Analysis of Coupled Reaction-Diffusion Equations for RNA Interactions.
Hohn, Maryann E; Li, Bo; Yang, Weihua
2015-05-01
We consider a system of coupled reaction-diffusion equations that models the interaction between multiple types of chemical species, particularly the interaction between one messenger RNA and different types of non-coding microRNAs in biological cells. We construct various modeling systems with different levels of complexity for the reaction, nonlinear diffusion, and coupled reaction and diffusion of the RNA interactions, respectively, with the most complex one being the full coupled reaction-diffusion equations. The simplest system consists of ordinary differential equations (ODE) modeling the chemical reaction. We present a derivation of this system using the chemical master equation and the mean-field approximation, and prove the existence, uniqueness, and linear stability of equilibrium solution of the ODE system. Next, we consider a single, nonlinear diffusion equation for one species that results from the slow diffusion of the others. Using variational techniques, we prove the existence and uniqueness of solution to a boundary-value problem of this nonlinear diffusion equation. Finally, we consider the full system of reaction-diffusion equations, both steady-state and time-dependent. We use the monotone method to construct iteratively upper and lower solutions and show that their respective limits are solutions to the reaction-diffusion system. For the time-dependent system of reaction-diffusion equations, we obtain the existence and uniqueness of global solutions. We also obtain some asymptotic properties of such solutions.
Analysis of Coupled Reaction-Diffusion Equations for RNA Interactions
Hohn, Maryann E.; Li, Bo; Yang, Weihua
2015-01-01
We consider a system of coupled reaction-diffusion equations that models the interaction between multiple types of chemical species, particularly the interaction between one messenger RNA and different types of non-coding microRNAs in biological cells. We construct various modeling systems with different levels of complexity for the reaction, nonlinear diffusion, and coupled reaction and diffusion of the RNA interactions, respectively, with the most complex one being the full coupled reaction-diffusion equations. The simplest system consists of ordinary differential equations (ODE) modeling the chemical reaction. We present a derivation of this system using the chemical master equation and the mean-field approximation, and prove the existence, uniqueness, and linear stability of equilibrium solution of the ODE system. Next, we consider a single, nonlinear diffusion equation for one species that results from the slow diffusion of the others. Using variational techniques, we prove the existence and uniqueness of solution to a boundary-value problem of this nonlinear diffusion equation. Finally, we consider the full system of reaction-diffusion equations, both steady-state and time-dependent. We use the monotone method to construct iteratively upper and lower solutions and show that their respective limits are solutions to the reaction-diffusion system. For the time-dependent system of reaction-diffusion equations, we obtain the existence and uniqueness of global solutions. We also obtain some asymptotic properties of such solutions. PMID:25601722
NASA Astrophysics Data System (ADS)
Jung, Na-Hyun; Han, Weon Shik; Han, Kyungdoe; Park, Eungyu
2015-05-01
Regional-scale advective, diffusive, and eruptive transport dynamics of CO2 and brine within a natural analogue in the northern Paradox Basin, Utah, were explored by integrating numerical simulations with soil CO2 flux measurements. Deeply sourced CO2 migrates through steeply dipping fault zones to the shallow aquifers predominantly as an aqueous phase. Dense CO2-rich brine mixes with regional groundwater, enhancing CO2 dissolution. Linear stability analysis reveals that CO2 could be dissolved completely within only ~500 years. Assigning lower permeability to the fault zones induces fault-parallel movement, feeds up-gradient aquifers with more CO2, and impedes down-gradient fluid flow, developing anticlinal CO2 traps at shallow depths (<300 m). The regional fault permeability that best reproduces field spatial CO2 flux variation is estimated 1 × 10-17 ≤ kh < 1 × 10-16 m2 and 5 × 10-16 ≤ kv < 1 × 10-15 m2. The anticlinal trap serves as an essential fluid source for eruption at Crystal Geyser. Geyser-like discharge sensitively responds to varying well permeability, radius, and CO2 recharge rate. The cyclic behavior of wellbore CO2 leakage decreases with time.
Knopman, Debra S.; Voss, Clifford I.
1987-01-01
The spatial and temporal variability of sensitivities has a significant impact on parameter estimation and sampling design for studies of solute transport in porous media. Physical insight into the behavior of sensitivities is offered through an analysis of analytically derived sensitivities for the one-dimensional form of the advection-dispersion equation. When parameters are estimated in regression models of one-dimensional transport, the spatial and temporal variability in sensitivities influences variance and covariance of parameter estimates. Several principles account for the observed influence of sensitivities on parameter uncertainty. (1) Information about a physical parameter may be most accurately gained at points in space and time. (2) As the distance of observation points from the upstream boundary increases, maximum sensitivity to velocity during passage of the solute front increases. (3) The frequency of sampling must be 'in phase' with the S shape of the dispersion sensitivity curve to yield the most information on dispersion. (4) The sensitivity to the dispersion coefficient is usually at least an order of magnitude less than the sensitivity to velocity. (5) The assumed probability distribution of random error in observations of solute concentration determines the form of the sensitivities. (6) If variance in random error in observations is large, trends in sensitivities of observation points may be obscured by noise. (7) Designs that minimize the variance of one parameter may not necessarily minimize the variance of other parameters.
Smoluchowski diffusion equation for active Brownian swimmers
NASA Astrophysics Data System (ADS)
Sevilla, Francisco J.; Sandoval, Mario
2015-05-01
We study the free diffusion in two dimensions of active Brownian swimmers subject to passive fluctuations on the translational motion and to active fluctuations on the rotational one. The Smoluchowski equation is derived from a Langevin-like model of active swimmers and analytically solved in the long-time regime for arbitrary values of the Péclet number; this allows us to analyze the out-of-equilibrium evolution of the positions distribution of active particles at all time regimes. Explicit expressions for the mean-square displacement and for the kurtosis of the probability distribution function are presented and the effects of persistence discussed. We show through Brownian dynamics simulations that our prescription for the mean-square displacement gives the exact time dependence at all times. The departure of the probability distribution from a Gaussian, measured by the kurtosis, is also analyzed both analytically and computationally. We find that for the inverse of Péclet numbers ≲0.1 , the distance from Gaussian increases as ˜t-2 at short times, while it diminishes as ˜t-1 in the asymptotic limit.
Ginzburg, Irina
2017-01-01
Impact of the unphysical tangential advective-diffusion constraint of the bounce-back (BB) reflection on the impermeable solid surface is examined for the first four moments of concentration. Despite the number of recent improvements for the Neumann condition in the lattice Boltzmann method-advection-diffusion equation, the BB rule remains the only known local mass-conserving no-flux condition suitable for staircase porous geometry. We examine the closure relation of the BB rule in straight channel and cylindrical capillary analytically, and show that it excites the Knudsen-type boundary layers in the nonequilibrium solution for full-weight equilibrium stencil. Although the d2Q5 and d3Q7 coordinate schemes are sufficient for the modeling of isotropic diffusion, the full-weight stencils are appealing for their advanced stability, isotropy, anisotropy and anti-numerical-diffusion ability. The boundary layers are not covered by the Chapman-Enskog expansion around the expected equilibrium, but they accommodate the Chapman-Enskog expansion in the bulk with the closure relation of the bounce-back rule. We show that the induced boundary layers introduce first-order errors in two primary transport properties, namely, mean velocity (first moment) and molecular diffusion coefficient (second moment). As a side effect, the Taylor-dispersion coefficient (second moment), skewness (third moment), and kurtosis (fourth moment) deviate from their physical values and predictions of the fourth-order Chapman-Enskog analysis, even though the kurtosis error in pure diffusion does not depend on grid resolution. In two- and three-dimensional grid-aligned channels and open-tubular conduits, the errors of velocity and diffusion are proportional to the diagonal weight values of the corresponding equilibrium terms. The d2Q5 and d3Q7 schemes do not suffer from this deficiency in grid-aligned geometries but they cannot avoid it if the boundaries are not parallel to the coordinate lines. In order
NASA Astrophysics Data System (ADS)
Ginzburg, Irina
2017-01-01
Impact of the unphysical tangential advective-diffusion constraint of the bounce-back (BB) reflection on the impermeable solid surface is examined for the first four moments of concentration. Despite the number of recent improvements for the Neumann condition in the lattice Boltzmann method-advection-diffusion equation, the BB rule remains the only known local mass-conserving no-flux condition suitable for staircase porous geometry. We examine the closure relation of the BB rule in straight channel and cylindrical capillary analytically, and show that it excites the Knudsen-type boundary layers in the nonequilibrium solution for full-weight equilibrium stencil. Although the d2Q5 and d3Q7 coordinate schemes are sufficient for the modeling of isotropic diffusion, the full-weight stencils are appealing for their advanced stability, isotropy, anisotropy and anti-numerical-diffusion ability. The boundary layers are not covered by the Chapman-Enskog expansion around the expected equilibrium, but they accommodate the Chapman-Enskog expansion in the bulk with the closure relation of the bounce-back rule. We show that the induced boundary layers introduce first-order errors in two primary transport properties, namely, mean velocity (first moment) and molecular diffusion coefficient (second moment). As a side effect, the Taylor-dispersion coefficient (second moment), skewness (third moment), and kurtosis (fourth moment) deviate from their physical values and predictions of the fourth-order Chapman-Enskog analysis, even though the kurtosis error in pure diffusion does not depend on grid resolution. In two- and three-dimensional grid-aligned channels and open-tubular conduits, the errors of velocity and diffusion are proportional to the diagonal weight values of the corresponding equilibrium terms. The d2Q5 and d3Q7 schemes do not suffer from this deficiency in grid-aligned geometries but they cannot avoid it if the boundaries are not parallel to the coordinate lines. In order
Gradient Driven Flow: Lattice Gas, Diffusion Equation and Measurement Scales
2001-01-01
03-200 1 Journal Article (refereed) 2001 4. TITLE AND SUBTITLE Sa. CONTRACT NUMBER Gradient Driven Flow : Lattice Gas, Diffusion Equation and...time regime, the collective motion exhibits an onset of oscillation. 15. SUBJECT TERMS Diffusion; Fick’s Law; Gradient Driven Flow ; Lattice Gas 16...Form 298 (Rev. 8-98) Prescribed by ANSI Std. Z39.18 20010907 062 Gradient driven flow : lattice gas, diffusion equation and measurement scales R.B
Ho, C.-L.; Lee, C.-C.
2016-01-15
We consider solvability of the generalized reaction–diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction–diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction–diffusion systems. Several representative examples of exactly solvable reaction–diffusion equations are presented.
NASA Astrophysics Data System (ADS)
Witherden, F. D.; Farrington, A. M.; Vincent, P. E.
2014-11-01
High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection-diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier-Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org). Catalogue identifier: AETY_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETY_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: New style BSD license No. of lines in
Symmetry classification of time-fractional diffusion equation
NASA Astrophysics Data System (ADS)
Naeem, I.; Khan, M. D.
2017-01-01
In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional diffusion equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional diffusion equation transforms to a partial differential equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional diffusion equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α. This study provides a new way of computing symmetries for a class of fractional differential equations.
NASA Astrophysics Data System (ADS)
Popova, E.; Srokosz, M.
2006-12-01
This paper examines the effect of vertical advection and vertical diffusion on the supply of nutrients to the euphotic zone. This is done using a high resolution coupled biological-physical model, that has previously been used to reproduce in situ and satellite observations of physical and biological variability at the Iceland Faeroes Front (IFF). Oligotrophic conditions are imposed in the model in order to examine the vertical flux of nutrients.The results show that, while instantaneous vertical advective fluxes of nutrients can be much larger than vertical diffusive ones, over a period of days the latter act consistently to supply nutrients to the euphotic zone. In contrast, the spatially and temporally varying nature of the vertical velocity field means that there is no consistent vertical advective flux of nutrients. This suggests that for real "messy" complex flows, such as the one modelled here, ageostrophic vertical velocities induced by eddies and frontal meanders may not play as important a role in supplying nutrient to the euphotic zone, and in enhancing biological production there, as has previously been thought.
Langevin equation approach to diffusion magnetic resonance imaging.
Cooke, Jennie M; Kalmykov, Yuri P; Coffey, William T; Kerskens, Christian M
2009-12-01
The normal phase diffusion problem in magnetic resonance imaging (MRI) is treated by means of the Langevin equation for the phase variable using only the properties of the characteristic function of Gaussian random variables. The calculation may be simply extended to anomalous diffusion using a fractional generalization of the Langevin equation proposed by Lutz [E. Lutz, Phys. Rev. E 64, 051106 (2001)] pertaining to the fractional Brownian motion of a free particle coupled to a fractal heat bath. The results compare favorably with diffusion-weighted experiments acquired in human neuronal tissue using a 3 T MRI scanner.
Stability of Stationary Solutions of the Multifrequency Radiation Diffusion Equations
Hald, O H; Shestakov, A I
2004-01-20
A nondimensional model of the multifrequency radiation diffusion equation is derived. A single material, ideal gas, equation of state is assumed. Opacities are proportional to the inverse of the cube of the frequency. Inclusion of stimulated emission implies a Wien spectrum for the radiation source function. It is shown that the solutions are uniformly bounded in time and that stationary solutions are stable. The spatially independent solutions are asymptotically stable, while the spatially dependent solutions of the linearized equations approach zero.
Diffusion MRI/NMR magnetization equations with relaxation times
NASA Astrophysics Data System (ADS)
de, Dilip; Daniel, Simon
2012-10-01
Bloch-Torrey diffusion magnetization equation ignores relaxation effects of magnetization. Relaxation times are important in any diffusion magnetization studies of perfusion in tissues(Brain and heart specially). Bloch-Torrey equation cannot therefore describe diffusion magnetization in a real-life situation where relaxation effects play a key role, characteristics of tissues under examination. This paper describes derivations of two equations for each of the y and z component diffusion NMR/MRI magnetization (separately) in a rotating frame of reference, where rf B1 field is applied along x direction and bias magnetic field(Bo) is along z direction. The two equations are expected to further advance the science & technology of Diffusion MRI(DMRI) and diffusion functional MRI(DFMRI). These two techniques are becoming increasingly important in the study and treatment of neurological disorders, especially for the management of patients with acute stroke. It is rapidly becoming a standard for white matter disorders, as diffusion tensor imaging (DTI) can reveal abnormalities in white matter fibre structure and provide models of brain connectivity.
Exact solutions for logistic reaction-diffusion equations in biology
NASA Astrophysics Data System (ADS)
Broadbridge, P.; Bradshaw-Hajek, B. H.
2016-08-01
Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.
Sharp, R.W. Jr.; Barton, R.T.
1981-01-21
A continuous rezoning procedure has been implemented in the computational cycle of a version of the HEMP two-dimensional, Lagrange, fluid dynamics code. The rezoning problem is divided into two steps. The first step requires the solving of ordinary Lagrange equations of motion; the second step consists of adding equipotential grid relaxation along with an advective remapping scheme.
A SIS reaction-diffusion-advection model in a low-risk and high-risk domain
NASA Astrophysics Data System (ADS)
Ge, Jing; Kim, Kwang Ik; Lin, Zhigui; Zhu, Huaiping
2015-11-01
A simplified SIS model is proposed and investigated to understand the impact of spatial heterogeneity of environment and advection on the persistence and eradication of an infectious disease. The free boundary is introduced to model the spreading front of the disease. The basic reproduction number associated with the diseases in the spatial setting is introduced. Sufficient conditions for the disease to be eradicated or to spread are given. Our result shows that if the spreading domain is high-risk at some time, the disease will continue to spread till the whole area is infected; while if the spreading domain is low-risk, the disease may be vanishing or keep spreading depending on the expanding capability and the initial number of the infective individuals. The spreading speeds are also given when spreading happens, numerical simulations are presented to illustrate the impacts of the advection and the expanding capability on the spreading fronts.
Santoro, P A; de Paula, J L; Lenzi, E K; Evangelista, L R
2011-09-21
The electrical response of an electrolytic cell in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order is analyzed. The boundary conditions at the electrodes limiting the sample are described by an integro-differential equation governing the kinetic at the interface. The analysis is carried out by supposing that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies the Poisson's equation. The results cover a rich variety of scenarios, including the ones connected to anomalous diffusion.
Image segmentation and edge enhancement with stabilized inverse diffusion equations.
Pollak, I; Willsky, A S; Krim, H
2000-01-01
We introduce a family of first-order multidimensional ordinary differential equations (ODEs) with discontinuous right-hand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" (SIDEs). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation. In an experiment, SIDE's are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is also demonstrated.
Multi-diffusive nonlinear Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Ribeiro, Mauricio S.; Casas, Gabriela A.; Nobre, Fernando D.
2017-02-01
Nonlinear Fokker-Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker-Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker-Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker-Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution.
Diffusion-equation method for crystallographic figure of merits.
Markvardsen, Anders J; David, William I F
2010-09-01
Global optimization methods play a significant role in crystallography, particularly in structure solution from powder diffraction data. This paper presents the mathematical foundations for a diffusion-equation-based optimization method. The diffusion equation is best known for describing how heat propagates in matter. However, it has also attracted considerable attention as the basis for global optimization of a multimodal function [Piela et al. (1989). J. Phys. Chem. 93, 3339-3346]. The method relies heavily on available analytical solutions for the diffusion equation. Here it is shown that such solutions can be obtained for two important crystallographic figure-of-merit (FOM) functions that fully account for space-group symmetry and allow the diffusion-equation solution to vary depending on whether atomic coordinates are fixed or not. The resulting expression is computationally efficient, taking the same order of floating-point operations to evaluate as the starting FOM function measured in terms of the number of atoms in the asymmetric unit. This opens the possibility of implementing diffusion-equation methods for crystallographic global optimization algorithms such as structure determination from powder diffraction data.
NASA Astrophysics Data System (ADS)
Cherniha, Roman; King, John R.; Kovalenko, Sergii
2016-07-01
Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona-Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.
NASA Astrophysics Data System (ADS)
Raghib, Michael; Levin, Simon; Kevrekidis, Ioannis
2010-05-01
2. The long-time behavior of the msd of the centroid walk scales linearly with time for naïve groups (diffusion), but shows a sharp transition to quadratic scaling (advection) for informed ones. These observations suggest that the mesoscopic variables of interest are the magnitude of the drift, the diffusion coefficient and the time-scales at which the anomalous and the asymptotic behavior respectively dominate transport, the latter being linked to the time scale at which the group reaches a decision. In order to estimate these summary statistics from the msd, we assumed that the configuration centroid follows an uncoupled Continuous Time Random Walk (CTRW) with smooth jump and waiting time pdf's. The mesoscopic transport equation for this type of random walk corresponds to an Advection-Diffusion Equation with Memory (ADEM). The introduction of the memory, and thus non-Markovian effects, is necessary in order to correctly account for the two time scales present. Although we were not able to calculate the memory directly from the individual-level rules, we show that it can estimated from a single, relatively short, simulation run using a Mittag-Leffler function as template. With this function it is possible to predict accurately the behavior of the msd, as well as the full pdf for the position of the centroid. The resulting ADEM is self-consistent in the sense that transport parameters estimated from the memory via a Kubo relationship coincide with those estimated from the moments of the jump size pdf of the associated CTRW for a large number of group sizes, proportions of informed individuals, and degrees of bias along the preferred direction. We also discuss the phase diagrams for the transport coefficients estimated from this method, where we notice velocity-precision trade-offs, where precision is a measure of the deviation of realized group orientations with respect to the informed direction. We also note that the time scale to collective decision is invariant
Moaty Sayed, A A; Hussein, M A; Becker, T
2010-04-01
Lattice Boltzmann models (LBM) are rapidly showing their ability to simulate a lot of fluid dynamics problems that previously required very complex approaches. This study presents a LBM for simulating diffusion-advection transport of substrate in a 2-D laminar flow. The model considers the substrate influx into a set of active cells placed inside the flow field. A new innovative method was used to simulate the cells activity using the LBM by means of Michaelis-Menten kinetics. The model is validated with some numerical benchmark problems and proved highly accurate results. After validation the model was used to simulate the transport of oxygen substrates that diffuse in water to feed a set of active cartilage cells inside a new designed bioreactor.
Leapfrog/Finite Element Method for Fractional Diffusion Equation
Zhao, Zhengang; Zheng, Yunying
2014-01-01
We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove an L 2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis. PMID:24955431
The Continuous Coagulation-FragmentationEquations with Diffusion
NASA Astrophysics Data System (ADS)
Laurençot, Philippe; Mischler, Stéphane
Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L1 in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.
Advection around ventilated U-shaped burrows: A model study
NASA Astrophysics Data System (ADS)
Brand, Andreas; Lewandowski, JöRg; Hamann, Enrico; Nützmann, Gunnar
2013-05-01
Advective transport in the porous matrix of sediments surrounding burrows formed by fauna such as Chironomus plumosus has been generally neglected. A positron emission tomography study recently revealed that the pumping activity of the midge larvae can indeed induce fluid flow in the sediment. We present a numerical model study which explores the conditions at which advective transport in the sediment becomes relevant. A 0.15 m deep U-shaped burrow with a diameter of 0.002 m within the sediment was represented in a 3-D domain. Fluid flow in the burrow was calculated using the Navier-Stokes equation for incompressible laminar flow in the burrow, and flow in the sediment was described by Darcy's law. Nonreactive and reactive transport scenarios were simulated considering diffusion and advection. The pumping activity of the model larva results in considerable advective flow in the sediment at reasonable high permeabilities with flow velocities of up to 7.0 × 10-6 m s-1 close to the larva for a permeability of 3 × 10-12 m2. At permeabilities below 7 × 10-13 m2 advection is negligible compared to diffusion. Reactive transport simulations using first-order kinetics for oxygen revealed that advective flux into the sediment downstream of the pumping larva enhances sedimentary uptake, while the advective flux into the burrow upstream of the larvae inhibits diffusive sedimentary uptake. Despite the fact that both effects cancel each other with respect to total solute uptake, the advection-induced asymmetry in concentration distribution can lead to a heterogeneous solute and redox distribution in the sediment relevant to complex reaction networks.
A numerical solution for the diffusion equation in hydrogeologic systems
Ishii, A.L.; Healy, R.W.; Striegl, R.G.
1989-01-01
The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)
Pullback attractors for nonclassical diffusion equations with delays
NASA Astrophysics Data System (ADS)
Zhu, Kaixuan; Sun, Chunyou
2015-09-01
In this paper, we prove the existence of pullback attractors in C H0 1 ( Ω ) for a nonclassical diffusion equation with delay term g(t, ut) which contains some hereditary characteristics. We consider two types of nonlinearity f: one is the case of critical growth and the other one is the polynomial growth of arbitrary order p - 1(p ≥ 2).
FDM study of ion exchange diffusion equation in glass
NASA Astrophysics Data System (ADS)
Zhou, Zigang; Yang, Yongjia; Wang, Qiang; Sun, Guangchun
2009-05-01
Ion-exchange technique in glass was developed to fabricate gradient refractive index optical devices. In this paper, the Finite Difference Method(FDM), which is used for the solution of ion-diffusion equation, is reported. This method transforms continual diffusion equation to separate difference equation. It unitizes the matrix of MATLAB program to solve the iteration process. The collation results under square boundary condition show that it gets a more accurate numerical solution. Compared to experiment data, the relative error is less than 0.2%. Furthermore, it has simply operation and kinds of output solutions. This method can provide better results for border-proliferation of the hexagonal and the channel devices too.
Fisher equation for anisotropic diffusion: simulating South American human dispersals.
Martino, Luis A; Osella, Ana; Dorso, Claudio; Lanata, José L
2007-09-01
The Fisher equation is commonly used to model population dynamics. This equation allows describing reaction-diffusion processes, considering both population growth and diffusion mechanism. Some results have been reported about modeling human dispersion, always assuming isotropic diffusion. Nevertheless, it is well-known that dispersion depends not only on the characteristics of the habitats where individuals are but also on the properties of the places where they intend to move, then isotropic approaches cannot adequately reproduce the evolution of the wave of advance of populations. Solutions to a Fisher equation are difficult to obtain for complex geometries, moreover, when anisotropy has to be considered and so few studies have been conducted in this direction. With this scope in mind, we present in this paper a solution for a Fisher equation, introducing anisotropy. We apply a finite difference method using the Crank-Nicholson approximation and analyze the results as a function of the characteristic parameters. Finally, this methodology is applied to model South American human dispersal.
On the entropy conditions for some flux limited diffusion equations
NASA Astrophysics Data System (ADS)
Caselles, V.
2011-04-01
In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.
Diffusive and dynamical radiating stars with realistic equations of state
NASA Astrophysics Data System (ADS)
Brassel, Byron P.; Maharaj, Sunil D.; Goswami, Rituparno
2017-03-01
We model the dynamics of a spherically symmetric radiating dynamical star with three spacetime regions. The local internal atmosphere is a two-component system consisting of standard pressure-free, null radiation and an additional string fluid with energy density and nonzero pressure obeying all physically realistic energy conditions. The middle region is purely radiative which matches to a third region which is the Schwarzschild exterior. A large family of solutions to the field equations are presented for various realistic equations of state. We demonstrate that it is possible to obtain solutions via a direct integration of the second order equations resulting from the assumption of an equation of state. A comparison of our solutions with earlier well known results is undertaken and we show that all these solutions, including those of Husain, are contained in our family. We then generalise our class of solutions to higher dimensions. Finally we consider the effects of diffusive transport and transparently derive the specific equations of state for which this diffusive behaviour is possible.
Simple jumping process with memory: Transport equation and diffusion
NASA Astrophysics Data System (ADS)
Kamińska, A.; Srokowski, T.
2004-06-01
We present a stochastic jumping process, defined in terms of jump-size probability density and jumping rate, which is a generalization of the well-known kangaroo process. The definition takes into account two process values: after and before the jump. Therefore, the process is able to preserve memory about its previous values. It possesses a simple stationary limit. Its master equation is interpreted as the kinetic equation with variable collision rate. The process can be easily applied to model systems which relax to distributions other than Maxwellian. The case of a constant jumping rate corresponds to the diffusion process, either normal or ballistic.
Support Operators Method for the Diffusion Equation in Multiple Materials
Winters, Andrew R.; Shashkov, Mikhail J.
2012-08-14
A second-order finite difference scheme for the solution of the diffusion equation on non-uniform meshes is implemented. The method allows the heat conductivity to be discontinuous. The algorithm is formulated on a one dimensional mesh and is derived using the support operators method. A key component of the derivation is that the discrete analog of the flux operator is constructed to be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resultant discrete operators in the fully discretized diffusion equation are symmetric and positive definite. The algorithm is generalized to operate on meshes with cells which have mixed material properties. A mechanism to recover intermediate temperature values in mixed cells using a limited linear reconstruction is introduced. The implementation of the algorithm is verified and the linear reconstruction mechanism is compared to previous results for obtaining new material temperatures.
Algorithm Refinement for Stochastic Partial Differential Equations. I. Linear Diffusion
NASA Astrophysics Data System (ADS)
Alexander, Francis J.; Garcia, Alejandro L.; Tartakovsky, Daniel M.
2002-10-01
A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. Results from a variety of numerical experiments are presented for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a nonstochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except in particle regions away from the interface. Extensions of the methodology to fluid mechanics applications are discussed.
The paper presents an analysis of steady-state diffusion in the soil for two different conditions of moisture. The model accounts for multiphase emanation and transport. When the position dependence of the moisture profile is taken into account, the model and measurements agree w...
Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity
NASA Astrophysics Data System (ADS)
Asvestas, M.; Sifalakis, A. G.; Papadopoulou, E. P.; Saridakis, Y. G.
2014-03-01
Motivated by proliferation-diffusion mathematical models for studying highly diffusive brain tumors, that also take into account the heterogeneity of the brain tissue, in the present work we consider a multi-domain linear reaction-diffusion equation with a discontinuous diffusion coefficient. For the solution of the problem at hand we implement Fokas transform method by directly following, and extending in this way, our recent work for a white-gray-white matter brain model pertaining to high grade gliomas. Fokas's novel approach for the solution of linear PDE problems, yields novel integral representations of the solution in the complex plane that, for appropriately chosen integration contours, decay exponentially fast and converge uniformly at the boundaries. Combining these method-inherent advantages with simple numerical quadrature rules, we produce an efficient method, with fast decaying error properties, for the solution of the discontinuous reaction-diffusion problem.
Reaction rates for a generalized reaction-diffusion master equation
Hellander, Stefan; Petzold, Linda
2016-01-19
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.
Reaction rates for a generalized reaction-diffusion master equation.
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
Reaction rates for a generalized reaction-diffusion master equation
Hellander, Stefan; Petzold, Linda
2016-01-19
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show inmore » two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules.« less
Time-fractional wave-diffusion equation in an inhomogeneous half-space
NASA Astrophysics Data System (ADS)
Liemert, André; Kienle, Alwin
2015-06-01
We consider the fundamental solution of the time-fractional wave-diffusion equation in a three-dimensional half-space medium which contains an inhomogeneity in form of a plane parallel layer. The corresponding Green’s function which is derived by means of the Fourier and Laplace transforms can be accurately and efficiently evaluated without recourse to the Mittag-Leffler or the Fox H-function. Moreover, it is shown that in the one-dimensional case the fundamental solution in an inhomogeneous half-space is no longer a probability density function. In addition, we consider the advection equation for the fractional Laplacian {{(-Δ )}\\frac{1{2}}} and the Caputo time-fractional derivative of orders 0\\lt β ≤slant 1 on a bounded domain. Simple algorithms for accurate evaluation of the M-Wright function {{M}β }(x) and the Mittag-Leffler function {{E}β }(-x) are enclosed at the end of this article.
On three explicit difference schemes for fractional diffusion and diffusion-wave equations
NASA Astrophysics Data System (ADS)
Quintana Murillo, Joaquín; Bravo Yuste, Santos
2009-10-01
Three explicit difference schemes for solving fractional diffusion and fractional diffusion-wave equations are studied. We consider these equations in both the Riemann-Liouville and the Caputo forms. We find that the Gorenflo et al (2000 J. Comput. Appl. Math. 118 175) and the Yuste-Acedo (2005 SIAM J. Numer. Anal. 42 1862) methods when applied to fractional diffusion equations are equivalent when BDF1 coefficients are used to discretize the fractional derivative operators, but that this is not the case for fractional diffusion-wave equations. The accuracy and stability of the three methods are studied. Surprisingly, the third method, that of Ciesielski-Leszczynski (2003 Proc. 15th Conf. on Computer Methods in Mechanics), although closely related to the Gorenflo et al method, is the least accurate, especially for short times. The stability analysis is carried out by means of a procedure close to the standard von Neumann method. We find that the stability bounds of the three methods are the same.
Lin, Guoxing
2015-10-01
Pulsed field gradient (PFG) diffusion measurement has a lot of applications in NMR and MRI. Its analysis relies on the ability to obtain the signal attenuation expressions, which can be obtained by averaging over the accumulating phase shift distribution (APSD). However, current theoretical models are not robust or require approximations to get the APSD. Here, a new formalism, an effective phase shift diffusion (EPSD) equation method is presented to calculate the APSD directly. This is based on the idea that the gradient pulse effect on the change of the APSD can be viewed as a diffusion process in the virtual phase space (VPS). The EPSD has a diffusion coefficient, K(β)(t)D rad(β)/s(α), where α is time derivative order and β is a space derivative order, respectively. The EPSD equations of VPS are built based on the diffusion equations of real space by replacing the diffusion coefficients and the coordinate system (from real space coordinate to virtual phase coordinate). Two different models, the fractal derivative model and the fractional derivative model from the literature were used to build the EPSD fractional diffusion equations. The APSD obtained from solving these EPSD equations were used to calculate the PFG signal attenuation. From the fractal derivative model the attenuation is exp(-γ(β)g(β)δ(β)Df1t(α)), a stretched exponential function (SEF) attenuation, while from the fractional derivative model the attenuation is Eα,1(-γ(β)g(β)δ(β)Df2t(α)), a Mittag-Leffler function (MLF) attenuation. The MLF attenuation can be reduced to SEF attenuation when α=1, and can be approximated as a SEF attenuation when the attenuation is small. Additionally, the effect of finite gradient pulse widths (FGPW) is calculated. From the fractal derivative model, the signal attenuation including FGPW effect is exp[ -Df1∫0(τ) K(β)(t)dt(α)]. The results obtained in this study are in good agreement with the results in literature. Several expressions that
Spectral analysis and structure preserving preconditioners for fractional diffusion equations
NASA Astrophysics Data System (ADS)
Donatelli, Marco; Mazza, Mariarosa; Serra-Capizzano, Stefano
2016-02-01
Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grünwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.
NASA Astrophysics Data System (ADS)
Ohmori, Shousuke; Yamazaki, Yoshihiro
2016-01-01
Ultradiscrete equations are derived from a set of reaction-diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.
NASA Technical Reports Server (NTRS)
Kooi, Henk; Beaumont, Christopher
1994-01-01
Experiments with a surface processes model of large-scale (1-1000 km) long-term (1-100 m.y.) erosional denudation are used to establish the controls on the evolution of a model escarpment that is related to the rifting of a continent. The mdoel describes changes in topographic form as a result of sumultaneous short- and long-range mass transport representing hillslope (diffusive) processes and fluvial transport (advection), repsectively. Fluvial entrainment is modeled as a first-order kinetic reaction which reflects the erodibility of the substrate, and therefore the fluvial system is not necessarily carrying at capacity. One dimensional and planform models demonstrate that the principal controls on the evolution of an initially steep model escarpment are (1) antecedent topography/drainage; (2) the timesale (or equivalently a length scale) in the fluvial entrainment reaction; (3) the flexural response of the lithosphere to denudation; and (4) the relative efficiencies of the short- and long-range transport processes. When rainfall and substrate lithology are uniform, a significant amount of discharge draining over the escarpment top causes it to degrade. Only when the top of the model escarpment coincides with a drainage divide can escarpment retreat occur for these conditions. An additional requirement for retreat of a model escarpment without decline is a long reaction time scale for fluvial entrainment. This corresponds to a substrate that is hard to detach by flucial erosion, and therefore to fluvial erosion that is not transport limited. Coninuous backtilting of an escarpment due ot flexural isostatic uplift in response to denudational unloading helps maintain the scarp top as a divide. It is essntial if the escarpment gradient is to be preserved during retreat in a uniform lithology. Low flexural rigidieties propote steep and slowly retreating escarpments. For given rainfall and substrate conditions, the morphology of a retraeating model escarpment is
Dynamic hysteresis modeling including skin effect using diffusion equation model
NASA Astrophysics Data System (ADS)
Hamada, Souad; Louai, Fatima Zohra; Nait-Said, Nasreddine; Benabou, Abdelkader
2016-07-01
An improved dynamic hysteresis model is proposed for the prediction of hysteresis loop of electrical steel up to mean frequencies, taking into account the skin effect. In previous works, the analytical solution of the diffusion equation for low frequency (DELF) was coupled with the inverse static Jiles-Atherton (JA) model in order to represent the hysteresis behavior for a lamination. In the present paper, this approach is improved to ensure the reproducibility of measured hysteresis loops at mean frequency. The results of simulation are compared with the experimental ones. The selected results for frequencies 50 Hz, 100 Hz, 200 Hz and 400 Hz are presented and discussed.
The Influence of Fractional Diffusion in Fisher-KPP Equations
NASA Astrophysics Data System (ADS)
Cabré, Xavier; Roquejoffre, Jean-Michel
2013-06-01
We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.
Limit Cycle Solutions of Reaction-Diffusion Equations.
1980-06-01
chernotactic bacteria: a theoretical analysis. .J. Theor. Biol., 30(1971), pp. 235-24 8. A. Kolm.ogorov, 1. Petrovsky, and N . Piskunov . Etude de 1’, quation de...equations of the form u + F(u)+K V (1.1) NN where u is an N -dimensional vector, K is a nonnegative-definite diffusion matrix, and F(u) is a vector...k6 c, the rate constants k1, ..., k6 are empirical but various relations exist between them (for instance, at equilibrium, amb n /c k f k2/k - k4 /k3
Chaotic dynamics and diffusion in a piecewise linear equation
NASA Astrophysics Data System (ADS)
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-01
Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
Chaotic dynamics and diffusion in a piecewise linear equation
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-15
Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.
NASA Astrophysics Data System (ADS)
Baechler, S.; Croisé, J.; Altmann, S.
2012-12-01
Chemico-osmosis is a recognized phenomenon taking place in clay mineral-rich sedimentary formations and a number of questions have been raised concerning its potential effects on pressure fields in and around underground radioactive waste repositories installed in such formations. Certain radioactive waste packages contain large quantities of nitrate salts whose release might result in the presence of highly concentrated salt solutions in the disposal cells, during their resaturation after closure of the facility. This would lead to large solute concentration gradients within the formation's porewater which could then potentially induce significant chemico-osmotic fluxes. In this paper, we assess the impact of chemico-osmotic fluxes on the water pressure during the post-closure period of a typical disposal cell for intermediate-level, long-lived bituminised radioactive waste in the Callovo-Oxfordian Clay formation. A numerical model of chemico-osmotic water flow and solute transport has been developed based on the work of Bader and Kooi (2005) [5], and including Bresler's dependence of osmotic efficiency on concentration and compaction state [9]. Model validity has been extended to highly concentrated solutions by incorporating a concentration-dependent activity coefficient, based on the Pitzer's equations. Results show that due to the strong dependence of the osmotic coefficient on concentration, the impact of chemico-osmosis on water flow and on the pressure field around the disposal cell is relatively low. A maximum overpressure of the order of 1 MPa was obtained. No difference in the simulation results were noticed for disposal cell solutions having concentrations higher than 1 M NaNO3. Differences between simulations were found to be almost entirely due to Bresler's relationship i.e., the model of the dependence between osmotic efficiency and concentration, and only slightly on the activity coefficient correction. Questions remain regarding the appropriate
NASA Astrophysics Data System (ADS)
He, Ying; Puckett, Elbridge Gerry; Billen, Magali I.
2017-02-01
Mineral composition has a strong effect on the properties of rocks and is an essentially non-diffusive property in the context of large-scale mantle convection. Due to the non-diffusive nature and the origin of compositionally distinct regions in the Earth the boundaries between distinct regions can be nearly discontinuous. While there are different methods for tracking rock composition in numerical simulations of mantle convection, one must consider trade-offs between computational cost, accuracy or ease of implementation when choosing an appropriate method. Existing methods can be computationally expensive, cause over-/undershoots, smear sharp boundaries, or are not easily adapted to tracking multiple compositional fields. Here we present a Discontinuous Galerkin method with a bound preserving limiter (abbreviated as DG-BP) using a second order Runge-Kutta, strong stability-preserving time discretization method for the advection of non-diffusive fields. First, we show that the method is bound-preserving for a point-wise divergence free flow (e.g., a prescribed circular flow in a box). However, using standard adaptive mesh refinement (AMR) there is an over-shoot error (2%) because the cell average is not preserved during mesh coarsening. The effectiveness of the algorithm for convection-dominated flows is demonstrated using the falling box problem. We find that the DG-BP method maintains sharper compositional boundaries (3-5 elements) as compared to an artificial entropy-viscosity method (6-15 elements), although the over-/undershoot errors are similar. When used with AMR the DG-BP method results in fewer degrees of freedom due to smaller regions of mesh refinement in the neighborhood of the discontinuity. However, using Taylor-Hood elements and a uniform mesh there is an over-/undershoot error on the order of 0.0001%, but this error increases to 0.01-0.10% when using AMR. Therefore, for research problems in which a continuous field method is desired the DG
Nonlinear dirac and diffusion equations in 1+1 dimensions from stochastic considerations
Maharana
2000-08-01
We generalize the method of obtaining fundamental linear partial differential equations such as the diffusion and Schrodinger equation, the Dirac, and the telegrapher's equation from a simple stochastic consideration to arrive at a certain nonlinear form of these equations. A group classification through a one-parameter group of transformations for two of these equations is also carried out.
Guiding brine shrimp through mazes by solving reaction diffusion equations
NASA Astrophysics Data System (ADS)
Singal, Krishma; Fenton, Flavio
Excitable systems driven by reaction diffusion equations have been shown to not only find solutions to mazes but to also to find the shortest path between the beginning and the end of the maze. In this talk we describe how we can use the Fitzhugh-Nagumo model, a generic model for excitable media, to solve a maze by varying the basin of attraction of its two fixed points. We demonstrate how two dimensional mazes are solved numerically using a Java Applet and then accelerated to run in real time by using graphic processors (GPUs). An application of this work is shown by guiding phototactic brine shrimp through a maze solved by the algorithm. Once the path is obtained, an Arduino directs the shrimp through the maze using lights from LEDs placed at the floor of the Maze. This method running in real time could be eventually used for guiding robots and cars through traffic.
Local multiplicative Schwarz algorithms for convection-diffusion equations
NASA Technical Reports Server (NTRS)
Cai, Xiao-Chuan; Sarkis, Marcus
1995-01-01
We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diffusion equations discretized by finite element or finite difference methods. The preconditioners consist of two components, namely, the usual two-level additive Schwarz preconditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the flow. We prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. We show by numerical examples that the new algorithms are less sensitive to the direction of the flow than either the classical multiplicative Schwarz algorithms, and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for fluid flow applications than the classical additive or multiplicative Schwarz algorithms.
A fast solver for systems of reaction-diffusion equations.
Garbey, M.; Kaper, H. G.; Romanyukha, N.
2001-04-20
In this paper we present a fast algorithm for the numerical solution of systems of reaction-diffusion equations, {partial_derivative}{sub t} u + a {center_dot} {del}u = {Delta}u + f(x,t,u), and x element of {Omega} contained in R{sup 3}, t > 0. Here, u is a vector-valued function, u triple bond u(x,t) element of R{sup m} is large, and the corresponding system of ODEs, {partial_derivative}{sub t}u = F(x,t,u), is stiff. Typical examples arise in air pollution studies, where a is the given wind field and the nonlinear function F models the atmospheric chemistry. The time integration of Eq. (1) is best handled by the method of characteristics. The problem is thus reduced to designing for the reaction-diffusion part a fast solver that has good stability properties for the given time step and does not require the computation of the full Jacobi matrix. An operator-splitting technique, even a high-order one, combining a fast nonlinear ODE solver with an efficient solver for the diffusion operator is less effective when the reaction term is stiff. In fact, the classical Strang splitting method may underperform a first-order source splitting method. The algorithm we propose in this paper uses an a posteriori filtering technique to stabilize the computation of the diffusion term. The algorithm parallelizes well, because the solution of the large system of ODEs is done pointwise; however, the integration of the chemistry may lead to load-balancing problems. The Tchebycheff acceleration technique proposed in offers an alternative that complements the approach presented here. To facilitate the presentation, we limit the discussion to domains {Omega} that either admit a regular discretization grid or decompose into subdomains that admit regular discretization grids. We describe the algorithm for one-dimensional domains in Section 2 and for multidimensional domains in Section 3. Section 4 briefly outlines future work.
Hard ellipses: Equation of state, structure, and self-diffusion.
Xu, Wen-Sheng; Li, Yan-Wei; Sun, Zhao-Yan; An, Li-Jia
2013-07-14
Despite their fundamental and practical interest, the physical properties of hard ellipses remain largely unknown. In this paper, we present an event-driven molecular dynamics study for hard ellipses and assess the effects of aspect ratio and area fraction on their physical properties. For state points in the plane of aspect ratio (1 ≤ k ≤ 9) and area fraction (0.01 ≤ φ ≤ 0.8), we identify three different phases, including isotropic, plastic, and nematic states. We analyze in detail the thermodynamic, structural, and self-diffusive properties in the formed various phases of hard ellipses. The equation of state (EOS) is shown for a wide range of aspect ratios and is compared with the scaled particle theory (SPT) for the isotropic states. We find that SPT provides a good description of the EOS for the isotropic phase of hard ellipses. At large fixed φ, the reduced pressure p increases with k in both the isotropic and the plastic phases and, interestingly, its dependence on k is rather weak in the nematic phase. We rationalize the thermodynamics of hard ellipses in terms of particle motions. The static structures of hard ellipses are then investigated both positionally and orientationally in the different phases. The plastic crystal is shown to form for aspect ratios up to k = 1.4, while appearance of the stable nematic phase starts approximately at k = 3. We quantitatively determine the locations of the isotropic-plastic (I-P) transition and the isotropic-nematic (I-N) transition by analyzing the bond-orientation correlations and the angular correlations, respectively. As expected, the I-P transition point is found to increase with k, while a larger k leads to a smaller area fraction where the I-N transition takes place. Moreover, our simulations strongly support that the two-dimensional nematic phase in hard ellipses has only quasi-long-range orientational order. The self-diffusion of hard ellipses is further explored and connections are revealed between
Background-Error Correlation Model Based on the Implicit Solution of a Diffusion Equation
2010-01-01
1 Background- Error Correlation Model Based on the Implicit Solution of a Diffusion Equation Matthew J. Carrier* and Hans Ngodock...4. TITLE AND SUBTITLE Background- Error Correlation Model Based on the Implicit Solution of a Diffusion Equation 5a. CONTRACT NUMBER 5b. GRANT...2001), which sought to model error correlations based on the explicit solution of a generalized diffusion equation. The implicit solution is
Hybrid diffusion-P3 equation in N-layered turbid media: steady-state domain.
Shi, Zhenzhi; Zhao, Huijuan; Xu, Kexin
2011-10-01
This paper discusses light propagation in N-layered turbid media. The hybrid diffusion-P3 equation is solved for an N-layered finite or infinite turbid medium in the steady-state domain for one point source using the extrapolated boundary condition. The Fourier transform formalism is applied to derive the analytical solutions of the fluence rate in Fourier space. Two inverse Fourier transform methods are developed to calculate the fluence rate in real space. In addition, the solutions of the hybrid diffusion-P3 equation are compared to the solutions of the diffusion equation and the Monte Carlo simulation. For the case of small absorption coefficients, the solutions of the N-layered diffusion equation and hybrid diffusion-P3 equation are almost equivalent and are in agreement with the Monte Carlo simulation. For the case of large absorption coefficients, the model of the hybrid diffusion-P3 equation is more precise than that of the diffusion equation. In conclusion, the model of the hybrid diffusion-P3 equation can replace the diffusion equation for modeling light propagation in the N-layered turbid media for a wide range of absorption coefficients.
NASA Astrophysics Data System (ADS)
Jung, Na-Hyun
This study investigated a natural analogue for CO2 leakage near Green River, Utah, aiming to understand the influence of various factors on CO2 leakage and to reliably predict underground CO2 behavior after injection for geologic CO2 sequestration. Advective, diffusive, and eruptive characteristics of CO2 leakage were assessed via a soil CO2 flux survey and numerical modeling. The field results show anomalous CO2 fluxes (> 10 g m-2 d-1 ) along the faults, particularly adjacent to CO2-driven cold springs and geysers (e.g., 36,259 g m-2 d-1 at Crystal Geyser), ancient travertines (e.g., 5,917 g m-2 d-1), joint zones in sandstone (e.g., 120 g m-2 d-1), and brine discharge zones (e.g., 5,515 g m-2 d-1). Combined with similar isotopic ratios of gas and progressive evolution of brine chemistry at springs and geysers, a gradual decrease of soil CO2 flux from the Little Grand Wash (LGW; ~36,259 g m -2 d-1) to Salt Wash (SW; ~1,428 g m-2 d-1) fault zones reveals the same CO2 origin and potential southward transport of CO2 over 10-20 km. The numerical simulations exhibit lateral transport of free CO2 and CO2-rich brine from the LGW to SW fault zones through the regional aquifers (e.g., Entrada, Navajo, Kayenta, Wingate, White Rim). CO2 travels predominantly as an aqueous phase (XCO2=~0.045) as previously suggested, giving rise to the convective instability that further accelerates CO2 dissolution. While the buoyant free CO2 always tends to ascend, a fraction of dense CO2-rich brine flows laterally into the aquifer and mixes with the formation fluids during upward migration along the fault. The fault always enhances advective CO2 transport regardless of its permeability (k). However, only low-k fault prevents unconditional upright migration of CO2 and induces fault-parallel movement, feeding the northern aquifers with more CO2. Low-k fault also impedes lateral southward fluid flow from the northern aquifers, developing anticlinal CO2 traps at shallow depths (<300 m). The
Accurate spectral numerical schemes for kinetic equations with energy diffusion
NASA Astrophysics Data System (ADS)
Wilkening, Jon; Cerfon, Antoine J.; Landreman, Matt
2015-08-01
We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker-Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.
Diffusion phenomenon for linear dissipative wave equations in an exterior domain
NASA Astrophysics Data System (ADS)
Ikehata, Ryo
Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.
Generalized space-time fractional diffusion equation with composite fractional time derivative
NASA Astrophysics Data System (ADS)
Tomovski, Živorad; Sandev, Trifce; Metzler, Ralf; Dubbeldam, Johan
2012-04-01
We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of the form δ(x)ṡ tΓ/(1-β) (β>0).
The Conley index along heteroclinic trajectories of reaction-diffusion equations
NASA Astrophysics Data System (ADS)
Jänig, A.
It is well known that hyperbolic equilibria of reaction-diffusion equations have the homotopy Conley index of a pointed sphere, the dimension of which is the Morse index of the equilibrium. A similar result concerning the homotopy Conley index along heteroclinic solutions of ordinary differential equations under the assumption that the respective stable and unstable manifolds intersect transversally, is due to McCord. This result has recently been generalized by Dancer to some reaction-diffusion equations by using finite-dimensional approximations. We extend McCord's result to reaction-diffusion equations. Additionally, an error in the original proof is corrected.
NASA Astrophysics Data System (ADS)
Sweilam, N. H.; Abou Hasan, M. M.
2016-08-01
This paper reports a new spectral algorithm for obtaining an approximate solution for the Lévy-Feller diffusion equation depending on Legendre polynomials and Chebyshev collocation points. The Lévy-Feller diffusion equation is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative. A new formula expressing explicitly any fractional-order derivatives, in the sense of Riesz-Feller operator, of Legendre polynomials of any degree in terms of Jacobi polynomials is proved. Moreover, the Chebyshev-Legendre collocation method together with the implicit Euler method are used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. Numerical results with comparisons are given to confirm the reliability of the proposed method for the Lévy-Feller diffusion equation.
Innovation diffusion equations on correlated scale-free networks
NASA Astrophysics Data System (ADS)
Bertotti, M. L.; Brunner, J.; Modanese, G.
2016-07-01
We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the total adoption in a predictable way. In a major departure from the standard model, we model a trickle-up process by introducing heterogeneous publicity coefficients (which can also be negative for the hubs, thus turning them into stiflers) and a stochastic term which represents the erratic generation of innovation at the periphery of the network. The results confirm the robustness of the Bass model and expand considerably its range of applicability.
Solutions of fractional reaction-diffusion equations in terms of the H-function
NASA Astrophysics Data System (ADS)
Haubold, H. J.; Mathai, A. M.; Saxena, R. K.
2007-12-01
This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction-diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form.
Harnack inequality and strong Feller property for stochastic fast-diffusion equations
NASA Astrophysics Data System (ADS)
Liu, Wei; Wang, Feng-Yu
2008-06-01
As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333-1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.
Manzini, Gianmarco; Cangiani, Andrea; Sutton, Oliver
2014-10-02
This document presents the results of a set of preliminary numerical experiments using several possible conforming virtual element approximations of the convection-reaction-diffusion equation with variable coefficients.
A modified diffusion equation for room-acoustic predication.
Jing, Yun; Xiang, Ning
2007-06-01
This letter presents a modified diffusion model using an Eyring absorption coefficient to predict the reverberation time and sound pressure distributions in enclosures. While the original diffusion model [Ollendorff, Acustica 21, 236-245 (1969); J. Picaut et al., Acustica 83, 614-621 (1997); Valeau et al., J. Acoust. Soc. Am. 119, 1504-1513 (2006)] usually has good performance for low absorption, the modified diffusion model yields more satisfactory results for both low and high absorption. Comparisons among the modified model, the original model, a geometrical-acoustics model, and several well-established theories in terms of reverberation times and sound pressure level distributions, indicate significantly improved prediction accuracy by the modification.
Horowitz, Jordan M.
2015-07-28
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.
Horowitz, Jordan M
2015-07-28
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.
Quantum position diffusion and its implications for the quantum linear Boltzmann equation
Kamleitner, I.; Cresser, J.
2010-01-15
We derive a quantum linear Boltzmann equation from first principles to describe collisional friction, diffusion, and decoherence in a unified framework. In doing so, we discover that the previously celebrated quantum contribution to position diffusion is not a true physical process, but rather an artifact of the use of a coarse-grained time scale necessary to derive Markovian dynamics.
Similarity solution to fractional nonlinear space-time diffusion-wave equation
NASA Astrophysics Data System (ADS)
Costa, F. Silva; Marão, J. A. P. F.; Soares, J. C. Alves; de Oliveira, E. Capelas
2015-03-01
In this article, the so-called fractional nonlinear space-time wave-diffusion equation is presented and discussed. This equation is solved by the similarity method using fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. Some particular cases are presented and the corresponding solutions are shown by means of 2-D and 3-D plots.
Arnold, J.; Kosson, D.S.; Garrabrants, A.; Meeussen, J.C.L.; Sloot, H.A. van der
2013-02-15
A robust numerical solution of the nonlinear Poisson-Boltzmann equation for asymmetric polyelectrolyte solutions in discrete pore geometries is presented. Comparisons to the linearized approximation of the Poisson-Boltzmann equation reveal that the assumptions leading to linearization may not be appropriate for the electrochemical regime in many cementitious materials. Implications of the electric double layer on both partitioning of species and on diffusive release are discussed. The influence of the electric double layer on anion diffusion relative to cation diffusion is examined.
Vlad, Marcel Ovidiu; Ross, John
2002-12-01
We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.
NASA Astrophysics Data System (ADS)
Yuan, Zhen; Zhang, Qizhi; Sobel, Eric; Jiang, Huabei
2009-09-01
In this study, a simplified spherical harmonics approximated higher order diffusion model is employed for 3-D diffuse optical tomography of osteoarthritis in the finger joints. We find that the use of a higher-order diffusion model in a stand-alone framework provides significant improvement in reconstruction accuracy over the diffusion approximation model. However, we also find that this is not the case in the image-guided setting when spatial prior knowledge from x-rays is incorporated. The results show that the reconstruction error between these two models is about 15 and 4%, respectively, for stand-alone and image-guided frameworks.
Pathways for Advective Transport
2001-01-19
the approach is given and an application to the Gulf of Mexico is described where the analysis precisely identifies the boundaries of coherent vortical structures as well as pathways for advective transport.
Fractional diffusion equation for an n-dimensional correlated Lévy walk.
Taylor-King, Jake P; Klages, Rainer; Fedotov, Sergei; Van Gorder, Robert A
2016-07-01
Lévy walks define a fundamental concept in random walk theory that allows one to model diffusive spreading faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n-dimensional correlated Lévy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short-range auto-correlated Lévy walks in the large time limit, and we solve it. Our derivation discloses different dynamical mechanisms leading to correlated Lévy walk diffusion in terms of quantities that can be measured experimentally.
Coarse-graining Brownian motion: from particles to a discrete diffusion equation.
de la Torre, J A; Español, Pep
2011-09-21
We study the process of coarse-graining in a simple model of diffusion of Brownian particles. At a detailed level of description, the system is governed by a Brownian dynamics of non-interacting particles. The coarse-level is described by discrete concentration variables defined in terms of Delaunay cells. These coarse variables obey a stochastic differential equation that can be understood as a discrete version of a diffusion equation. We study different models for the two basic building blocks of this equation which are the free energy function and the diffusion matrix. The free energy function is shown to be non-additive due to the overlapping of cells in the Delaunay construction. The diffusion matrix is state dependent in principle, but for near-equilibrium situations it is shown that it may be safely evaluated at the equilibrium value of the concentration field.
Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime
NASA Astrophysics Data System (ADS)
Liang, Jin-Rong; Wang, Jun; Lǔ, Long-Jin; Gu, Hui; Qiu, Wei-Yuan; Ren, Fu-Yao
2012-01-01
In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X α, H ( t)= X H ( S α ( t)), 0< α, H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equation and Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X α, H ( t) and the corresponding Black-Scholes formula for the fair prices of European option.
An asymptotic induced numerical method for the convection-diffusion-reaction equation
NASA Technical Reports Server (NTRS)
Scroggs, Jeffrey S.; Sorensen, Danny C.
1988-01-01
A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run time results demonstrate the viability of the method.
Shestakov, Aleksei I.
2013-06-15
We derive time-dependent multifrequency diffusion equations for homogeneous, refractive lossy media. The equations are applicable for a domain composed of several materials with distinct refractive indexes. In such applications, the fundamental radiation variable, the intensity I, is discontinuous across material interfaces. The diffusion equations evolve a variable ξ, the integral of I over all directions divided by the square of the refractive index. Attention is focused on boundary and internal interface conditions for ξ. For numerical solutions using finite elements, it is shown that at material interfaces, the usual diffusion coefficient 1/3κ of the multifrequency equation, where κ is the opacity, is modified by a tensor diffusion term consisting of integrals of the reflectivity. Numerical results are presented. For a single material simulation, the ξ equations yield the same result as diffusion equations that evolve the spectral radiation energy density. A second simulation solves a test problem that models radiation transport in a domain comprised of materials with different refractive indexes. Results qualitatively agree with those previously published.
Spatially Periodic Domain Structure in Coupled Reaction--Diffusion Equations for Segment Formation
NASA Astrophysics Data System (ADS)
Sakaguchi, Hidetsugu
2012-02-01
Segment formation is an important process of pattern formation in the developing vertebrate embryo. The mechanism of such pattern formation is considered to be different from that of Turing instability. We propose reaction--diffusion equations generating traveling pulses and coupled reaction--diffusion equations for two genes that generate a domain structure. Next, we construct a synthetic model for segment formation by combining the coupled reaction--diffusion equations. A spatially periodic domain structure is found in the numerical simulation of the model equation. It is shown that the wavelength of the spatially periodic pattern and the proportion of the sizes of the anterior and posterior domains in each segment can be controlled by adjusting some system parameters.
Fa, Kwok Sau
2015-02-15
An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems. - Highlights: • Calculation of the correlation function. • The correlation function is connected to the survival probability. • The model can be applied to the internal dynamics of proteins.
Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel
NASA Astrophysics Data System (ADS)
Gómez-Aguilar, J. F.
2017-01-01
In this paper, using the fractional operators with Mittag-Leffler kernel in Caputo and Riemann-Liouville sense the space-time fractional diffusion equation is modified, the fractional equation will be examined separately; with fractional spatial derivative and fractional temporal derivative. For the study cases, the order considered is 0 < β , γ ≤ 1 respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation, these parameters related to equation results in a fractal space-time geometry provide a new family of solutions for the diffusive processes. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, viscoelastic materials, material heterogeneities and media with different scales.
Numerical approximation of Levy-Feller diffusion equation and its probability interpretation
NASA Astrophysics Data System (ADS)
Zhang, H.; Liu, F.; Anh, V.
2007-09-01
In this paper, we consider the Levy-Feller fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order and skewness [theta] ([theta][less-than-or-equals, slant]min{[alpha],2-[alpha]}). We construct two new discrete schemes of the Cauchy problem for the above equation with 0<[alpha]<1 and 1<[alpha][less-than-or-equals, slant]2, respectively. We investigate their probabilistic interpretation and the domain of attraction of the corresponding stable Levy distribution. Furthermore, we present a numerical analysis for the Levy-Feller fractional diffusion equation with 1<[alpha]<2 in a bounded spatial domain. Finally, we present a numerical example to evaluate our theoretical analysis.
NASA Astrophysics Data System (ADS)
Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick; Ortensi, Javier; Baker, Benjamin; Laboure, Vincent; Wang, Congjian; DeHart, Mark; Martineau, Richard
2017-06-01
This work presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the SN transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form is based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. While NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in
Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations
NASA Astrophysics Data System (ADS)
Lu, Jianfang; Fang, Jinwei; Tan, Sirui; Shu, Chi-Wang; Zhang, Mengping
2016-07-01
We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure [28], in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.
Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker-Planck and Fractional Equations
NASA Astrophysics Data System (ADS)
Boon, Jean Pierre; Lutsko, James F.
2017-03-01
The temporal Fokker-Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker-Planck equation for the first passage distribution function f_j(r,t) of a particle moving on a substrate with time delays τ _j. Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability P_j is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, P_j ∝ f_j^{ν - 1}, the generalized Fokker-Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, P_j ∝ τ _j^{-1-α } (with 0< α < 2), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.
Long-time behavior of spreading solutions of Schrödinger and diffusion equations.
Anteneodo, C; Dias, J C; Mendes, R S
2006-05-01
We investigate the asymptotic time behavior of the solutions of a large class of linear differential equations that generalize the free-particle Schrödinger and diffusion equations, containing the standard ones as particular cases. We find general scalings that depend only on characteristic features of both the arbitrary initial condition and the Green function associated with the evolution equation. Basically, the amplitude of a long-time solution can be expressed in terms of low order moments of the initial condition (if finite) and low order spatial derivatives of the Green function. These derivatives can also be of the fractional type, which naturally arise when moments are divergent. We apply our results to a large class of differential equations that includes the fractional Schrödinger and Lévy diffusion equations. In particular, we show that, except for threshold cases, the amplitude of a packet may follow the asymptotic law t-alpha, with arbitrary positive alpha.
Group iterative methods for the solution of two-dimensional time-fractional diffusion equation
NASA Astrophysics Data System (ADS)
Balasim, Alla Tareq; Ali, Norhashidah Hj. Mohd.
2016-06-01
Variety of problems in science and engineering may be described by fractional partial differential equations (FPDE) in relation to space and/or time fractional derivatives. The difference between time fractional diffusion equations and standard diffusion equations lies primarily in the time derivative. Over the last few years, iterative schemes derived from the rotated finite difference approximation have been proven to work well in solving standard diffusion equations. However, its application on time fractional diffusion counterpart is still yet to be investigated. In this paper, we will present a preliminary study on the formulation and analysis of new explicit group iterative methods in solving a two-dimensional time fractional diffusion equation. These methods were derived from the standard and rotated Crank-Nicolson difference approximation formula. Several numerical experiments were conducted to show the efficiency of the developed schemes in terms of CPU time and iteration number. At the request of all authors of the paper an updated version of this article was published on 7 July 2016. The original version supplied to AIP Publishing contained an error in Table 1 and References 15 and 16 were incomplete. These errors have been corrected in the updated and republished article.
Shumaker, D E; Woodward, C S
2005-05-03
In this paper, the authors investigate performance of a fully implicit formulation and solution method of a diffusion-reaction system modeling radiation diffusion with material energy transfer and a fusion fuel source. In certain parameter regimes this system can lead to a rapid conversion of potential energy into material energy. Accuracy in time integration is essential for a good solution since a major fraction of the fuel can be depleted in a very short time. Such systems arise in a number of application areas including evolution of a star and inertial confinement fusion. Previous work has addressed implicit solution of radiation diffusion problems. Recently Shadid and coauthors have looked at implicit and semi-implicit solution of reaction-diffusion systems. In general they have found that fully implicit is the most accurate method for difficult coupled nonlinear equations. In previous work, they have demonstrated that a method of lines approach coupled with a BDF time integrator and a Newton-Krylov nonlinear solver could efficiently and accurately solve a large-scale, implicit radiation diffusion problem. In this paper, they extend that work to include an additional heating term in the material energy equation and an equation to model the evolution of the reactive fuel density. This system now consists of three coupled equations for radiation energy, material energy, and fuel density. The radiation energy equation includes diffusion and energy exchange with material energy. The material energy equation includes reaction heating and exchange with radiation energy, and the fuel density equation includes its depletion due to the fuel consumption.
Application of a diffusion-desorption rate equation model in astrochemistry.
He, Jiao; Vidali, Gianfranco
2014-01-01
Desorption and diffusion are two of the most important processes on interstellar grain surfaces; knowledge of them is critical for the understanding of chemical reaction networks in the interstellar medium (ISM). However, a lack of information on desorption and diffusion is preventing further progress in astrochemistry. To obtain desorption energy distributions of molecules from the surfaces of ISM-related materials, one usually carries out adsorption-desorption temperature programmed desorption (TPD) experiments, and uses rate equation models to extract desorption energy distributions. However, the often-used rate equation models fail to adequately take into account diffusion processes and thus are only valid in situations where adsorption is strongly localized. As adsorption-desorption experiments show that adsorbate molecules tend to occupy deep adsorption sites before occupying shallow ones, a diffusion process must be involved. Thus, it is necessary to include a diffusion term in the model that takes into account the morphology of the surface as obtained from analyses of TPD experiments. We take the experimental data of CO desorption from the MgO(100) surface and of D2 desorption from amorphous solid water ice as examples to show how a diffusion-desorption rate equation model explains the redistribution of adsorbate molecules among different adsorption sites. We extract distributions of desorption energies and diffusion energy barriers from TPD profiles. These examples are contrasted with a system where adsorption is strongly localized--HD from an amorphous silicate surface. Suggestions for experimental investigations are provided.
Shadow Systems and Attractors in Reaction-Diffusion Equations,
1987-04-01
the shadow system of (1.1), (1.2), we mean the system au/at = DIAu + f(u,¢) (1.3) dt/dt = In1-1 fn g(u(,x), )dx in (I with the boundary condition (1.4...r,O . The functions ut,%t arc supposed t(-) belong to the spaces C(q-rOJX, . C(-r,0],YAa) . The shadow s5 stem is 8u 8t = DIAu + f(ut.z t ) in 0 dz...au/an = 0 , aw/an = 0 in &a We are going to consider this equation as a perturbation of the system au/at = DIAu + f(u,z) (2.4) dz/dt = In ŕ f g(u,z)dx
Self-similar solutions for a nonlinear radiation diffusion equation
Garnier, Josselin; Malinie, Guy; Saillard, Yves; Cherfils-Clerouin, Catherine
2006-09-15
This paper considers the hydrodynamic equations with nonlinear conduction when the internal energy and the opacity have power-law dependences in the density and in the temperature. This system models the situation in which a dense solid is brought into contact with a thermal bath. It supports self-similar solutions that depend on the surface temperature. The self-similar solution can exhibit a shock wave followed by an ablation front if the surface temperature does not increase too fast in time, but it can exhibit a heat front followed by an isothermal shock otherwise. These flows are carefully studied in order to clarify the role of the initial solid density in the energy absorption and the ablation process. Comparisons with numerical simulations show excellent agreement.
A fractional diffusion equation model for cancer tumor
NASA Astrophysics Data System (ADS)
Iyiola, Olaniyi Samuel; Zaman, F. D.
2014-10-01
In this article, we consider cancer tumor models and investigate the need for fractional order derivative as compared to the classical first order derivative in time. Three different cases of the net killing rate are taken into account including the case where net killing rate of the cancer cells is dependent on the concentration of the cells. At first, we use a relatively new analytical technique called q-Homotopy Analysis Method on the resulting time-fractional partial differential equations to obtain analytical solution in form of convergent series with easily computable components. Our numerical analysis enables us to give some recommendations on the appropriate order (fractional) of derivative in time to be used in modeling cancer tumor.
Nonlinear diffusion-wave equation for a gas in a regenerator subject to temperature gradient
NASA Astrophysics Data System (ADS)
Sugimoto, N.
2015-10-01
This paper derives an approximate equation for propagation of nonlinear thermoacoustic waves in a gas-filled, circular pore subject to temperature gradient. The pore radius is assumed to be much smaller than a thickness of thermoviscous diffusion layer, and the narrow-tube approximation is used in the sense that a typical axial length associated with temperature gradient is much longer than the radius. Introducing three small parameters, one being the ratio of the pore radius to the thickness of thermoviscous diffusion layer, another the ratio of a typical speed of thermoacoustic waves to an adiabatic sound speed and the other the ratio of a typical magnitude of pressure disturbance to a uniform pressure in a quiescent state, a system of fluid dynamical equations for an ideal gas is reduced asymptotically to a nonlinear diffusion-wave equation by using boundary conditions on a pore wall. Discussion on a temporal mean of an excess pressure due to periodic oscillations is included.
Fourier spectral method for higher order space fractional reaction-diffusion equations
NASA Astrophysics Data System (ADS)
Pindza, Edson; Owolabi, Kolade M.
2016-11-01
Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction-diffusion equations. The proposed method is based on an exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial dimensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we introduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimensional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.
NASA Astrophysics Data System (ADS)
Gal, Ciprian G.; Warma, Mahamadi
2016-08-01
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction-diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.
Simpson, Matthew J; Sharp, Jesse A; Morrow, Liam C; Baker, Ruth E
2015-01-01
Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction-diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction-diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction-diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially-confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially-confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.
1981-08-01
bifurcation of trajectories connecting saddle points from stationary solutions is studied. As an application, reaction-diffusion models in one space...to reaction- diffusion models are discussed. We obtain solutions, which may be considered *one-dimensional analogues of the patterns of concentric...Oscillating sina,!ar solutions connected with Hopf bifurcations We consider a general chemical reaction model given by an equation 2au _2u (6.1) =F( ,u) + D
Bailey, T S; Adams, M L; Yang, B; Zika, M R
2005-07-15
We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.
Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains
NASA Astrophysics Data System (ADS)
Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F.
2017-02-01
In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION
Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.
2013-01-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian. PMID:23772179
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature.
Gillespie, Dirk
2015-01-01
The numerical integration of the time-dependent spherically-symmetric radial diffusion equation from a point source is considered. The flux through the source can vary in time, possibly stochastically based on the concentration produced by the source itself. Fick's one-dimensional diffusion equation is integrated over a time interval by considering a source term and a propagation term. The source term adds new particles during the time interval, while the propagation term diffuses the concentration profile of the previous time step. The integral in the propagation term is evaluated numerically using a combination of a new diffusion-specific Gaussian quadrature and interpolation on a diffusion-specific grid. This attempts to balance accuracy with the least number of points for both integration and interpolation. The theory can also be extended to include a simple reaction-diffusion equation in the limit of high buffer concentrations. The method is unconditionally stable. In fact, not only does it converge for any time step Δt, the method offers one advantage over other methods because Δt can be arbitrarily large; it is solely defined by the timescale on which the flux source turns on and off.
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature
Gillespie, Dirk
2015-01-01
The numerical integration of the time-dependent spherically-symmetric radial diffusion equation from a point source is considered. The flux through the source can vary in time, possibly stochastically based on the concentration produced by the source itself. Fick’s one-dimensional diffusion equation is integrated over a time interval by considering a source term and a propagation term. The source term adds new particles during the time interval, while the propagation term diffuses the concentration profile of the previous time step. The integral in the propagation term is evaluated numerically using a combination of a new diffusion-specific Gaussian quadrature and interpolation on a diffusion-specific grid. This attempts to balance accuracy with the least number of points for both integration and interpolation. The theory can also be extended to include a simple reaction-diffusion equation in the limit of high buffer concentrations. The method is unconditionally stable. In fact, not only does it converge for any time step Δt, the method offers one advantage over other methods because Δt can be arbitrarily large; it is solely defined by the timescale on which the flux source turns on and off. PMID:26208111
A moving mesh finite difference method for equilibrium radiation diffusion equations
Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
Xie, Jiaquan; Huang, Qingxue; Yang, Xia
2016-01-01
In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent.
On the Dynamics of Some Discretizations of Convection-Diffusion Equations
NASA Technical Reports Server (NTRS)
Sweby, Peter K.; Yee, H. C.; Rai, Man Mohan (Technical Monitor)
1995-01-01
Numerical discretizations of differential equations which model physical processes can possess dynamics quite different from that of the equations themselves. Recently the emphasis has been on the the dynamics of numerical discretizations for Ordinary Differential Equations (ODEs). For Partial Differential Equations (PDEs) using a method of lines approach the situation is more complex. First, the spatial discretisation may introduce dynamics not present in the original equations; second, the solution of the resulting system of ODEs is open to the modified dynamics of the ODE solver used. These two effects may interact in a complex manner. In this talk we present some results of our recent work on the dynamics of discretizations of convection-diffusion equations, including those produced using Total Variation Diminishing (TVD) schemes and adaptive grid techniques. A more general overview of the area may be found on our accompanying poster presentation.
Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains
NASA Astrophysics Data System (ADS)
Li, Dingshi; Wang, Bixiang; Wang, Xiaohu
2017-02-01
This paper deals with the limiting behavior of stochastic reaction-diffusion equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. We first prove the existence, uniqueness and periodicity of pullback tempered random attractors for the equations in an (n + 1)-dimensional narrow domain, and then establish the upper semicontinuity of these attractors when a family of (n + 1)-dimensional thin domains collapses onto an n-dimensional domain.
NASA Astrophysics Data System (ADS)
Fallon, A.; Leonard, R. H.; Sackett, C. A.
2015-10-01
Atom interferometers using Bose-Einstein condensates are fundamentally limited by a phase diffusion process that arises from atomic interactions. Numerical solution of the Gross-Pitaevskii equation is here used to accurately calculate the diffusion rate for a Bragg interferometer. It is seen to agree with a Thomas-Fermi approximation at large atom numbers and a perturbative approximation at low atom numbers. Most experiments to date operate in the crossover region between these two regimes. Even in this case, it is found that that the diffusion time is governed in a simple way by the ratio of the healing length to the size of the condensate.
NASA Astrophysics Data System (ADS)
Zamani, K.; Bombardelli, F.
2011-12-01
Almost all natural phenomena on Earth are highly nonlinear. Even simplifications to the equations describing nature usually end up being nonlinear partial differential equations. Transport (ADR) equation is a pivotal equation in atmospheric sciences and water quality. This nonlinear equation needs to be solved numerically for practical purposes so academicians and engineers thoroughly rely on the assistance of numerical codes. Thus, numerical codes require verification before they are utilized for multiple applications in science and engineering. Model verification is a mathematical procedure whereby a numerical code is checked to assure the governing equation is properly solved as it is described in the design document. CFD verification is not a straightforward and well-defined course. Only a complete test suite can uncover all the limitations and bugs. Results are needed to be assessed to make a distinction between bug-induced-defect and innate limitation of a numerical scheme. As Roache (2009) said, numerical verification is a state-of-the-art procedure. Sometimes novel tricks work out. This study conveys the synopsis of the experiences we gained during a comprehensive verification process which was done for a transport solver. A test suite was designed including unit tests and algorithmic tests. Tests were layered in complexity in several dimensions from simple to complex. Acceptance criteria defined for the desirable capabilities of the transport code such as order of accuracy, mass conservation, handling stiff source term, spurious oscillation, and initial shape preservation. At the begining, mesh convergence study which is the main craft of the verification is performed. To that end, analytical solution of ADR equation gathered. Also a new solution was derived. In the more general cases, lack of analytical solution could be overcome through Richardson Extrapolation and Manufactured Solution. Then, two bugs which were concealed during the mesh convergence
Navarro, Juan M; Escolano, José; Cobos, Maximo; López, José J
2013-03-01
The diffusion equation model was used for room acoustic simulations to predict the sound pressure level and the reverberation time. The technical literature states that the diffusion equation method accurately models the late portion of the room impulse response if the energy is sufficiently scattered. This work provides conclusions on the validity of the diffusion equation model for rooms with homogeneous dimensions in relation to the scattering coefficients of the boundaries. A systematic evaluation was conducted out to determine the ranges of the absorption and scattering coefficient values that result in low noticeable differences between the predictions from a geometrical acoustic model and those from the diffusion equation model.
Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation
NASA Astrophysics Data System (ADS)
Sorokin, M. V.; Dubinko, V. I.; Borodin, V. A.
2017-01-01
The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.
The Dirichlet problem for the diffusion equation in the exterior of non-closed Lipschitz surfaces
NASA Astrophysics Data System (ADS)
Krutitskii, P. A.
2012-09-01
We study the Dirichlet problem for the stationary diffusion equation in the exterior of non-closed Lipschitz surfaces in R3. The Dirichlet problem for a Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.
Breakdown of the reaction-diffusion master equation with nonelementary rates.
Smith, Stephen; Grima, Ramon
2016-05-01
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.
A numerical study of the steady scalar convective diffusion equation for small viscosity
NASA Technical Reports Server (NTRS)
Giles, M. B.; Rose, M. E.
1983-01-01
A time-independent convection diffusion equation is studied by means of a compact finite difference scheme and numerical solutions are compared to the analytic inviscid solutions. The correct internal and external boundary layer behavior is observed, due to an inherent feature of the scheme which automatically produces upwind differencing in inviscid regions and the correct viscous behavior in viscous regions.
A fast finite volume method for conservative space-fractional diffusion equations in convex domains
NASA Astrophysics Data System (ADS)
Jia, Jinhong; Wang, Hong
2016-04-01
We develop a fast finite volume method for variable-coefficient, conservative space-fractional diffusion equations in convex domains via a volume-penalization approach. The method has an optimal storage and an almost linear computational complexity. The method retains second-order accuracy without requiring a Richardson extrapolation. Numerical results are presented to show the utility of the method.
The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion
Guo, Ran; Du, Jiulin
2015-08-15
We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution.
Wright, A K; Baxter, J E
1976-01-01
An iterative numerical technique is presented which allows the semiaxes for prolate and oblate ellipsoids to be determined from the Perrin equations for rotational and translational diffusion constants. The use of this inversion technique is illustrated by application to the proteins: lysozyme, bovine serum albumin, human transferrin, and bovine rhodopsin solubilized in digitonin. PMID:938731
Solution of a Two-Dimensional Diffusion Equation Using an Advanced Spreadsheet Program.
ERIC Educational Resources Information Center
Kharab, Abdelwahab
1997-01-01
Spreadsheet programs are used increasingly by engineering students to solve problems, especially problems requiring repetitive calculations, as they provide rapid and simple numerical solutions. This article shows how advanced spreadsheet programs are used in the learning of numerical solutions of two-dimensional diffusion equation using the…
Multigrid solution of the convection-diffusion equation with high-Reynolds number
Zhang, Jun
1996-12-31
A fourth-order compact finite difference scheme is employed with the multigrid technique to solve the variable coefficient convection-diffusion equation with high-Reynolds number. Scaled inter-grid transfer operators and potential on vectorization and parallelization are discussed. The high-order multigrid method is unconditionally stable and produces solution of 4th-order accuracy. Numerical experiments are included.
High speed and flexible PEB 3D diffusion simulation based on Sylvester equation
NASA Astrophysics Data System (ADS)
Lin, Pei-Chun; Chen, Charlie Chung-Ping
2013-04-01
Post exposure bake (PEB) Diffusion effect is one of the most difficult issues in modeling chemically amplified resists. These model equations result in a system of nonlinear partial differential equations describing the time rate of change reaction and diffusion. Verifying such models are difficult, so numerical simulations are needed to solve the model equations. In this paper, we propose a high speed 3D resist image simulation algorithm based on a novel method to solve the PEB Diffusion equation. Our major discovery is that the matrix formulation of the diffusion equation under the Crank- Nicolson scheme can be derived into a special form, AX+XB=C, where the X matrix is a 3D resist image after diffusion effect, A and B matrices contain the diffusion coefficients and the space relationship between directions x, y and z. These matrices are sparse, symmetric and diagonal dominant. The C matrix is the last time-step resist image. The Sylvester equation can be reduced to another form as (I⊗A + BT⊗I) X =C, in which the operator ⊗ is the Kronecker product notation. Compared with a traditional convolution method, our method is more useful in a way that boundary conditions can be more flexible. From our experimental results, we see that the error of the convolution method can be as high as 30% at borders of the design pattern. Furthermore, since the PEB temperature may not be uniform at multi-zone PEB, the convolution method might not be directly applicable in this scenario. Our method is about 20 times faster than the convolution method for a single time step (2 seconds) as illustrated in the attached figure. To simulate 50 seconds of the flexible PEB diffusion process, our method only takes 210 seconds with a convolution set up for a 1240×1240 working area. We use the typical 45nm immersion lithography in our work. The exposure wavelength is set to 193nm; the NA is 1.3775; and the diffusion coefficient is 1.455×10-17m2/s at PEB temperature 150°C along with PEB
An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit
NASA Astrophysics Data System (ADS)
Wang, Li; Yan, Bokai
2016-05-01
We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro-macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro-micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.
Manzini, Gianmarco; Cangiani, Andrea; Sutton, Oliver
2014-10-02
This document describes the conforming formulations for virtual element approximation of the convection-reaction-diffusion equation with variable coefficients. Emphasis is given to construction of the projection operators onto polynomial spaces of appropriate order. These projections make it possible the virtual formulation to achieve any order of accuracy. We present the construction of the internal and the external formulation. The difference between the two is in the way the projection operators act on the derivatives (laplacian, gradient) of the partial differential equation. For the diffusive regime we prove the well-posedness of the external formulation and we derive an estimate of the approximation error in the H^{1}-norm. For the convection-dominated case, the streamline diffusion stabilization (aka SUPG) is also discussed.
Image Reconstruction for Diffuse Optical Tomography Based on Radiative Transfer Equation
Han, Bo; Tang, Jinping
2015-01-01
Diffuse optical tomography is a novel molecular imaging technology for small animal studies. Most known reconstruction methods use the diffusion equation (DA) as forward model, although the validation of DA breaks down in certain situations. In this work, we use the radiative transfer equation as forward model which provides an accurate description of the light propagation within biological media and investigate the potential of sparsity constraints in solving the diffuse optical tomography inverse problem. The feasibility of the sparsity reconstruction approach is evaluated by boundary angular-averaged measurement data and internal angular-averaged measurement data. Simulation results demonstrate that in most of the test cases the reconstructions with sparsity regularization are both qualitatively and quantitatively more reliable than those with standard L2 regularization. Results also show the competitive performance of the split Bregman algorithm for the DOT image reconstruction with sparsity regularization compared with other existing L1 algorithms. PMID:25648064
Gorpas, Dimitris; Andersson-Engels, Stefan
2012-12-01
The solution of the forward problem in fluorescence molecular imaging strongly influences the successful convergence of the fluorophore reconstruction. The most common approach to meeting this problem has been to apply the diffusion approximation. However, this model is a first-order angular approximation of the radiative transfer equation, and thus is subject to some well-known limitations. This manuscript proposes a methodology that confronts these limitations by applying the radiative transfer equation in spatial regions in which the diffusion approximation gives decreased accuracy. The explicit integro differential equations that formulate this model were solved by applying the Galerkin finite element approximation. The required spatial discretization of the investigated domain was implemented through the Delaunay triangulation, while the azimuthal discretization scheme was used for the angular space. This model has been evaluated on two simulation geometries and the results were compared with results from an independent Monte Carlo method and the radiative transfer equation by calculating the absolute values of the relative errors between these models. The results show that the proposed forward solver can approximate the radiative transfer equation and the Monte Carlo method with better than 95% accuracy, while the accuracy of the diffusion approximation is approximately 10% lower.
Shestakov, A I; Vignes, R M; Stolken, J S
2010-01-05
Starting from the radiation transport equation for homogeneous, refractive lossy media, we derive the corresponding time-dependent multifrequency diffusion equations. Zeroth and first moments of the transport equation couple the energy density, flux and pressure tensor. The system is closed by neglecting the temporal derivative of the flux and replacing the pressure tensor by its diagonal analogue. The system is coupled to a diffusion equation for the matter temperature. We are interested in modeling annealing of silica (SiO{sub 2}). We derive boundary conditions at a planar air-silica interface taking account of reflectivities. The spectral dimension is discretized into a finite number of intervals leading to a system of multigroup diffusion equations. Three simulations are presented. One models cooling of a silica slab, initially at 2500 K, for 10 s. The other two are 1D and 2D simulations of irradiating silica with a CO{sub 2} laser, {lambda} = 10.59 {micro}m. In 2D, we anneal a disk (radius = 0.4, thickness = 0.4 cm) with a laser, Gaussian profile (r{sub 0} = 0.5 mm for 1/e decay).
NASA Astrophysics Data System (ADS)
Li, Gongsheng; Zhang, Dali; Jia, Xianzheng; Yamamoto, Masahiro
2013-06-01
This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion.
Jin, Shi; Xiu, Dongbin; Zhu, Xueyu
2015-05-15
In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretization. Our stochastic AP schemes are extensions of the well-developed deterministic AP schemes. To handle the random inputs, we employ generalized polynomial chaos (gPC) expansion and combine it with stochastic Galerkin procedure. We apply the gPC Galerkin scheme to a set of representative hyperbolic and transport equations and establish the AP property in the stochastic setting. We then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic AP schemes.
Shankar, Varun; Wright, Grady B; Kirby, Robert M; Fogelson, Aaron L
2016-06-01
In this paper, we present a method based on Radial Basis Function (RBF)-generated Finite Differences (FD) for numerically solving diffusion and reaction-diffusion equations (PDEs) on closed surfaces embedded in ℝ (d) . Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.
Prediction equations for diffusing capacity (transfer factor) of lung for North Indians
Chhabra, Sunil Kumar; Kumar, Rajeev; Gupta, Uday A
2016-01-01
Background: Prediction equations for diffusing capacity of lung for carbon monoxide (DLCO), alveolar volume (VA), and DLCO/VA using the current standardization guidelines are not available for Indian population. The present study was carried out to develop equations for these parameters for North Indian adults and examine the ethnic diversity in predictions. Materials and Methods: DLCO was measured by single-breath technique and VA by single-breath helium dilution using standardized methodology in 357 (258 males, 99 females) normal nonsmoker adult North Indians and DLCO/VA was computed. The subjects were randomized into training and test datasets for development of prediction equations by multiple linear regressions and for validation, respectively. Results: For males, the following equations were developed: DLCO, −7.813 + 0.318 × ht −0.624 × age + 0.00552 × age2; VA, −8.152 + 0.087 × ht −0.019 × wt; DLCO/VA, 7.315 − 0.037 × age. For females, the equations were: DLCO, −44.15 + 0.449 × ht −0.099 × age; VA, −6.893 + 0.068 × ht. A statistically acceptable prediction equation was not obtained for DLCO/VA in females. It was therefore computed from predicted DLCO and predicted VA. All equations were internally valid. Predictions of DLCO by Indian equations were lower than most Caucasian predictions in both males and females and greater than the Chinese predictions for males. Conclusion: This study has developed validated prediction equations for DLCO, VA, and DLCO/VA in North Indians. Substantial ethnic diversity exists in predictions for DLCO and VA with Caucasian equations generally yielding higher values than the Indian or Chinese equations. However, DLCO/VA predicted by the Indian equations is slightly higher than that by other equations. PMID:27625439
A cross-diffusion system derived from a Fokker-Planck equation with partial averaging
NASA Astrophysics Data System (ADS)
Jüngel, Ansgar; Zamponi, Nicola
2017-02-01
A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.
Singular solution of the Feller diffusion equation via a spectral decomposition
NASA Astrophysics Data System (ADS)
Gan, Xinjun; Waxman, David
2015-01-01
Feller studied a branching process and found that the distribution for this process approximately obeys a diffusion equation [W. Feller, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley and Los Angeles, 1951), pp. 227-246]. This diffusion equation and its generalizations play an important role in many scientific problems, including, physics, biology, finance, and probability theory. We work under the assumption that the fundamental solution represents a probability density and should account for all of the probability in the problem. Thus, under the circumstances where the random process can be irreversibly absorbed at the boundary, this should lead to the presence of a Dirac delta function in the fundamental solution at the boundary. However, such a feature is not present in the standard approach (Laplace transformation). Here we require that the total integrated probability is conserved. This yields a fundamental solution which, when appropriate, contains a term proportional to a Dirac delta function at the boundary. We determine the fundamental solution directly from the diffusion equation via spectral decomposition. We obtain exact expressions for the eigenfunctions, and when the fundamental solution contains a Dirac delta function at the boundary, every eigenfunction of the forward diffusion operator contains a delta function. We show how these combine to produce a weight of the delta function at the boundary which ensures the total integrated probability is conserved. The solution we present covers cases where parameters are time dependent, thereby greatly extending its applicability.
Singular solution of the Feller diffusion equation via a spectral decomposition.
Gan, Xinjun; Waxman, David
2015-01-01
Feller studied a branching process and found that the distribution for this process approximately obeys a diffusion equation [W. Feller, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley and Los Angeles, 1951), pp. 227-246]. This diffusion equation and its generalizations play an important role in many scientific problems, including, physics, biology, finance, and probability theory. We work under the assumption that the fundamental solution represents a probability density and should account for all of the probability in the problem. Thus, under the circumstances where the random process can be irreversibly absorbed at the boundary, this should lead to the presence of a Dirac delta function in the fundamental solution at the boundary. However, such a feature is not present in the standard approach (Laplace transformation). Here we require that the total integrated probability is conserved. This yields a fundamental solution which, when appropriate, contains a term proportional to a Dirac delta function at the boundary. We determine the fundamental solution directly from the diffusion equation via spectral decomposition. We obtain exact expressions for the eigenfunctions, and when the fundamental solution contains a Dirac delta function at the boundary, every eigenfunction of the forward diffusion operator contains a delta function. We show how these combine to produce a weight of the delta function at the boundary which ensures the total integrated probability is conserved. The solution we present covers cases where parameters are time dependent, thereby greatly extending its applicability.
Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation.
Pattanayak, D; Yodh, A
1999-04-12
A new integro-differential equation for diffuse photon density waves (DPDW) is derived within the diffusion approximation. The new equation applies to inhomogeneous bounded turbid media. Interestingly, it does not contain any terms involving gradients of the light diffusion coefficient. The integro-differential equation for diffusive waves is used to develop a 3D-slice imaging algorithm based the on angular spectrum representation in the parallel plate geometry. The algorithm may be useful for near infrared optical imaging of breast tissue, and is applicable to other diagnostics such as ultrasound and microwave imaging.
Anomalous scaling of a scalar field advected by turbulence
Kraichnan, R.H.
1995-12-31
Recent work leading to deduction of anomalous scaling exponents for the inertial range of an advected passive field from the equations of motion is reviewed. Implications for other turbulence problems are discussed.
NASA Astrophysics Data System (ADS)
Dubrovskii, V. G.; Hervieu, Yu. Yu.
2014-09-01
In this work, we present a theoretical analysis of the diffusion-induced growth of "vapor-liquid-solid" nanowires, based on the stationary equations with generalized boundary conditions. We discuss why and how the earlier results are modified when the adatom chemical potential is discontinuous at the nanowire base. Several simplified models for the adatom diffusion flux are discussed, yielding the 1 /Rp radius dependence of the length, with p ranging from 0.5 to 2. The self-consistent approach is used to couple the diffusion transport with the kinetics of 2D nucleation under the droplet. This leads to a new growth equation that contains only two dimensional parameters and the power exponents p and q, where q=1 or 2 depends on the nucleus position. We show that this equation describes the size-dependent depression of the growth rate of narrow nanowires much better than the Gibbs-Thomson correction in several important cases. Overall, our equation fits very well the experimental data on the length-radius correlations of III-V and group IV nanowires obtained by different epitaxy techniques.
NASA Astrophysics Data System (ADS)
Liang, Yingjie; Chen, Wen; Magin, Richard L.
2016-07-01
Analytical solutions to the fractional diffusion equation are often obtained by using Laplace and Fourier transforms, which conveniently encode the order of the time and the space derivatives (α and β) as non-integer powers of the conjugate transform variables (s, and k) for the spectral and the spatial frequencies, respectively. This study presents a new solution to the fractional diffusion equation obtained using the Laplace transform and expressed as a Fox's H-function. This result clearly illustrates the kinetics of the underlying stochastic process in terms of the Laplace spectral frequency and entropy. The spectral entropy is numerically calculated by using the direct integration method and the adaptive Gauss-Kronrod quadrature algorithm. Here, the properties of spectral entropy are investigated for the cases of sub-diffusion and super-diffusion. We find that the overall spectral entropy decreases with the increasing α and β, and that the normal or Gaussian case with α = 1 and β = 2, has the lowest spectral entropy (i.e., less information is needed to describe the state of a Gaussian process). In addition, as the neighborhood over which the entropy is calculated increases, the spectral entropy decreases, which implies a spatial averaging or coarse graining of the material properties. Consequently, the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion. Future studies should be designed to examine changes of spectral entropy in physical, chemical and biological systems undergoing phase changes, chemical reactions and tissue regeneration.
Dynamical invariants in a non-Markovian quantum-state-diffusion equation
NASA Astrophysics Data System (ADS)
Luo, Da-Wei; Pyshkin, P. V.; Lam, Chi-Hang; Yu, Ting; Lin, Hai-Qing; You, J. Q.; Wu, Lian-Ao
2015-12-01
We find dynamical invariants for open quantum systems described by the non-Markovian quantum-state-diffusion (QSD) equation. In stark contrast to closed systems where the dynamical invariant can be identical to the system density operator, these dynamical invariants no longer share the equation of motion for the density operator. Moreover, the invariants obtained with a biorthonormal basis can be used to render an exact solution to the QSD equation and the corresponding non-Markovian dynamics without using master equations or numerical simulations. Significantly we show that we can apply these dynamical invariants to reverse engineering a Hamiltonian that is capable of driving the system to the target state, providing a different way to design control strategy for open quantum systems.
Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation
Barchielli, A.; Pellegrini, C.
2010-11-15
The ''correlated-projection technique'' has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.
Calculation of the neutron diffusion equation by using Homotopy Perturbation Method
NASA Astrophysics Data System (ADS)
Koklu, H.; Ersoy, A.; Gulecyuz, M. C.; Ozer, O.
2016-03-01
The distribution of the neutrons in a nuclear fuel element in the nuclear reactor core can be calculated by the neutron diffusion theory. It is the basic and the simplest approximation for the neutron flux function in the reactor core. In this study, the neutron flux function is obtained by the Homotopy Perturbation Method (HPM) that is a new and convenient method in recent years. One-group time-independent neutron diffusion equation is examined for the most solved geometrical reactor core of spherical, cubic and cylindrical shapes, in the frame of the HPM. It is observed that the HPM produces excellent results consistent with the existing literature.
Analytical solutions to matrix diffusion problems
Kekäläinen, Pekka
2014-10-06
We report an analytical method to solve in a few cases of practical interest the equations which have traditionally been proposed for the matrix diffusion problem. In matrix diffusion, elements dissolved in ground water can penetrate the porous rock surronuding the advective flow paths. In the context of radioactive waste repositories this phenomenon provides a mechanism by which the area of rock surface in contact with advecting elements is greatly enhanced, and can thus be an important delay mechanism. The cases solved are relevant for laboratory as well for in situ experiments. Solutions are given as integral representations well suited for easy numerical solution.
A Bloch-Torrey Equation for Diffusion in a Deforming Media
Rohmer, Damien; Gullberg, Grant T.
2006-12-29
Diffusion Tensor Magnetic Resonance Imaging (DTMRI)technique enables the measurement of diffusion parameters and therefore,informs on the structure of the biological tissue. This technique isapplied with success to the static organs such as brain. However, thediffusion measurement on the dynamically deformable organs such as thein-vivo heart is a complex problem that has however a great potential inthe measurement of cardiac health. In order to understand the behavior ofthe Magnetic Resonance (MR)signal in a deforming media, the Bloch-Torreyequation that leads the MR behavior is expressed in general curvilinearcoordinates. These coordinates enable to follow the heart geometry anddeformations through time. The equation is finally discretized andpresented in a numerical formulation using implicit methods, in order toget a stable scheme that can be applied to any smooth deformations.Diffusion process enables the link between the macroscopic behavior ofmolecules and themicroscopic structure in which they evolve. Themeasurement of diffusion in biological tissues is therefore of majorimportance in understanding the complex underlying structure that cannotbe studied directly. The Diffusion Tensor Magnetic ResonanceImaging(DTMRI) technique enables the measurement of diffusion parametersand therefore provides information on the structure of the biologicaltissue. This technique has been applied with success to static organssuch as the brain. However, diffusion measurement of dynamicallydeformable organs such as the in-vivo heart remains a complex problem,which holds great potential in determining cardiac health. In order tounderstand the behavior of the magnetic resonance (MR) signal in adeforming media, the Bloch-Torrey equation that defines the MR behavioris expressed in general curvilinear coordinates. These coordinates enableus to follow the heart geometry and deformations through time. Theequation is finally discretized and presented in a numerical formulationusing
A variational meshfree method for solving time-discrete diffusion equations
NASA Astrophysics Data System (ADS)
Krottje, Johannes K.
2006-08-01
A meshfree method is developed for solving time-discrete diffusion equations that arise in models in brain research. Important criteria for a suitable method are flexibility with respect to domain geometry and the ability to work with very small moving sources requiring easy refinement possibilities. One part of the work concerns a meshfree discretization of the modified Helmholtz equation based on the related minimization problem and a local least-squares function approximation. In a second part, a node choosing algorithm is presented that moves around randomly distributed nodes for optimizing the node distribution and varying the node density as needed. The method is illustrated by two numerical tests.
Introducing graded meshes in the numerical approximation of distributed-order diffusion equations
NASA Astrophysics Data System (ADS)
Morgado, M. L.; Rebelo, M.
2016-10-01
In this paper we deal with the numerical approximation of initial-boundary value problems to the diffusion equation with distributed order in time. As it is widely known, the solutions of fractional differential equations may present a singularity at t = 0 and therefore in these cases, standard finite difference schemes usually suffer a convergence order reduction with respect to time discretization. In order to overcome this, here we propose a finite difference scheme with a graded time mesh, constructed in such a way that the time step-size is smaller near the potential singular point. Numerical results are presented and compared with those obtained with finite difference schemes with uniform meshes.
The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit
Szoke, A.; Brooks, E. D.
2016-07-12
We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.
Sinc-Chebyshev Collocation Method for a Class of Fractional Diffusion-Wave Equations
Mao, Zhi; Xiao, Aiguo; Yu, Zuguo; Shi, Long
2014-01-01
This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed. PMID:24977177
Bailey, Teresa S. Adams, Marvin L. Yang, Brian Zika, Michael R.
2008-04-01
We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer's vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer's. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.
NASA Astrophysics Data System (ADS)
Jia, Jinhong; Wang, Hong
2015-07-01
Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O (N3) of computational work per time step and O (N2) of memory to store where N is the number of spatial grid points in the discretization. In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O (N) of memory and O (Nlog N) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.
Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains
NASA Astrophysics Data System (ADS)
Guo, Gang; Li, Kun; Wang, Yuhui
2015-01-01
We investigate the solutions of a modified fractional diffusion equation which has a secondary fractional time derivative acting on a diffusion operator. We obtain analytical solutions for the modified equation in the finite and semi-infinite domains subject to absorbing boundary conditions. Most of the results have been derived by using the Laplace transform, the Fourier Cosine transform, the Mellin transform and the properties of Fox H function. We show that the semi-infinite solution can be expressed using an infinite series of Fox H functions similar to the infinite case, while the finite solution requires double infinite series including both Fox H functions and trigonometric functions instead of one infinite series. The characteristic crossover between more and less anomalous behaviour as well as the effect of absorbing boundary conditions are clearly demonstrated according to the analytical solutions.
A comparison study of solving diffusion equations with different algorithm methods
NASA Astrophysics Data System (ADS)
Huang, Houbing; Wang, Xueyun; Ma, Xingqiao
2016-12-01
A comparison study for solving diffusion equations with different algorithm methods is studied to understand the oxygen vacancy defect transport under the electric field. We compare computational efficiency and numerical accuracy with different algorithm methods, including finite difference, finite element (COMSOL), and Fourier-Chebysev spectral methods. All the results of oxygen vacancy distribution under an electric field from different algorithm methods are compared with the analytical solution results. Two kinds of boundary conditions are used in solving diffusion equations and the absolute error of different methods are discussed. The main purpose of these results is to provide guidance for studying the role of point defect transport in the degradation and breakdown of devices.
Modeling the advection of discontinuous quantities in Geophysical flows using Particle Level Sets
NASA Astrophysics Data System (ADS)
Aleksandrov, V.; Samuel, H.; Evonuk, M.
2010-12-01
Advection is one of the major processes that commonly acts on various scales in nature (core formation, mantle convective stirring, multi-phase flows in magma chambers, salt diapirism ...). While this process can be modeled numerically by solving conservation equations, various geodynamic scenarios involve advection of quantities with sharp discontinuities. Unfortunately, in these cases modeling numerically pure advection becomes very challenging, in particular because sharp discontinuities lead to numerical instabilities, which prevent the local use of high order numerical schemes. Several approaches have been used in computational geodynamics in order to overcome this difficulty, with variable amounts of success. Despite the use of correcting filters or non-oscillatory, shock-preserving schemes, Eulerian (fixed grid) techniques generally suffer from artificial numerical diffusion. Lagrangian approaches (dynamic grids or particles) tend to be more popular in computational geodynamics because they are not prone to excessive numerical diffusion. However, these approaches are generally computationally expensive, especially in 3D, and can suffer from spurious statistical noise. As an alternative to these aforementioned approaches, we have applied a relatively recent Particle Level set method [Enright et al., 2002] for modeling advection of quantities with the presence of sharp discontinuities. We have tested this improved method, which combines the best of Eulerian and Lagrangian approaches, against well known benchmarks and classical Geodynamic flows. In each case the Particle Level Set method accuracy equals or is better than other Eulerian and Lagrangian methods, and leads to significantly smaller computational cost, in particular in three-dimensional flows, where the reduction of computational time for modeling advection processes is most needed.
NASA Astrophysics Data System (ADS)
Samuel, Henri
2010-05-01
Advection is one of the major processes that commonly acts on various scales in nature (core formation, mantle convective stirring, multi-phase flows in magma chambers, salt diapirism ...). While this process can be modeled numerically by solving conservation equations, various geodynamic scenarios involve advection of quantities with sharp discontinuities. Unfortunately, in these cases modeling numerically pure advection becomes very challenging, in particular because sharp discontinuities lead to numerical instabilities, which prevent the local use of high order numerical schemes. Several approaches have been used in computational geodynamics in order to overcome this difficulty, with variable amounts of success. Despite the use of correcting filters or non-oscillatory, shock-preserving schemes, Eulerian (fixed grid) techniques generally suffer from artificial numerical diffusion. Lagrangian approaches (dynamic grids or particles) tend to be more popular in computational geodynamics because they are not prone to excessive numerical diffusion. However, these approaches are generally computationally expensive, especially in 3D, and can suffer from spurious statistical noise. As an alternative to these aforementioned approaches, I have applied a relatively recent Particle Level set method [Enright et al., 2002] for modeling advection of quantities with the presence of sharp discontinuities. I have adapted this improved method, which combines the best of Eulerian and Lagrangian approaches, and I have tested it against well known benchmarks and classical Geodynamic flows. In each case the Particle Level Set method accuracy equals or is better than other Eulerian and Lagrangian methods, and leads to significantly smaller computational cost, in particular in three-dimensional flows, where the reduction of computational time for modeling advection processes is most needed.
Numerical study of the inverse problem for the diffusion-reaction equation using optimization method
NASA Astrophysics Data System (ADS)
Soboleva, O. V.; Brizitskii, R. V.
2016-04-01
The model of transfer of substance with mixed boundary condition is considered. The inverse extremum problem of identification of the main coefficient in a nonstationary diffusion-reaction equation is formulated. The numerical algorithm based on the Newton-method of nonlinear optimization and finite difference discretization for solving this extremum problem is developed and realized on computer. The results of numerical experiments are discussed.
NASA Astrophysics Data System (ADS)
Winkelmann, Stefanie; Schütte, Christof
2016-12-01
Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We consider the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments, and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented.
NASA Astrophysics Data System (ADS)
Guo, Gang; Chen, Bin; Zhao, Xinjun; Zhao, Fang; Wang, Quanmin
2015-09-01
We investigate the first passage time (FPT) distribution for accelerating subdiffusion governed by the modified fractional diffusion equation which has a secondary fractional time derivative acting on a diffusion operator. For the FPT problem subject to absorbing barrier condition, we obtain exact analytical expressions for the FPT distribution as well as its Laplace transform in the semi-infinite interval. Most of the results have been derived by using the Laplace transform, the Fourier Cosine transform, the Mellin transform and the properties of the Fox H-function. In contrast to the Laplace transform of the FPT distribution which can be expressed elegantly and neatly, the exact solution for the FPT distribution requires an infinite series of Fox H-functions instead of a single Fox H-function. Numerical result reveals that the crossover between the two distinct scaling regimes is apparent only when the discrepancy between the two diffusion exponents becomes more pronounced.
Accelerated molecular dynamics and equation-free methods for simulating diffusion in solids.
Deng, Jie; Zimmerman, Jonathan A.; Thompson, Aidan Patrick; Brown, William Michael; Plimpton, Steven James; Zhou, Xiao Wang; Wagner, Gregory John; Erickson, Lindsay Crowl
2011-09-01
Many of the most important and hardest-to-solve problems related to the synthesis, performance, and aging of materials involve diffusion through the material or along surfaces and interfaces. These diffusion processes are driven by motions at the atomic scale, but traditional atomistic simulation methods such as molecular dynamics are limited to very short timescales on the order of the atomic vibration period (less than a picosecond), while macroscale diffusion takes place over timescales many orders of magnitude larger. We have completed an LDRD project with the goal of developing and implementing new simulation tools to overcome this timescale problem. In particular, we have focused on two main classes of methods: accelerated molecular dynamics methods that seek to extend the timescale attainable in atomistic simulations, and so-called 'equation-free' methods that combine a fine scale atomistic description of a system with a slower, coarse scale description in order to project the system forward over long times.
Equation of Diffusion of a Composite Mixture into a Composite Medium
NASA Astrophysics Data System (ADS)
Kravchuk, A. S.; Kravchuk, A. I.; Popova, T. S.
2016-07-01
The equation of diffusion of a composite mixture into a composite medium has been obtained for the first time. The assumption used is that the macropoint of the medium, i.e., an elementary macrovolume, in which the statistical parameters of distribution of inhomogeneities coincide with the corresponding values assigned for the medium as a whole, is small compared to the geometric dimensions of the volume considered. The ″Reuss-Voigt fork″ has been obtained for determining the limits of the change in the diffusion coefficient. Thereafter the fork is narrowed to the ″Kravchuk-Tarasyuk fork.″ Effective diffusion coefficients are obtained as an arithmetic mean value of the Kravchuk-Tarasyuk fork. The found averaged physical parameters can be used in solving specific physical problems for inhomogeneous media.
NASA Astrophysics Data System (ADS)
Saxena, R. K.; Mathai, A. M.; Haubold, H. J.
2015-10-01
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function ϕ (x, t) which is a nonlinear function governing reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.
Three-dimensional analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities are derived in a systematic fashion. For homogeneous Neumann (total reflection), Dirichlet (total adsorpti...
Three-dimensional analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities are derived in a systematic fashion. For homogeneous Neumann (total reflection), Dirichlet (total adsorpti...
NASA Astrophysics Data System (ADS)
Sierakowski, Adam J.; Lukassen, Laura J.
2016-11-01
In the shear flow of non-Brownian particles, we describe the long-time diffusive processes stochastically using a Fokker-Planck equation. Previous work has indicated that a Fokker-Planck equation coupling the probability densities of position and velocity spaces may be appropriate for describing this phenomenon. The stochastic description, integrated over velocity space to obtain a reduced position-space Fokker-Planck equation, contains unknown space diffusion coefficients. In this work, we use the Physalis method for simulating disperse particle flows to verify the colored-noise velocity space model (an Ornstein-Uhlenbeck process) by comparing the simulated long-time diffusion rate with the diffusion rate proposed by the theory. We then use the simulated data to calculate the unknown space diffusion coefficients that appear in the reduced position-space Fokker-Planck equation and summarize the results. This study was partially supported by US NSF Grant CBET1335965.
Fast numerical solution for fractional diffusion equations by exponential quadrature rule
NASA Astrophysics Data System (ADS)
Zhang, Lu; Sun, Hai-Wei; Pang, Hong-Kui
2015-10-01
After spatial discretization to the fractional diffusion equation by the shifted Grünwald formula, it leads to a system of ordinary differential equations, where the resulting coefficient matrix possesses the Toeplitz-like structure. An exponential quadrature rule is employed to solve such a system of ordinary differential equations. The convergence by the proposed method is theoretically studied. In practical computation, the product of a Toeplitz-like matrix exponential and a vector is calculated by the shift-invert Arnoldi method. Meanwhile, the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method. Numerical results are given to demonstrate the efficiency of the proposed method.
A deterministic particle method for one-dimensional reaction-diffusion equations
NASA Technical Reports Server (NTRS)
Mascagni, Michael
1995-01-01
We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
NASA Astrophysics Data System (ADS)
Maas, Jan; Matthes, Daniel
2016-07-01
We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary d≥slant 1 . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.
Simulations of diffusion-reaction equations with implications to turbulent combustion modeling
NASA Technical Reports Server (NTRS)
Girimaji, Sharath S.
1993-01-01
An enhanced diffusion-reaction reaction system (DRS) is proposed as a statistical model for the evolution of multiple scalars undergoing mixing and reaction in an isotropic turbulence field. The DRS model is close enough to the scalar equations in a reacting flow that other statistical models of turbulent mixing that decouple the velocity field from scalar mixing and reaction (e.g. mapping closure model, assumed-pdf models) cannot distinguish the model equations from the original equations. Numerical simulations of DRS are performed for three scalars evolving from non-premixed initial conditions. A simple one-step reversible reaction is considered. The data from the simulations are used (1) to study the effect of chemical conversion on the evolution of scalar statistics, and (2) to evaluate other models (mapping-closure model, assumed multivariate beta-pdf model).
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains
NASA Astrophysics Data System (ADS)
Bonforte, Matteo; Vázquez, Juan Luis
2015-10-01
We investigate quantitative properties of the nonnegative solutions to the nonlinear fractional diffusion equation, , posed in a bounded domain, , with m > 1 for t > 0. As we use one of the most common definitions of the fractional Laplacian , 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. In addition, we obtain similar estimates for fractional semilinear elliptic equations. Either the standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.
NASA Astrophysics Data System (ADS)
Schlick, Conor P.; Christov, Ivan C.; Umbanhowar, Paul B.; Ottino, Julio M.; Lueptow, Richard M.
2013-05-01
We present an accurate and efficient computational method for solving the advection-diffusion equation in time-periodic chaotic flows. The method uses operator splitting, which allows the advection and diffusion steps to be treated independently. Taking advantage of flow periodicity, the advection step is solved using a mapping method, and diffusion is "added" discretely after each iteration of the advection map. This approach results in the construction of a composite mapping matrix over an entire period of the chaotic advection-diffusion process and provides a natural framework for the analysis of mixing. To test the approach, we consider two-dimensional time-periodic sine flow. By comparing the numerical solutions obtained by our method to reference solutions, we find qualitative agreement for large time steps (structure of concentration profile) and quantitative agreement for small time steps (low error). Further, we study the interplay between mixing through chaotic advection and mixing through diffusion leading to an analytical model for the evolution of the intensity of segregation with time. Additionally, we demonstrate that our operator splitting mapping approach can be readily extended to three dimensions.
Hellander, Andreas; Lawson, Michael J; Drawert, Brian; Petzold, Linda
2014-06-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity.
NASA Astrophysics Data System (ADS)
Plante, Ianik
2016-01-01
The exact Green's function of the diffusion equation (GFDE) is often considered to be the gold standard for the simulation of partially diffusion-controlled reactions. As the GFDE with angular dependency is quite complex, the radial GFDE is more often used. Indeed, the exact GFDE is expressed as a Legendre expansion, the coefficients of which are given in terms of an integral comprising Bessel functions. This integral does not seem to have been evaluated analytically in existing literature. While the integral can be evaluated numerically, the Bessel functions make the integral oscillate and convergence is difficult to obtain. Therefore it would be of great interest to evaluate the integral analytically. The first term was evaluated previously, and was found to be equal to the radial GFDE. In this work, the second term of this expansion was evaluated. As this work has shown that the first two terms of the Legendre polynomial expansion can be calculated analytically, it raises the question of the possibility that an analytical solution exists for the other terms.
NASA Astrophysics Data System (ADS)
Chen, Xueli; Yang, Defu; Qu, Xiaochao; Hu, Hao; Liang, Jimin; Gao, Xinbo; Tian, Jie
2012-06-01
Bioluminescence tomography (BLT) has been successfully applied to the detection and therapeutic evaluation of solid cancers. However, the existing BLT reconstruction algorithms are not accurate enough for cavity cancer detection because of neglecting the void problem. Motivated by the ability of the hybrid radiosity-diffusion model (HRDM) in describing the light propagation in cavity organs, an HRDM-based BLT reconstruction algorithm was provided for the specific problem of cavity cancer detection. HRDM has been applied to optical tomography but is limited to simple and regular geometries because of the complexity in coupling the boundary between the scattering and void region. In the provided algorithm, HRDM was first applied to three-dimensional complicated and irregular geometries and then employed as the forward light transport model to describe the bioluminescent light propagation in tissues. Combining HRDM with the sparse reconstruction strategy, the cavity cancer cells labeled with bioluminescent probes can be more accurately reconstructed. Compared with the diffusion equation based reconstruction algorithm, the essentiality and superiority of the HRDM-based algorithm were demonstrated with simulation, phantom and animal studies. An in vivo gastric cancer-bearing nude mouse experiment was conducted, whose results revealed the ability and feasibility of the HRDM-based algorithm in the biomedical application of gastric cancer detection.
NASA Technical Reports Server (NTRS)
Lenardic, A.; Kaula, W. M.
1993-01-01
Effective numerical treatment of multicomponent viscous flow problems involving the advection of sharp interfaces between materials of differing physical properties requires correction techniques to prevent spurious diffusion and dispersion. We develop a particular algorithm, based on modern shock-capture techniques, employing a two-step nonlinear method. The first step involves the global application of a high-order upwind scheme to a hyperbolic advection equation used to model the distribution of distinct material components in a flow field. The second step is corrective and involves the application of a global filter designed to remove dispersion errors that result from the advection of discontinuities (e.g., material interfaces) by high-order, minimally dissipative schemes. The filter introduces no additional diffusion error. Nonuniform viscosity across a material interface is allowed for by the implementation of a compositionally weighted-inverse interface viscosity scheme. The combined method approaches the optimal accuracy of modern shock-capture techniques with a minimal increase in computational time and memory. A key advantage of this method is its simplicity to incorporate into preexisting codes be they finite difference, element, or volume of two or three dimensions.
A Mapping method for mixing with diffusion
NASA Astrophysics Data System (ADS)
Schlick, Conor P.; Christov, Ivan C.; Umbanhowar, Paul B.; Ottino, Julio M.; Lueptow, Richard M.
2012-11-01
We present an accurate and efficient computational method for solving the advection-diffusion equation in time-periodic chaotic flows. The method uses operator splitting which allows advection and diffusion steps to be treated independently. Taking advantage of flow periodicity, the advection step is solved with a mapping method, and diffusion is added discretely after each iteration of the advection map. This approach allows for a ``composite'' mapping matrix to be constructed for an entire period of a chaotic advection-diffusion process, which provides a natural approach to the spectral analysis of mixing. To test the approach, we consider the two-dimensional time-periodic sine flow. When compared to the exact solution for this simple velocity field, the operator splitting method exhibits qualitative agreement (overall concentration structure) for large time steps and is quantitatively accurate (average and maximum error) for small time steps. We extend the operator splitting approach to three-dimensional chaotic flows. Funded by NSF Grant CMMI-1000469. Present affiliation: Princeton University. Supported by NSF Grant DMS-1104047.
Efficient mass transport by optical advection
Kajorndejnukul, Veerachart; Sukhov, Sergey; Dogariu, Aristide
2015-01-01
Advection is critical for efficient mass transport. For instance, bare diffusion cannot explain the spatial and temporal scales of some of the cellular processes. The regulation of intracellular functions is strongly influenced by the transport of mass at low Reynolds numbers where viscous drag dominates inertia. Mimicking the efficacy and specificity of the cellular machinery has been a long time pursuit and, due to inherent flexibility, optical manipulation is of particular interest. However, optical forces are relatively small and cannot significantly modify diffusion properties. Here we show that the effectiveness of microparticle transport can be dramatically enhanced by recycling the optical energy through an effective optical advection process. We demonstrate theoretically and experimentally that this new advection mechanism permits an efficient control of collective and directional mass transport in colloidal systems. The cooperative long-range interaction between large numbers of particles can be optically manipulated to create complex flow patterns, enabling efficient and tunable transport in microfluidic lab-on-chip platforms. PMID:26440069
Thermal diffusion segregation in granular binary mixtures described by the Enskog equation
NASA Astrophysics Data System (ADS)
Garzó, Vicente
2011-05-01
The diffusion induced by a thermal gradient in a granular binary mixture is analyzed here in the context of the (inelastic) Enskog equation. Although the Enskog equation neglects velocity correlations among particles that are about to collide, it retains the spatial correlations arising from volume exclusion effects and thus is expected to be applicable for moderate densities. In the steady state with gradients only along a given direction, a segregation criterion is obtained from the thermal diffusion factor Λ by measuring the amount of segregation parallel to the thermal gradient. As expected, the sign of the factor Λ provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the parameters of the mixture (the masses and sizes of particles, concentration, solid volume fraction and coefficients of restitution). The form of the phase diagrams for the BNE/RBNE transition is illustrated in detail for several systems, with special emphasis on the significant role played by the inelasticity of collisions. In particular, an effect already found in dilute gases (segregation in a binary mixture of identical masses and sizes but different coefficients of restitution) is extended to dense systems. A comparison with recent computer simulation results reveals good qualitative agreement at the level of the thermal diffusion factor. The present analysis generalizes to arbitrary concentration previous theoretical results derived in the tracer limit case.
Simplified approach for calculating moments of action for linear reaction-diffusion equations.
Ellery, Adam J; Simpson, Matthew J; McCue, Scott W; Baker, Ruth E
2013-11-01
The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time underapproximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.
Simulation of a fast diffuse optical tomography system based on radiative transfer equation
NASA Astrophysics Data System (ADS)
Motevalli, S. M.; Payani, A.
2016-12-01
Studies show that near-infrared (NIR) light (light with wavelength between 700nm and 1300nm) undergoes two interactions, absorption and scattering, when it penetrates a tissue. Since scattering is the predominant interaction, the calculation of light distribution in the tissue and the image reconstruction of absorption and scattering coefficients are very complicated. Some analytical and numerical methods, such as radiative transport equation and Monte Carlo method, have been used for the simulation of light penetration in tissue. Recently, some investigators in the world have tried to develop a diffuse optical tomography system. In these systems, NIR light penetrates the tissue and passes through the tissue. Then, light exiting the tissue is measured by NIR detectors placed around the tissue. These data are collected from all the detectors and transferred to the computational parts (including hardware and software), which make a cross-sectional image of the tissue after performing some computational processes. In this paper, the results of the simulation of an optical diffuse tomography system are presented. This simulation involves two stages: a) Simulation of the forward problem (or light penetration in the tissue), which is performed by solving the diffusion approximation equation in the stationary state using FEM. b) Simulation of the inverse problem (or image reconstruction), which is performed by the optimization algorithm called Broyden quasi-Newton. This method of image reconstruction is faster compared to the other Newton-based optimization algorithms, such as the Levenberg-Marquardt one.
Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift
Zhao, Lei; Yue, Xingye; Waxman, David
2013-01-01
A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size. PMID:23749318
NASA Astrophysics Data System (ADS)
Korneev, V. G.
2016-11-01
Efficiency of the error control of numerical solutions of partial differential equations entirely depends on the two factors: accuracy of an a posteriori error majorant and the computational cost of its evaluation for some test function/vector-function plus the cost of the latter. In the paper consistency of an a posteriori bound implies that it is the same in the order with the respective unimprovable a priori bound. Therefore, it is the basic characteristic related to the first factor. The paper is dedicated to the elliptic diffusion-reaction equations. We present a guaranteed robust a posteriori error majorant effective at any nonnegative constant reaction coefficient (r.c.). For a wide range of finite element solutions on a quasiuniform meshes the majorant is consistent. For big values of r.c. the majorant coincides with the majorant of Aubin (1972), which, as it is known, for relatively small r.c. (< ch -2 ) is inconsistent and looses its sense at r.c. approaching zero. Our majorant improves also some other majorants derived for the Poisson and reaction-diffusion equations.
Diffusive Barrier and Getter Under Waste Packages VA Reference Design Feature Evaluations
MacNeil, K.
1999-05-24
This technical document evaluates those aspects of the diffusive barrier and getter features which have the potential for enhancing the performance of the Viability Assessment Reference Design and are also directly related to the key attributes for the repository safety strategy of that design. The effects of advection, hydrodynamic dispersion, and diffusion on the radionuclide migration rates through the diffusive barrier were determined through the application of the one-dimensional, advection/dispersion/diffusion equation. The results showed that because advective flow described by the advection-dispersion equation dominates, the diffusive barrier feature alone would not be effective in retarding migration of radiocuclides. However, if the diffusive barrier were combined with one or more features that reduced the potential for advection, then transport of radionuclides would be dominated by diffusion and their migration from the EBS would be impeded. Apatite was chosen as the getter material used for this report. Two getter configurations were developed, Case 1 and Case 2. As in the evaluation of the diffusive barrier, the effects of advection, hydrodynamic dispersion, and diffusion on the migration of radionuclides through the getter are evaluated. However, in addition to these mechanisms, the one-dimensional advection/dispersion/diffusion model is modified to include the effect of sorption on radionuclide migration rates through the sorptive medium (getter). As a result of sorption, the longitudinal dispersion coefficient, and the average linear velocity are effectively reduced by the retardation factor. The retardation factor is a function of the getter material's dry bulk density, sorption coefficient and moisture content. The results of the evaluation showed that a significant delay in breakthrough through the getter can be achieved if the thickness of the getter barrier is increased.
Retinal Image Enhancement Using Robust Inverse Diffusion Equation and Self-Similarity Filtering
Fu, Shujun; Xu, Lingzhong; Zhao, Kun; Zhang, Caiming
2016-01-01
As a common ocular complication for diabetic patients, diabetic retinopathy has become an important public health problem in the world. Early diagnosis and early treatment with the help of fundus imaging technology is an effective control method. In this paper, a robust inverse diffusion equation combining a self-similarity filtering is presented to detect and evaluate diabetic retinopathy using retinal image enhancement. A flux corrected transport technique is used to control diffusion flux adaptively, which eliminates overshoots inherent in the Laplacian operation. Feature preserving denoising by the self-similarity filtering ensures a robust enhancement of noisy and blurry retinal images. Experimental results demonstrate that this algorithm can enhance important details of retinal image data effectively, affording an opportunity for better medical interpretation and subsequent processing. PMID:27388503
Diffusion-equation representations of landform evolution in the simplest circumstances: Appendix C
Hanks, Thomas C.
2009-01-01
Most of us here know that the diffusion equation has also been used to describe the evolution through time of scarp-like landforms, including fault scarps, shoreline scarps, or a set of marine terraces. The methods, models, and data employed in such studies have been described in the literature many times over the past 25 years. For most situations, everything you will ever need (or want) to know can be found in Hanks et al. (1984) and Hanks (2000), the latter being a review of numerous studies of the 1980s and 1990s and a summary of available estimates of the mass diffusivity κ. The geometric parameterization of scarp-like landforms is shown in Figure 1.
Shear-induced diffusion of non-Brownian suspensions using a colored noise Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Lukassen, Laura; Oberlack, Martin
2013-11-01
In the Literature, shear-induced diffusion resulting from hydrodynamic interactions between particles, is described as a long-time diffusion. In contrast to the well-known Brownian diffusion which is described by a white noise force, several authors report that the former type of diffusion exhibits the particularity of a much longer correlation time of velocities. Further, Fokker-Planck equations describing this process of shear-induced diffusion have mostly been derived in position space. We present a considerably extended framework of the shear-induced diffusion problem, which essentially relies on the Markov process assumption under the consideration of long correlation times. Applying the mathematical machinery of Markov processes and Fokker-Planck equations, we conclude that this process may only be properly modelled by a Fokker-Planck approach if written in both position and velocity space. With this complementation we observe, that the long correlation times enter as a colored noise velocity. As a result, the Fokker-Planck equation also needs to be extended and we derive the Fokker-Planck equation for the shear-induced diffusion problem following the definitions of a colored noise Fokker-Planck equation. Graduate School of Excellence Computational Engineering.
LAYER DEPENDENT ADVECTION IN CMAQ
The advection methods used in CMAQ require that the Courant-Friedrichs-Lewy (CFL) condition be satisfied for numerical stability and accuracy. In CMAQ prior to version 4.3, the ADVSTEP algorithm established CFL-safe synchronization and advection timesteps that were uniform throu...
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
Bakhtiyari-Ramezani, M. Alinejad, N.; Mahmoodi, J.
2015-11-15
In the fusion devices, ions, H atoms, and H{sub 2} molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H{sub 2} molecules, and desorption of the recombined H{sub 2} molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.
An algorithm for solving the fractional convection diffusion equation with nonlinear source term
NASA Astrophysics Data System (ADS)
Momani, Shaher
2007-10-01
In this paper an algorithm based on Adomian's decomposition method is developed to approximate the solution of the nonlinear fractional convection-diffusion equation {∂αu}/{∂tα}={∂2u}/{∂x2}-c{∂u}/{∂x}+Ψ(u)+f(x,t),0
A finite volume method for two-sided fractional diffusion equations on non-uniform meshes
NASA Astrophysics Data System (ADS)
Simmons, Alex; Yang, Qianqian; Moroney, Timothy
2017-04-01
We derive a finite volume method for two-sided fractional diffusion equations with Riemann-Liouville derivatives in one spatial dimension. The method applies to non-uniform meshes, with arbitrary nodal spacing. The discretisation utilises the integral definition of the fractional derivatives, and we show that it leads to a diagonally dominant matrix representation, and a provably stable numerical scheme. Being a finite volume method, the numerical scheme is fully conservative, and the ability to locally refine the mesh can produce solutions with more accuracy for the same number of nodes compared to a uniform mesh, as we demonstrate numerically.
Multi-Dimensional Asymptotically Stable 4th Order Accurate Schemes for the Diffusion Equation
NASA Technical Reports Server (NTRS)
Abarbanel, Saul; Ditkowski, Adi
1996-01-01
An algorithm is presented which solves the multi-dimensional diffusion equation on co mplex shapes to 4th-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions fail.
NASA Astrophysics Data System (ADS)
Chakrabarti, Anindya S.
2016-01-01
We present a model of technological evolution due to interaction between multiple countries and the resultant effects on the corresponding macro variables. The world consists of a set of economies where some countries are leaders and some are followers in the technology ladder. All of them potentially gain from technological breakthroughs. Applying Lotka-Volterra (LV) equations to model evolution of the technology frontier, we show that the way technology diffuses creates repercussions in the partner economies. This process captures the spill-over effects on major macro variables seen in the current highly globalized world due to trickle-down effects of technology.
On the Long Time Simulation of Reaction-Diffusion Equations with Delay
Zhang, Chengjian
2014-01-01
For a consistent numerical method to be practically useful, it is widely accepted that it must preserve the asymptotic stability of the original continuous problem. However, in this study, we show that it may lead to unreliable numerical solutions in long time simulation even if a classical numerical method gives a larger stability region than that of the original continuous problem. Some numerical experiments on the reaction-diffusion equations with delay are presented to confirm our findings. Finally, some open problems on the subject are proposed. PMID:24672296
Scalable implicit methods for reaction-diffusion equations in two and three space dimensions
Veronese, S.V.; Othmer, H.G.
1996-12-31
This paper describes the implementation of a solver for systems of semi-linear parabolic partial differential equations in two and three space dimensions. The solver is based on a parallel implementation of a non-linear Alternating Direction Implicit (ADI) scheme which uses a Cartesian grid in space and an implicit time-stepping algorithm. Various reordering strategies for the linearized equations are used to reduce the stride and improve the overall effectiveness of the parallel implementation. We have successfully used this solver for large-scale reaction-diffusion problems in computational biology and medicine in which the desired solution is a traveling wave that may contain rapid transitions. A number of examples that illustrate the efficiency and accuracy of the method are given here; the theoretical analysis will be presented.
NASA Astrophysics Data System (ADS)
Huang, Juntao; Hu, Zexi; Yong, Wen-An
2016-04-01
In this paper, we present a kind of second-order curved boundary treatments for the lattice Boltzmann method solving two-dimensional convection-diffusion equations with general nonlinear Robin boundary conditions. The key idea is to derive approximate boundary values or normal derivatives on computational boundaries, with second-order accuracy, by using the prescribed boundary condition. Once the approximate information is known, the second-order bounce-back schemes can be perfectly adopted. Our boundary treatments are validated with a number of numerical examples. The results show the utility of our boundary treatments and very well support our theoretical predications on the second-order accuracy thereof. The idea is quite universal. It can be directly generalized to 3-dimensional problems, multiple-relaxation-time models, and the Navier-Stokes equations.
Modeling velocity in gradient flows with coupled-map lattices with advection.
Lind, Pedro G; Corte-Real, João; Gallas, Jason A C
2002-07-01
We introduce a simple model to investigate large scale behavior of gradient flows based on a lattice of coupled maps which, in addition to the usual diffusive term, incorporates advection, as an asymmetry in the coupling between nearest neighbors. This diffusive-advective model predicts traveling patterns to have velocities obeying the same scaling as wind velocities in the atmosphere, regarding the advective parameter as a sort of geostrophic wind. In addition, the velocity and wavelength of traveling wave solutions are studied. In general, due to the presence of advection, two regimes are identified: for strong diffusion the velocity varies linearly with advection, while for weak diffusion a power law is found with a characteristic exponent proportional to the diffusion.
Kwok, Sau Fa
2012-08-15
A Langevin equation with multiplicative white noise and its corresponding Fokker-Planck equation are considered in this work. From the Fokker-Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. - Highlights: Black-Right-Pointing-Pointer Fokker-Planck equation corresponding to the Langevin equation with mul- tiplicative white noise is presented. Black-Right-Pointing-Pointer Transformation of diffusion processes into the Wiener process in different prescriptions is provided. Black-Right-Pointing-Pointer The prescription parameter is associated with the growth rate for a Gompertz-type model.
Theory of advection-driven long range biotic transport
Technology Transfer Automated Retrieval System (TEKTRAN)
We propose a simple mechanistic model to examine the effects of advective flow on the spread of fungal diseases spread by wind-blown spores. The model is defined by a set of two coupled non-linear partial differential equations for spore densities. One equation describes the long-distance advectiv...
PLUS family: a set of computer programs to evaluate analytical solutions of the diffusion equation
Montan, D.N.
1986-02-01
This report is intended to describe, document and provide instructions for the use of a set of computer programs commonly referred to as the PLUS family. These programs were designed to numerically evaluate simple analytic solutions of the diffusion equation. The original member of the family, a program called PLUS, was written to provide calculational support for a study of the storage of nuclear waste in geological media. Originally, PLUS computed temperture changes at points in space and time due to a finite length line source. The need to handle arrays of sources led to modifications. To this end, PLUS was changed to subroutine status and new programs were written: CELERY, to control the I/O chores; STALKS, with storage space for an arbitrary array of sources; MIDNITE, to produce thermal contours in and/or about the array. The newest members of the family, TWIGS, and DAYLITE were created to do some of the things (uniform arrays of identical sources) that STALKS and MIDNITE could do but without the need for additional storage space. The original design of these programs was for thermal calculations, however, diffusion equation is used in the study of a number of other field, for example, fluid flow in porous media (hydrology, petroleum reservoirs), and the PLUS family may find other possible homes. 9 refs., 13 figs.
NASA Astrophysics Data System (ADS)
Yuste, S. B.; Abad, E.; Escudero, C.
2016-09-01
We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time τ (t ) , which we define via the relation τ ˙=1 /a2 , where a (t ) is the expansion scale factor. If the medium expansion is driven by a power law [a (t ) ∝tγ with γ >0 ] , then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent γ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value γ =1 /2 . The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long-time limit.
Derlet, P. M.; Gilbert, M. R.; Dudarev, S. L.
2011-10-01
Nanoscale prismatic loops are modeled via a partial stochastic differential equation that describes an overdamped continuum elastic string, with a view to describing both the internal and collective dynamics of the loop as a function of temperature. Within the framework of the Langevin equation, expressions are derived that relate the empirical parameters of the model, the friction per unit length, and the elastic stiffness per unit length, to observables that can be obtained directly via molecular-dynamics simulations of interstitial or vacancy prismatic loop mobility. The resulting expressions naturally exhibit the properties that the collective diffusion coefficient of the loop (i) scales inversely with the square root of the number of interstitials, a feature that has been observed in both atomistic simulation and in situ TEM investigations of loop mobility, and (ii) the collective diffusion coefficient is not at all dependent on the internal interactions within the loop, thus qualitatively rationalizing past simulation results showing that the characteristic migration energy barrier is comparable to that of a single interstitial, and cluster migration is a result of individual (but correlated) interstitial activity.
Tönjes, Ralf; Blasius, Bernd
2009-01-01
The Kuramoto phase-diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction-diffusion system. Synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity, the synchronization frequency. The formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings. Concentric waves in synchronization also occur in heterogeneous systems, where the local frequencies are distributed randomly. We present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system. The nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second-order perturbation terms. We apply the theory to simple topologies, like a line or sphere, and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium. We show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions. By changing a system parameter, the synchronized state may become quasidegenerate. We demonstrate how perturbation theory fails at such a critical point.
Yuste, S B; Abad, E; Escudero, C
2016-09-01
We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time τ(t), which we define via the relation τ[over ̇]=1/a^{2}, where a(t) is the expansion scale factor. If the medium expansion is driven by a power law [a(t)∝t^{γ} with γ>0], then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent γ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value γ=1/2. The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long-time limit.
Classical non-Markovian Boltzmann equation
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
Classical non-Markovian Boltzmann equation
NASA Astrophysics Data System (ADS)
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
NASA Astrophysics Data System (ADS)
Anand, S.; Mayya, Y. S.
2011-08-01
Coagulation and condensation/evaporation combined with atmospheric dispersion are the main processes responsible for the evolution of aerosol particle size distributions and number concentrations emitted from localized sources. A crucial question is: what fraction of freshly emitted particles survive intra-coagulation effect to persist in the atmosphere and become available for further interaction with background aerosols?. The difficulty in estimating this quantity, designated as the number survival fraction, arises due chiefly to the joint action of atmospheric diffusion with nonlinear coagulation effects which are computationally intensive to handle. We provide a simplified approach to evaluate this quantity in the context of instantaneous (puff) and continuous (plume) releases based on a reduction of the respective coagulation-diffusion equations under the assumption of a constant coagulation kernel ( K). The condensation/evaporation processes, being number conserving, are not included in the study. The approach consists of constructing moment equations for the evolution of number concentration and variance of the spatial extension of puff or plume in terms of either time or downstream distance. The puff model, applicable to instantaneous releases is solved within a 3-D, spherically symmetric framework, under an additional assumption of a constant diffusion coefficient ( D) which renders itself amenable to a closed form solution that provides a benchmark for developing the solution to the plume model. The latter case, corresponding to continuous releases, is discussed within a 2-D framework under the assumptions of constant advection velocity ( U) and space dependent diffusion coefficient expressed in terms of turbulent energy dissipation rate ( ɛ). The study brings out the special effect of the coagulation-induced flattening of the spatial concentration profiles because of which particle sizes will be larger at the centre of a Gaussian puff. For a puff of
Singh, Brajesh K; Srivastava, Vineet K
2015-04-01
The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.
Singh, Brajesh K.; Srivastava, Vineet K.
2015-01-01
The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639
Threshold thickness for applying diffusion equation in thin tissue optical imaging
NASA Astrophysics Data System (ADS)
Zhang, Yunyao; Zhu, Jingping; Cui, Weiwen; Nie, Wei; Li, Jie; Xu, Zhenghong
2014-08-01
We investigated the suitability of the semi-infinite model of the diffusion equation when using diffuse optical imaging (DOI) to image thin tissues with double boundaries. Both diffuse approximation and Monte Carlo methods were applied to simulate light propagation in the thin tissue model with variable optical parameters and tissue thicknesses. A threshold value of the tissue thickness was defined as the minimum thickness in which the semi-infinite model exhibits the same reflected intensity as that from the double-boundary model and was generated as the final result. In contrast to our initial hypothesis that all optical properties would affect the threshold thickness, our results show that only absorption coefficient is the dominant parameter and the others are negligible. The threshold thickness decreases from 1 cm to 4 mm as the absorption coefficient grows from 0.01 mm-1 to 0.2 mm-1. A look-up curve was derived to guide the selection of the appropriate model during the optical diagnosis of thin tissue cancers. These results are useful in guiding the development of the endoscopic DOI for esophageal, cervical and colorectal cancers, among others.
Studies of the accuracy of time integration methods for reaction-diffusion equations
NASA Astrophysics Data System (ADS)
Ropp, David L.; Shadid, John N.; Ober, Curtis C.
2004-03-01
In this study we present numerical experiments of time integration methods applied to systems of reaction-diffusion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the competing component time scales. Nearly balanced systems can produce a significant coupling of the physical mechanisms and introduce a slow dynamical time scale of interest. These problems provide a challenging test for this evaluation and tend to reveal subtle differences between the various methods. The methods we consider include first- and second-order semi-implicit, fully implicit, and operator-splitting techniques. The test problems include a prototype propagating nonlinear reaction-diffusion wave, a non-equilibrium radiation-diffusion system, a Brusselator chemical dynamics system and a blow-up example. In this evaluation we demonstrate a "split personality" for the operator-splitting methods that we consider. While operator-splitting methods often obtain very good accuracy, they can also manifest a serious degradation in accuracy due to stability problems.
Sound energy decay in coupled spaces using a parametric analytical solution of a diffusion equation.
Luizard, Paul; Polack, Jean-Dominique; Katz, Brian F G
2014-05-01
Sound field behavior in performance spaces is a complex phenomenon. Issues regarding coupled spaces present additional concerns due to sound energy exchanges. Coupled volume concert halls have been of increasing interest in recent decades because this architectural principle offers the possibility to modify the hall's acoustical environment in a passive way by modifying the coupling area. Under specific conditions, the use of coupled reverberation chambers can provide non-exponential sound energy decay in the main room, resulting in both high clarity and long reverberation which are antagonistic parameters in a single volume room. Previous studies have proposed various sound energy decay models based on statistical acoustics and diffusion theory. Statistical acoustics assumes a perfectly uniform sound field within a given room whereas measurements show an attenuation of energy with increasing source-receiver distance. While previously proposed models based on diffusion theory use numerical solvers, the present study proposes a heuristic model of sound energy behavior based on an analytical solution of the commonly used diffusion equation and physically justified approximations. This model is validated by means of comparisons to scale model measurements and numerical geometrical acoustics simulations, both applied to the same simple concert hall geometry.
Fast multigrid solution of the advection problem with closed characteristics
Yavneh, I.; Venner, C.H.; Brandt, A.
1996-12-31
The numerical solution of the advection-diffusion problem in the inviscid limit with closed characteristics is studied as a prelude to an efficient high Reynolds-number flow solver. It is demonstrated by a heuristic analysis and numerical calculations that using upstream discretization with downstream relaxation-ordering and appropriate residual weighting in a simple multigrid V cycle produces an efficient solution process. We also derive upstream finite-difference approximations to the advection operator, whose truncation terms approximate {open_quotes}physical{close_quotes} (Laplacian) viscosity, thus avoiding spurious solutions to the homogeneous problem when the artificial diffusivity dominates the physical viscosity.
Estimation of molecular diffusivity in aqueous solution of acetonitrile by the Wilke-Chang equation.
Miyabe, Kanji
2011-10-01
It was tried to estimate the molecular diffusivity (D(m)) of solutes in the mixtures of acetonitrile (ACN) and water by the Wilke-Chang equation. Although the information about association coefficient (α) is necessary for the calculation, it has never been proposed for ACN. The value of α was estimated as 1.37 from D(m) of benzene in ACN at 303 K experimentally measured by the peak parking method. The values of α, i.e. 2.6, 1.9, 1.5, and 1.0, which have respectively been proposed for four solvents, i.e. water, methanol, ethanol, and benzene, were correlated with two physico-chemical parameters of the solvents, i.e. solubility parameter and E(T) value. The α value for ACN was plotted around the two correlations, indicating its appropriateness. The values of D(m) calculated by the Wilke-Chang equation using the α value for ACN were compared with those measured by the peak parking method and the Aris-Taylor method in aqueous solutions of ACN. The mean square deviation of the estimation of D(m) was calculated as 8.8 and 14%. It was demonstrated that the Wilke-Chang equation can be used for estimating D(m) with a reasonable accuracy in the mixtures consisting of ACN and water.
High Order Semi-Lagrangian Advection Scheme
NASA Astrophysics Data System (ADS)
Malaga, Carlos; Mandujano, Francisco; Becerra, Julian
2014-11-01
In most fluid phenomena, advection plays an important roll. A numerical scheme capable of making quantitative predictions and simulations must compute correctly the advection terms appearing in the equations governing fluid flow. Here we present a high order forward semi-Lagrangian numerical scheme specifically tailored to compute material derivatives. The scheme relies on the geometrical interpretation of material derivatives to compute the time evolution of fields on grids that deform with the material fluid domain, an interpolating procedure of arbitrary order that preserves the moments of the interpolated distributions, and a nonlinear mapping strategy to perform interpolations between undeformed and deformed grids. Additionally, a discontinuity criterion was implemented to deal with discontinuous fields and shocks. Tests of pure advection, shock formation and nonlinear phenomena are presented to show performance and convergence of the scheme. The high computational cost is considerably reduced when implemented on massively parallel architectures found in graphic cards. The authors acknowledge funding from Fondo Sectorial CONACYT-SENER Grant Number 42536 (DGAJ-SPI-34-170412-217).
Thermally driven advection for radioxenon transport from an underground nuclear explosion
NASA Astrophysics Data System (ADS)
Sun, Yunwei; Carrigan, Charles R.
2016-05-01
Barometric pumping is a ubiquitous process resulting in migration of gases in the subsurface that has been studied as the primary mechanism for noble gas transport from an underground nuclear explosion (UNE). However, at early times following a UNE, advection driven by explosion residual heat is relevant to noble gas transport. A rigorous measure is needed for demonstrating how, when, and where advection is important. In this paper three physical processes of uncertain magnitude (oscillatory advection, matrix diffusion, and thermally driven advection) are parameterized by using boundary conditions, system properties, and source term strength. Sobol' sensitivity analysis is conducted to evaluate the importance of all physical processes influencing the xenon signals. This study indicates that thermally driven advection plays a more important role in producing xenon signals than oscillatory advection and matrix diffusion at early times following a UNE, and xenon isotopic ratios are observed to have both time and spatial dependence.
NASA Astrophysics Data System (ADS)
Cho, Guangsup; Uhm, Han Sup
2016-10-01
The time-dependent solution of diffusion equation by the Fourier integration provides the axial diffusion velocity of a plasma packet, which is a key element of the plasma propagation in a plasma jet operated by the several tens of kHz. The plasma diffusion velocity is higher than the order of un ˜ 10 m/s at a high electric-field region of plasma generation and it is about the order of un ˜ 10 m/s at the plasma column of a low field region in a jet-nozzle inside. Meanwhile, the diffusion velocity is slower than the order of un ˜ 10 m/s in the open-air space where the plasma density flattens due to its radial expansion. Using these diffusion velocity data, the group-velocity of plasma diffusion wave-packet is given by ug ˜ cs2/un, a combination of the diffusion velocity un and the acoustic velocity cs. The experimental results of the plasma propagation can be verified with the plasma propagation in a form of the wave-packet whose propagation velocity is 104 m/s in a tube inside and is as fast as 105 m/s in the open-air space, thereby reconfirming that the theory of a plasma diffusion-wave is the origin of the plasma propagation in a plasma jet.
NASA Astrophysics Data System (ADS)
Guner, Ozkan; Bekir, Ahmet; Unsal, Omer; Cevikel, Adem C.
2017-01-01
In this paper, we pay attention to the analytical method named, ansatz method for finding the exact solutions of the variable-coefficient modified KdV equation and variable coefficient diffusion-reaction equation. As a result the singular 1-soliton solution is obtained. These solutions are important for the explanation of some practical physical problems. The obtained results show that these methods provides a powerful mathematical tool for solving nonlinear equations with variable coefficients. This method can be extended to solve other variable coefficient nonlinear partial differential equations.
NASA Technical Reports Server (NTRS)
Srivastava, R. C.; Coen, J. L.
1992-01-01
The traditional explicit growth equation has been widely used to calculate the growth and evaporation of hydrometeors by the diffusion of water vapor. This paper reexamines the assumptions underlying the traditional equation and shows that large errors (10-30 percent in some cases) result if it is used carelessly. More accurate explicit equations are derived by approximating the saturation vapor-density difference as a quadratic rather than a linear function of the temperature difference between the particle and ambient air. These new equations, which reduce the error to less than a few percent, merit inclusion in a broad range of atmospheric models.
Slowing Allee effect versus accelerating heavy tails in monostable reaction diffusion equations
NASA Astrophysics Data System (ADS)
Alfaro, Matthieu
2017-02-01
We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between ‘no acceleration’ and ’acceleration’. This implies in particular that, for tails exponentially unbounded but lighter than algebraic, acceleration never occurs in the presence of an Allee effect. This is in sharp contrast with the KPP situation [20]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.
Frank, T D; Friedrich, R; Beek, P J
2006-11-01
We address two questions that are central to understanding human motor control variability: what kind of dynamical components contribute to motor control variability (i.e., deterministic and/or random ones), and how are those components structured? To this end, we derive a stochastic order parameter equation for isometric force production from experimental data using drift-diffusion estimates. We show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity. A structural analysis reveals that the deterministic component consists of a linear control loop, while the random component involves a noise source that scales with force output. In addition, we present evidence for the existence of a subject-independent overall noise level of human isometric force production.
Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
NASA Astrophysics Data System (ADS)
Jamelot, Erell; Ciarlet, Patrick
2013-05-01
Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method for the mixed neutron diffusion equations, discretized with Raviart-Thomas-Nédélec finite elements. This method is based on the Schwarz iterative algorithm with Robin interface conditions to handle communications. We analyse this method from the continuous point of view to the discrete point of view, and we give some numerical results in a realistic highly heterogeneous 3D configuration. Computations are carried out with the MINOS solver of the APOLLO3® neutronics code. APOLLO3 is a registered trademark in France.
Numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory
NASA Technical Reports Server (NTRS)
Ramos, J. I.
1987-01-01
A review of numerical methods for one-dimensional reaction-diffusion equations arising in combustion theory is presented. The methods reviewed include explicit, implicit, quasi-linearization, time linearization, operator-splitting, random walk and finite-element techniques and methods of lines. Adaptive and nonadaptive procedures are also reviewed. These techniques are applied first to solve two model problems which have exact traveling wave solutions with which the numerical results can be compared. This comparison is performed in terms of both the wave profile and computed wave speed. It is shown that the computed wave speed is not a good indicator of the accuracy of a particular method. A fourth-order time-linearized, Hermitian compact operator technique is found to be the most accurate method for a variety of time and space sizes.
A Radiation Chemistry Code Based on the Greens Functions of the Diffusion Equation
NASA Technical Reports Server (NTRS)
Plante, Ianik; Wu, Honglu
2014-01-01
Ionizing radiation produces several radiolytic species such as.OH, e-aq, and H. when interacting with biological matter. Following their creation, radiolytic species diffuse and chemically react with biological molecules such as DNA. Despite years of research, many questions on the DNA damage by ionizing radiation remains, notably on the indirect effect, i.e. the damage resulting from the reactions of the radiolytic species with DNA. To simulate DNA damage by ionizing radiation, we are developing a step-by-step radiation chemistry code that is based on the Green's functions of the diffusion equation (GFDE), which is able to follow the trajectories of all particles and their reactions with time. In the recent years, simulations based on the GFDE have been used extensively in biochemistry, notably to simulate biochemical networks in time and space and are often used as the "gold standard" to validate diffusion-reaction theories. The exact GFDE for partially diffusion-controlled reactions is difficult to use because of its complex form. Therefore, the radial Green's function, which is much simpler, is often used. Hence, much effort has been devoted to the sampling of the radial Green's functions, for which we have developed a sampling algorithm This algorithm only yields the inter-particle distance vector length after a time step; the sampling of the deviation angle of the inter-particle vector is not taken into consideration. In this work, we show that the radial distribution is predicted by the exact radial Green's function. We also use a technique developed by Clifford et al. to generate the inter-particle vector deviation angles, knowing the inter-particle vector length before and after a time step. The results are compared with those predicted by the exact GFDE and by the analytical angular functions for free diffusion. This first step in the creation of the radiation chemistry code should help the understanding of the contribution of the indirect effect in the
Concentration polarization, surface currents, and bulk advection in a microchannel
NASA Astrophysics Data System (ADS)
Nielsen, Christoffer P.; Bruus, Henrik
2014-10-01
We present a comprehensive analysis of salt transport and overlimiting currents in a microchannel during concentration polarization. We have carried out full numerical simulations of the coupled Poisson-Nernst-Planck-Stokes problem governing the transport and rationalized the behavior of the system. A remarkable outcome of the investigations is the discovery of strong couplings between bulk advection and the surface current; without a surface current, bulk advection is strongly suppressed. The numerical simulations are supplemented by analytical models valid in the long channel limit as well as in the limit of negligible surface charge. By including the effects of diffusion and advection in the diffuse part of the electric double layers, we extend a recently published analytical model of overlimiting current due to surface conduction.
NASA Astrophysics Data System (ADS)
Lin, Jin-Sheng; Hildemann, Lynn M.
A generalized mathematical scheme is developed to simulate the turbulent dispersion of pollutants which are adsorbed or deposit to the ground. The scheme is an analytical (exact) solution of the atmospheric diffusion equation with height-dependent wind speed and eddy diffusivities, and with a Robin-type boundary condition at the ground. Unlike published solutions of similar problems where complex or non-programmable (e.g., hypergeometric or Kummer) functions are obtained, the analytical solution proposed herein consists of two previously derived Green's functions (modified Bessel functions) expressed in an integral form that is amenable to numerical integration. In the case of invariant wind speed and turbulent eddies with height (i.e., Gaussian deposition plume), the solution reduces to an equivalent well-known heat conduction solution. The physical behavior represented by the Green's functions comprising the solution can be interpreted. This generalized scheme can be modified further to account for inversion effects or other meteorological conditions. The solution derived is useful for examining the accuracy and performance of sophisticated numerical dispersion models, and is particularly suitable for modeling the transport of pollutants undergoing strong surface adsorption or high depositional losses.
Non-dispersive carrier transport in molecularly doped polymers and the convection-diffusion equation
NASA Astrophysics Data System (ADS)
Tyutnev, A. P.; Parris, P. E.; Saenko, V. S.
2015-08-01
We reinvestigate the applicability of the concept of trap-free carrier transport in molecularly doped polymers and the possibility of realistically describing time-of-flight (TOF) current transients in these materials using the classical convection-diffusion equation (CDE). The problem is treated as rigorously as possible using boundary conditions appropriate to conventional time of flight experiments. Two types of pulsed carrier generation are considered. In addition to the traditional case of surface excitation, we also consider the case where carrier generation is spatially uniform. In our analysis, the front electrode is treated as a reflecting boundary, while the counter electrode is assumed to act either as a neutral contact (not disturbing the current flow) or as an absorbing boundary at which the carrier concentration vanishes. As expected, at low fields transient currents exhibit unusual behavior, as diffusion currents overwhelm drift currents to such an extent that it becomes impossible to determine transit times (and hence, carrier mobilities). At high fields, computed transients are more like those typically observed, with well-defined plateaus and sharp transit times. Careful analysis, however, reveals that the non-dispersive picture, and predictions of the CDE contradict both experiment and existing disorder-based theories in important ways, and that the CDE should be applied rather cautiously, and even then only for engineering purposes.
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions.
Hu, Zexi; Huang, Juntao; Yong, Wen-An
2016-04-01
In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015)JCTPAH0021-999110.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions
NASA Astrophysics Data System (ADS)
Hu, Zexi; Huang, Juntao; Yong, Wen-An
2016-04-01
In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015), 10.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.
NASA Astrophysics Data System (ADS)
Tang, Min; Wang, Yihong
2017-02-01
In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of 1012. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular 1 / ɛ terms can be replaced by O (1) terms after the integration, which yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed. Uniform convergence with respect to the anisotropy strength 1 / ɛ can be observed numerically and the condition number does not scale with the anisotropy.
NASA Astrophysics Data System (ADS)
Kekenes-Huskey, P. M.; Gillette, A. K.; McCammon, J. A.
2014-05-01
The macroscopic diffusion constant for a charged diffuser is in part dependent on (1) the volume excluded by solute "obstacles" and (2) long-range interactions between those obstacles and the diffuser. Increasing excluded volume reduces transport of the diffuser, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. We previously demonstrated [P. M. Kekenes-Huskey et al., Biophys. J. 105, 2130 (2013)] using homogenization theory that the configuration of molecular-scale obstacles can both hinder diffusion and induce diffusional anisotropy for small ions. As the density of molecular obstacles increases, van der Waals (vdW) and electrostatic interactions between obstacle and a diffuser become significant and can strongly influence the latter's diffusivity, which was neglected in our original model. Here, we extend this methodology to include a fixed (time-independent) potential of mean force, through homogenization of the Smoluchowski equation. We consider the diffusion of ions in crowded, hydrophilic environments at physiological ionic strengths and find that electrostatic and vdW interactions can enhance or depress effective diffusion rates for attractive or repulsive forces, respectively. Additionally, we show that the observed diffusion rate may be reduced independent of non-specific electrostatic and vdW interactions by treating obstacles that exhibit specific binding interactions as "buffers" that absorb free diffusers. Finally, we demonstrate that effective diffusion rates are sensitive to distribution of surface charge on a globular protein, Troponin C, suggesting that the use of molecular structures with atomistic-scale resolution can account for electrostatic influences on substrate transport. This approach offers new insight into the influence of molecular-scale, long-range interactions on transport of charged species, particularly for diffusion-influenced signaling events occurring in crowded
Kekenes-Huskey, P. M.; Gillette, A. K.; McCammon, J. A.
2014-05-07
The macroscopic diffusion constant for a charged diffuser is in part dependent on (1) the volume excluded by solute “obstacles” and (2) long-range interactions between those obstacles and the diffuser. Increasing excluded volume reduces transport of the diffuser, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. We previously demonstrated [P. M. Kekenes-Huskey et al., Biophys. J. 105, 2130 (2013)] using homogenization theory that the configuration of molecular-scale obstacles can both hinder diffusion and induce diffusional anisotropy for small ions. As the density of molecular obstacles increases, van der Waals (vdW) and electrostatic interactions between obstacle and a diffuser become significant and can strongly influence the latter's diffusivity, which was neglected in our original model. Here, we extend this methodology to include a fixed (time-independent) potential of mean force, through homogenization of the Smoluchowski equation. We consider the diffusion of ions in crowded, hydrophilic environments at physiological ionic strengths and find that electrostatic and vdW interactions can enhance or depress effective diffusion rates for attractive or repulsive forces, respectively. Additionally, we show that the observed diffusion rate may be reduced independent of non-specific electrostatic and vdW interactions by treating obstacles that exhibit specific binding interactions as “buffers” that absorb free diffusers. Finally, we demonstrate that effective diffusion rates are sensitive to distribution of surface charge on a globular protein, Troponin C, suggesting that the use of molecular structures with atomistic-scale resolution can account for electrostatic influences on substrate transport. This approach offers new insight into the influence of molecular-scale, long-range interactions on transport of charged species, particularly for diffusion-influenced signaling events occurring in
Kekenes-Huskey, P M; Gillette, A K; McCammon, J A
2014-05-07
The macroscopic diffusion constant for a charged diffuser is in part dependent on (1) the volume excluded by solute "obstacles" and (2) long-range interactions between those obstacles and the diffuser. Increasing excluded volume reduces transport of the diffuser, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. We previously demonstrated [P. M. Kekenes-Huskey et al., Biophys. J. 105, 2130 (2013)] using homogenization theory that the configuration of molecular-scale obstacles can both hinder diffusion and induce diffusional anisotropy for small ions. As the density of molecular obstacles increases, van der Waals (vdW) and electrostatic interactions between obstacle and a diffuser become significant and can strongly influence the latter's diffusivity, which was neglected in our original model. Here, we extend this methodology to include a fixed (time-independent) potential of mean force, through homogenization of the Smoluchowski equation. We consider the diffusion of ions in crowded, hydrophilic environments at physiological ionic strengths and find that electrostatic and vdW interactions can enhance or depress effective diffusion rates for attractive or repulsive forces, respectively. Additionally, we show that the observed diffusion rate may be reduced independent of non-specific electrostatic and vdW interactions by treating obstacles that exhibit specific binding interactions as "buffers" that absorb free diffusers. Finally, we demonstrate that effective diffusion rates are sensitive to distribution of surface charge on a globular protein, Troponin C, suggesting that the use of molecular structures with atomistic-scale resolution can account for electrostatic influences on substrate transport. This approach offers new insight into the influence of molecular-scale, long-range interactions on transport of charged species, particularly for diffusion-influenced signaling events occurring in crowded
NASA Astrophysics Data System (ADS)
Ding, Hongxia; Chen, Shangbin; Zeng, Shuai; Zeng, Shaoqun; Liu, Qian; Luo, Qingming
2008-12-01
Spreading depression (SD) shows as propagating suppression of electrical activity, which relates with migraine and focal cerebral ischaemia. The putative mechanism of SD is the reaction-diffusion hypothesis involving potassium ions. In part inspired by optical imaging of two SD waves collision, we aimed to show the merged and large wavefront but not annihilation during collision by experimental and computational study. This paper modified Reggia et al established bistable equation with recovery to compute and visualize SD. Firstly, the media tissue of SD was assumed as one-dimensional continuum. The Crank-Nicholson method was used to solve the modified equations with recovery term. Then, the computation results were extended to two-dimensional space by symmetry. One individual SD was visualized as a concentric wave initiating from the stimulation point. The mergence but not annihilation of two colliding waves of SD was demonstrated. In addition, the dynamics of SD depending on the parameters was studied and presented. The results allied SD with the emerging concepts of volume transmission. This work not only supplied a paradigm to compute and visualize SD but also became a tool to explore the mechanisms of SD.
Abi Mansour, Andrew; Al Ghoul, Mazen
2011-08-01
In this paper we investigate the dynamics of front propagation in the family of reactions (nA + mB (k)→ C) with initially segregated reactants in one dimension using hyperbolic reaction-diffusion equations with the mean-field approximation for the reaction rate. This leads to different dynamics than those predicted by their parabolic counterpart. Using perturbation techniques, we focus on the initial and intermediate temporal behavior of the center and width of the front and derive the different time scaling exponents. While the solution of the parabolic system yields a short time scaling as t(1/2) for the front center, width, and global reaction rate, the hyperbolic system exhibits linear scaling for those quantities. Moreover, those scaling laws are shown to be independent of the stoichiometric coefficients n and m. The perturbation results are compared with the full numerical solutions of the hyperbolic equations. The crossover time at which the hyperbolic regime crosses over to the parabolic regime is also studied. Conditions for static and moving fronts are also derived and numerically validated.
Numerical Calculation and Exergy Equations of Spray Heat Exchanger Attached to a Main Fan Diffuser
NASA Astrophysics Data System (ADS)
Cui, H.; Wang, H.; Chen, S.
2015-04-01
In the present study, the energy depreciation rule of spray heat exchanger, which is attached to a main fan diffuser, is analyzed based on the second law of thermodynamics. Firstly, the exergy equations of the exchanger are deduced. The equations are numerically calculated by the fourth-order Runge-Kutta method, and the exergy destruction is quantitatively effected by the exchanger structure parameters, working fluid (polluted air, i.e., PA; sprayed water, i.e., SW) initial state parameters and the ambient reference parameters. The results are showed: (1) heat transfer is given priority to latent transfer at the bottom of the exchanger, and heat transfer of convection and is equivalent to that of condensation in the upper. (2) With the decrease of initial temperature of SW droplet, the decrease of PA velocity or the ambient reference temperature, and with the increase of a SW droplet size or initial PA temperature, exergy destruction both increase. (3) The exergy efficiency of the exchanger is 72.1 %. An approach to analyze the energy potential of the exchanger may be provided for engineering designs.
Bifurcation analysis of brown tide by reaction-diffusion equation using finite element method
Kawahara, Mutsuto; Ding, Yan
1997-03-01
In this paper, we analyze the bifurcation of a biodynamics system in a two-dimensional domain by virtue of reaction-diffusion equations. The discretization method in space is the finite element method. The computational algorithm for an eigenspectrum is described in detail. On the basis of an analysis of eigenspectra according to Helmholtz`s equation, the discrete spectra in regards to the physical variables are numerically obtained in two-dimensional space. In order to investigate this mathematical model in regards to its practical use, we analyzed the stability of two cases, i.e., hydranth regeneration in the marine hydroid Tubularia and a brown tide in a harbor in Japan. By evaluating the stability according to the linearized stability definition, the critical parameters for outbreaks of brown tide can be theoretically determined. In addition, results for the linear combination of eigenspectrum coincide with the distribution of the observed brown tide. Its periodic characteristic was also verified. 10 refs., 8 figs., 5 tabs.
BUOYANT ADVECTION OF GASES IN UNSATURATED SOIL
Seely, Gregory E.; Falta, Ronald W.; Hunt, James R.
2010-01-01
In unsaturated soil, methane and volatile organic compounds can significantly alter the density of soil gas and induce buoyant gas flow. A series of laboratory experiments was conducted in a two-dimensional, homogeneous sand pack with gas permeabilities ranging from 110 to 3,000 darcy. Pure methane gas was injected horizontally into the sand and steady-state methane profiles were measured. Experimental results are in close agreement with a numerical model that represents the advective and diffusive components of methane transport. Comparison of simulations with and without gravitational acceleration permits identification of conditions where buoyancy dominates methane transport. Significant buoyant flow requires a Rayleigh number greater than 10 and an injected gas velocity sufficient to overcome dilution by molecular diffusion near the source. These criteria allow the extension of laboratory results to idealized field conditions for methane as well as denser-than-air vapors produced by volatilizing nonaqueous phase liquids trapped in unsaturated soil. PMID:20396624
Wright, A. Kent; Duncan, Robert C.; Beekman, Karen A.
1973-01-01
The rotational diffusion coefficients R1 and R3 for ellipsoids of revolution are shown to represent another pair of hydrodynamic data to obtain size and shape with theories by Sadron and Scheraga-Mandelkern. An iterative numerical technique is presented which allows the semiaxes to be determined from the Perrin equations for rotational diffusion constants. The use of this inversion technique is illustrated by application to literature data from dielectric dispersion studies. PMID:4726879
Drawert, Brian; Lawson, Michael J.; Petzold, Linda; Khammash, Mustafa
2010-01-01
We have developed a computational framework for accurate and efficient simulation of stochastic spatially inhomogeneous biochemical systems. The new computational method employs a fractional step hybrid strategy. A novel formulation of the finite state projection (FSP) method, called the diffusive FSP method, is introduced for the efficient and accurate simulation of diffusive transport. Reactions are handled by the stochastic simulation algorithm. PMID:20170209
SU-E-T-270: Diffusion Synthetic Acceleration for Linear Boltzmann Transport Equation
Chen, G; Hong, X; Gao, H
2015-06-15
Purpose: Linear Boltzmann transport equation (LBTE) is as accurate as the Monte Carlo method (MC) for dose calculation in photon/particle therapy (LBTE is a deterministic and Eulerian formulation and MC is a statistical and Lagrangian description). An advantage of LBTE is that numerous acceleration techniques can be utilized for acceleration. This work is to explore the acceleration of LBTE via diffusion synthetic acceleration (DSA). Methods: For simplicity, two-dimensional, steady-state, and within-group LBTE is considered with two angular dimensions and two spatial dimensions. The discrete ordinate method is developed for solving this integro-differential equation. The angular variables are discretized using a level-symmetric quadrature set on the unit sphere. The spatial variables are discretized on the structured grid based on the diamond scheme. The source-iteration method (SI) is used to solve the discretized system.Since SI is slow in optically thick and highly scattering regime. DSA is developed to accelerate SI. The motivation for DSA is that diffusion equation (DE) is a good approximation of LBTE in the above regime. However, DE is much cheaper than LBTE computationally since DE only involves spatial variables. Thus, in each DSA iteration, DSA adds to the SI step a computationally-negligible DE step, i.e., to first solve DE with the SI residual as source term, and then compensate the SI solution with DE solution. Results: DSA was benchmarked and compared with SI. The difference between two methods was within 0.12% which verifies the accuracy of DSA, while DSA demonstrated the great advantage in speed, e.g., the reduction of iteration number to 6% and 4% respectively for cases with 100 and 1,000 scattering-absorption ratio that commonly occur in clinical dose calculation. Conclusion: DSA has been developed as one of many possible means for accelerating the numerical solver of LBTE for dose calculation. The authors were partially supported by the NSFC
Roxin, Alex; Ledberg, Anders
2008-01-01
The response behaviors in many two-alternative choice tasks are well described by so-called sequential sampling models. In these models, the evidence for each one of the two alternatives accumulates over time until it reaches a threshold, at which point a response is made. At the neurophysiological level, single neuron data recorded while monkeys are engaged in two-alternative choice tasks are well described by winner-take-all network models in which the two choices are represented in the firing rates of separate populations of neurons. Here, we show that such nonlinear network models can generally be reduced to a one-dimensional nonlinear diffusion equation, which bears functional resemblance to standard sequential sampling models of behavior. This reduction gives the functional dependence of performance and reaction-times on external inputs in the original system, irrespective of the system details. What is more, the nonlinear diffusion equation can provide excellent fits to behavioral data from two-choice decision making tasks by varying these external inputs. This suggests that changes in behavior under various experimental conditions, e.g. changes in stimulus coherence or response deadline, are driven by internal modulation of afferent inputs to putative decision making circuits in the brain. For certain model systems one can analytically derive the nonlinear diffusion equation, thereby mapping the original system parameters onto the diffusion equation coefficients. Here, we illustrate this with three model systems including coupled rate equations and a network of spiking neurons. PMID:18369436
Constraints upon water advection in sediments of the Mariana Trough
Abbott, D.H.; Menke, W.; Morin, R.
1983-02-10
Thermal gradient measurements, consolidation tests, and pore water compositions from the Mariana Trough imply that water is moving through the sediments in areas with less than about 100 m of sediment cover. The maximum advection rates implied by the thermal measurements and consolidation tests may be as high as 10/sup -5/ cm s/sup -1/ but are most commonly in the range of 1 to 5 x 10/sup -6/ cm s/sup -1/. Theoretical calculations of the effect of the highest advection rates upon carbonate dissolution indicate that dissolution may be impeded or enhanced (depending upon the direction of flow) by a factor of 2 to 5 times the rate for diffusion alone. The average percentage of carbonate is consistently higher in two cores from the area with no advection or upward advection than the average percentage of carbonate in three cores from the area with downward advection. This increase in average amount of carbonate in cores with upward moving water or no movement cannot be attributed solely to differences in water depth or in amount of terrigenous dilution. If the sediment column acts as a passive boundary layer, then the water velocities necessary to affect chemical gradients of silica are in the range 10/sup -9/ to 10/sup -10/ cm s/sup -1/. However, if dissolution of silica occurs within the sediment column, then the advection velocities needed to affect chemical gradients are at least 3 x 10/sup -8/ cm s/sup -1/ and may be as high as 3 x 10/sup -6/ cm s/sup -1/. This order of magnitude increase in advection velocities when chemical reactions occur within the sediments is probably applicable to other cations in addition to silica. If so, then the advection velocities needed to affect heat flow (>10/sup -8/ cm s/sup -1/) and pore water chemical gradients are much nearer in magnitude than previously assumed.
A Streamline-Upwind Model for Filling Front Advection in Powder Injection Moulding
NASA Astrophysics Data System (ADS)
Larsen, Guillaume; Cheng, Zhi Qiang; Barriere, Thierry; Liu, Bao Sheng; Gelin, Jean-Claude
2010-06-01
The filling process of powder injection molding is modeled by the flows of two variably adjacent domains in the mold cavity. The feedstock is filled into the cavity while the air is expelled out by the injected feedstock [1]. Eulerian description is adopted. The filling patterns are determined by the solution of an advection equation, governed by the velocity field in both the feedstock flow and air flow [2]. In the real physics, the advance of filling front depends mainly on the flow of feedstock that locates behind the front. The flow of air in front of the injected material plays in fact no meaningful effect. However, the actual algorithm for solution of the advection equation takes equally the importance for both the flow of viscous feedstock and that of the slight air. Under such a condition, the injection flow of feedstock in simulation may be misdirected unrealistically by the velocity field in the air portion of the mold cavity. To correct this defect, an upwind scheme is proposed to reinforce the effect of upwind flow and reduce the effect of downstream flow. The present paper involves the investigation of an upwind algorithm for simulation of the filling state during powder injection molding. A Petrov-Galerkin upwind based method (SUPG) is adopted for numerical simulation of the transport equation instead of the Taylor-Galerkin method in previous work. In the proposed implementation of the Streamline-Upwind/Petrov-Galerkin (SUPG) approach. A stabilization method is used to prevent oscillations in the convection-dominated problems. It consists in the introduction of an artificial diffusion in streamline direction. Suitable modification of the test function is the important issue. It ensures the stable simulation of filling process and results in the more realistic prediction of filling patterns. The implementation of upwind scheme in mould filling state simulation, based on an advection equation and the whole velocity field of feedstock and air flow, makes
Striated populations in disordered environments with advection
NASA Astrophysics Data System (ADS)
Chotibut, Thiparat; Nelson, David R.; Succi, Sauro
2017-01-01
Growth in static and controlled environments such as a Petri dish can be used to study the spatial population dynamics of microorganisms. However, natural populations such as marine microbes experience fluid advection and often grow up in heterogeneous environments. We investigate a generalized Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation describing single species population subject to a constant flow field and quenched random spatially inhomogeneous growth rates with a fertile overall growth condition. We analytically and numerically demonstrate that the non-equilibrium steady-state population density develops a flow-driven striation pattern. The striations are highly asymmetric with a longitudinal correlation length that diverges linearly with the flow speed and a transverse correlation length that approaches a finite velocity-independent value. Linear response theory is developed to study the statistics of the steady states. Theoretical predictions show excellent agreement with the numerical steady states of the generalized FKPP equation obtained from Lattice Boltzmann simulations. These findings suggest that, although the growth disorder can be spatially uncorrelated, correlated population structures with striations emerge naturally at sufficiently strong advection.
Bachand, P.A.M.; S. Bachand,; Fleck, Jacob A.; Anderson, Frank E.; Windham-Myers, Lisamarie
2014-01-01
The current state of science and engineering related to analyzing wetlands overlooks the importance of transpiration and risks data misinterpretation. In response, we developed hydrologic and mass budgets for agricultural wetlands using electrical conductivity (EC) as a natural conservative tracer. We developed simple differential equations that quantify evaporation and transpiration rates using flowrates and tracer concentrations atwetland inflows and outflows. We used two ideal reactormodel solutions, a continuous flowstirred tank reactor (CFSTR) and a plug flow reactor (PFR), to bracket real non-ideal systems. From those models, estimated transpiration ranged from 55% (CFSTR) to 74% (PFR) of total evapotranspiration (ET) rates, consistent with published values using standard methods and direct measurements. The PFR model more appropriately represents these nonideal agricultural wetlands in which check ponds are in series. Using a fluxmodel, we also developed an equation delineating the root zone depth at which diffusive dominated fluxes transition to advective dominated fluxes. This relationship is similar to the Peclet number that identifies the dominance of advective or diffusive fluxes in surface and groundwater transport. Using diffusion coefficients for inorganic mercury (Hg) and methylmercury (MeHg) we calculated that during high ET periods typical of summer, advective fluxes dominate root zone transport except in the top millimeters below the sediment–water interface. The transition depth has diel and seasonal trends, tracking those of ET. Neglecting this pathway has profound implications: misallocating loads along different hydrologic pathways; misinterpreting seasonal and diel water quality trends; confounding Fick's First Law calculations when determining diffusion fluxes using pore water concentration data; and misinterpreting biogeochemicalmechanisms affecting dissolved constituent cycling in the root zone. In addition,our understanding of internal
Bachand, P A M; Bachand, S; Fleck, J; Anderson, F; Windham-Myers, L
2014-06-15
The current state of science and engineering related to analyzing wetlands overlooks the importance of transpiration and risks data misinterpretation. In response, we developed hydrologic and mass budgets for agricultural wetlands using electrical conductivity (EC) as a natural conservative tracer. We developed simple differential equations that quantify evaporation and transpiration rates using flow rates and tracer concentrations at wetland inflows and outflows. We used two ideal reactor model solutions, a continuous flow stirred tank reactor (CFSTR) and a plug flow reactor (PFR), to bracket real non-ideal systems. From those models, estimated transpiration ranged from 55% (CFSTR) to 74% (PFR) of total evapotranspiration (ET) rates, consistent with published values using standard methods and direct measurements. The PFR model more appropriately represents these non-ideal agricultural wetlands in which check ponds are in series. Using a flux model, we also developed an equation delineating the root zone depth at which diffusive dominated fluxes transition to advective dominated fluxes. This relationship is similar to the Peclet number that identifies the dominance of advective or diffusive fluxes in surface and groundwater transport. Using diffusion coefficients for inorganic mercury (Hg) and methylmercury (MeHg) we calculated that during high ET periods typical of summer, advective fluxes dominate root zone transport except in the top millimeters below the sediment-water interface. The transition depth has diel and seasonal trends, tracking those of ET. Neglecting this pathway has profound implications: misallocating loads along different hydrologic pathways; misinterpreting seasonal and diel water quality trends; confounding Fick's First Law calculations when determining diffusion fluxes using pore water concentration data; and misinterpreting biogeochemical mechanisms affecting dissolved constituent cycling in the root zone. In addition, our understanding of
NASA Astrophysics Data System (ADS)
Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C.
2016-08-01
Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.
A Minimum-Residual Finite Element Method for the Convection-Diffusion Equation
2013-05-01
outflow. (a) Solution u (b) u (c) Error representation e Fig. 5.6. Advection skew to mesh on a uniform quadratic 32 ⇥ 32 grid of triangles. The fine...Report (SAR) 18. NUMBER OF PAGES 20 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified...er. Specifically, the SUPG method can be interpreted as a modification of standard test functions1, biasing them in the upwind direction based on mesh
NASA Astrophysics Data System (ADS)
Haspot, Boris
2016-06-01
We consider the compressible Navier-Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier-Stokes equations using solutions of the pressureless Navier-Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667-674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are {C^{∞}} on {(0,T)× {R}N} for any {T > 0}. Finally we show the convergence of the global weak solution of compressible Navier-Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to {μ(ρ)=ρ^{α}} with {α > 1}. Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density {ρ0}.
Chang, Justin; Karra, Satish; Nakshatrala, Kalyana B.
2016-07-26
It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, usedmore » for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Furthermore, our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.« less
Chang, Justin; Karra, Satish; Nakshatrala, Kalyana B.
2016-07-26
It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Furthermore, our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.
A remark on the Beale-Kato-Majda criterion for the 3D MHD equations with zero magnetic diffusivity
NASA Astrophysics Data System (ADS)
Gala, Sadek; Ragusa, Maria Alessandra
2016-06-01
In this work, we show that a smooth solution of the 3D MHD equations with zero magnetic diffusivity in the whole space ℝ3 breaks down if and only if a certain norm of the magnetic field blows up at the same time.
A generalized mathematical scheme is developed to simulate the turbulent dispersion of pollutants which are adsorbed or deposit to the ground. The scheme is an analytical (exact) solution of the atmospheric diffusion equation with height-dependent wind speed a...
The intrinsic periodic fluctuation of forest: a theoretical model based on diffusion equation
NASA Astrophysics Data System (ADS)
Zhou, J.; Lin, G., Sr.
2015-12-01
Most forest dynamic models predict the stable state of size structure as well as the total basal area and biomass in mature forest, the variation of forest stands are mainly driven by environmental factors after the equilibrium has been reached. However, although the predicted power-law size-frequency distribution does exist in analysis of many forest inventory data sets, the estimated distribution exponents are always shifting between -2 and -4, and has a positive correlation with the mean value of DBH. This regular pattern can not be explained by the effects of stochastic disturbances on forest stands. Here, we adopted the partial differential equation (PDE) approach to deduce the systematic behavior of an ideal forest, by solving the diffusion equation under the restricted condition of invariable resource occupation, a periodic solution was gotten to meet the variable performance of forest size structure while the former models with stable performance were just a special case of the periodic solution when the fluctuation frequency equals zero. In our results, the number of individuals in each size class was the function of individual growth rate(G), mortality(M), size(D) and time(T), by borrowing the conclusion of allometric theory on these parameters, the results perfectly reflected the observed "exponent-mean DBH" relationship and also gave a logically complete description to the time varying form of forest size-frequency distribution. Our model implies that the total biomass of a forest can never reach a stable equilibrium state even in the absence of disturbances and climate regime shift, we propose the idea of intrinsic fluctuation property of forest and hope to provide a new perspective on forest dynamics and carbon cycle research.
Wang, Y.
2013-07-01
Nonlinear diffusion acceleration (NDA) can improve the performance of a neutron transport solver significantly especially for the multigroup eigenvalue problems. The high-order transport equation and the transport-corrected low-order diffusion equation form a nonlinear system in NDA, which can be solved via a Picard iteration. The consistency of the correction of the low-order equation is important to ensure the stabilization and effectiveness of the iteration. It also makes the low-order equation preserve the scalar flux of the high-order equation. In this paper, the consistent correction for a particular discretization scheme, self-adjoint angular flux (SAAF) formulation with discrete ordinates method (S{sub N}) and continuous finite element method (CFEM) is proposed for the multigroup neutron transport equation. Equations with the anisotropic scatterings and a void treatment are included. The Picard iteration with this scheme has been implemented and tested with RattleS{sub N}ake, a MOOSE-based application at INL. Convergence results are presented. (authors)
NASA Astrophysics Data System (ADS)
Zhao, Haixia; Gao, Jinghuai; Peng, Jigen
2017-01-01
The frequency-dependent seismic anomalies related to hydrocarbon reservoirs have lately attracted wide interest. The diffusive-viscous model was proposed to explain these anomalies. When an incident diffusive-viscous wave strikes a boundary between two different media, it is reflected and transmitted. The equation for the reflection coefficient is quite complex and laborious, so it does not provide an intuitive understanding of how different amplitude relates to the parameters of the media and how variation of a particular parameter affects the reflection coefficient. In this paper, we firstly derive a two-term (intercept-gradient) and three-term (intercept-gradient-curvature) approximation to the reflection coefficient of the plane diffusive-viscous wave without any assumptions. Then, we study the limitations of the obtained approximations by comparing the approximate value of the reflection coefficient with its exact value. Our results show that the two approximations match well with the exact solutions within the incident angle of 35°. Finally, we analyze the effects of diffusive and viscous attenuation parameters, velocity and density in the diffusive-viscous wave equation on the intercept, gradient and curvature terms in the approximations. The results show that the diffusive attenuation parameter has a big impact on them, while the viscous attenuation parameter is insensitive to them; the velocity and density have a significant influence on the normal reflections and they distinctly affect the intercept, gradient and curvature term at lower acoustic impedance.
Rényi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations
NASA Astrophysics Data System (ADS)
Carrillo, J. A.; Toscani, G.
2014-12-01
We investigate the large-time asymptotics of nonlinear diffusion equations ut = Δup in dimension n ⩾ 1, in the exponent interval p > n/(n + 2), when the initial datum u0 is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative Rényi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at each time t > 0 with respect to their own second moment, as proposed by Carrillo et al (2006 Arch. Ration. Mech. Anal. 180 127-49) and Toscani (2005 J. Evol. Eqns 5 185-203). The analysis shows that, in the range p > max((n - 1)/n, n/(n + 2)), the relative Rényi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-Newman entropy. The result follows by a suitable use of sharp Gagliardo-Nirenberg-Sobolev inequalities considered by Dolbeault and Toscani (2013 Ann. Inst. Henri Poincare (C) Non Linear Anal. 30 917-34), and their information-theoretical proof (Savaré and Toscani 2014 IEEE Trans. Inform. Theory 60 2687-93), known as concavity of Rényi entropy power.
Lattice Boltzmann methods for some 2-D nonlinear diffusion equations:Computational results
Elton, B.H.; Rodrigue, G.H. . Dept. of Applied Science Lawrence Livermore National Lab., CA ); Levermore, C.D. . Dept. of Mathematics)
1990-01-01
In this paper we examine two lattice Boltzmann methods (that are a derivative of lattice gas methods) for computing solutions to two two-dimensional nonlinear diffusion equations of the form {partial derivative}/{partial derivative}t u = v ({partial derivative}/{partial derivative}x D(u){partial derivative}/{partial derivative}x u + {partial derivative}/{partial derivative}y D(u){partial derivative}/{partial derivative}y u), where u = u({rvec x},t), {rvec x} {element of} R{sup 2}, v is a constant, and D(u) is a nonlinear term that arises from a Chapman-Enskog asymptotic expansion. In particular, we provide computational evidence supporting recent results showing that the methods are second order convergent (in the L{sub 1}-norm), conservative, conditionally monotone finite difference methods. Solutions computed via the lattice Boltzmann methods are compared with those computed by other explicit, second order, conservative, monotone finite difference methods. Results are reported for both the L{sub 1}- and L{sub {infinity}}-norms.
NASA Astrophysics Data System (ADS)
Shishkin, G. I.; Shishkina, L. P.
2015-03-01
An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of [InlineMediaObject not available: see fulltext.], where N + 1 and N 0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.
NASA Astrophysics Data System (ADS)
Shishkin, G. I.; Shishkina, L. P.
2010-12-01
For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the ( x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of ɛ-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges ɛ-uniformly in the maximum norm at the rate of O( N -2ln2 N + N {0/-1}), where N + 1 and N 0 + 1 are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme convergesɛ-uniformly at the rate of O( N -4ln4 N + N {0/-2}). For fixed values of the parameter, the convergence rate is O( N -4 + N {0/-2}).
Roberts, Elijah; Stone, John E.; Luthey-Schulten, Zaida
2013-01-01
Spatial stochastic simulation is a valuable technique for studying reactions in biological systems. With the availability of high-performance computing, the method is poised to allow integration of data from structural, single-molecule, and biochemical studies into coherent computational models of cells. Here we introduce the Lattice Microbes software package for simulating such cell models on high-performance computing systems. The software performs either well-stirred or spatially resolved stochastic simulations with approximated cytoplasmic crowding in a fast and efficient manner. Our new algorithm efficiently samples the reaction-diffusion master equation using NVIDIA GPUs and is shown to be two orders of magnitude faster than exact sampling for large systems while maintaining an accuracy of ∼0.1%. Display of cell models and animation of reaction trajectories involving millions of molecules is facilitated using a plug-in to the popular VMD visualization platform. The Lattice Microbes software is open source and available for download at http://www.scs.illinois.edu/schulten/lm. PMID:23007888
NASA Astrophysics Data System (ADS)
Sayed, Shehrin; Hong, Seokmin; Datta, Supriyo
We will present a general semiclassical theory for an arbitrary channel with spin-orbit coupling (SOC), that uses four electrochemical potential (U + , D + , U - , and D -) depending on the sign of z-component of the spin (up (U) , down (D)) and the sign of the x-component of the group velocity (+ , -) . This can be considered as an extension of the standard spin diffusion equation that uses two electrochemical potentials for up and down spin states, allowing us to take into account the unique coupling between charge and spin degrees of freedom in channels with SOC. We will describe applications of this model to answer a number of interesting questions in this field such as: (1) whether topological insulators can switch magnets, (2) how the charge to spin conversion is influenced by the channel resistivity, and (3) how device structures can be designed to enhance spin injection. This work was supported by FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.
Jia, Jingfei
2015-01-01
It is well known that radiative transfer equation (RTE) provides more accurate tomographic results than its diffusion approximation (DA). However, RTE-based tomographic reconstruction codes have limited applicability in practice due to their high computational cost. In this article, we propose a new efficient method for solving the RTE forward problem with multiple light sources in an all-at-once manner instead of solving it for each source separately. To this end, we introduce here a novel linear solver called block biconjugate gradient stabilized method (block BiCGStab) that makes full use of the shared information between different right hand sides to accelerate solution convergence. Two parallelized block BiCGStab methods are proposed for additional acceleration under limited threads situation. We evaluate the performance of this algorithm with numerical simulation studies involving the Delta-Eddington approximation to the scattering phase function. The results show that the single threading block RTE solver proposed here reduces computation time by a factor of 1.5~3 as compared to the traditional sequential solution method and the parallel block solver by a factor of 1.5 as compared to the traditional parallel sequential method. This block linear solver is, moreover, independent of discretization schemes and preconditioners used; thus further acceleration and higher accuracy can be expected when combined with other existing discretization schemes or preconditioners. PMID:26345531
Simulation of the radiolysis of water using Green's functions of the diffusion equation.
Plante, I; Cucinotta, F A
2015-09-01
Radiation chemistry is of fundamental importance in the understanding of the effects of ionising radiation, notably with regard to DNA damage by indirect effect (e.g. damage by ·OH radicals created by the radiolysis of water). In the recent years, Green's functions of the diffusion equation (GFDEs) have been used extensively in biochemistry, notably to simulate biochemical networks in time and space. In the present work, an approach based on the GFDE will be used to refine existing models on the indirect effect of ionising radiation on DNA. As a starting point, the code RITRACKS (relativistic ion tracks) will be used to simulate the radiation track structure and calculate the position of all radiolytic species formed during irradiation. The chemical reactions between these radiolytic species and with DNA will be done by using an efficient Monte Carlo sampling algorithm for the GFDE of reversible reactions with an intermediate state that has been developed recently. These simulations should help the understanding of the contribution of the indirect effect in the formation of DNA damage, particularly with regards to the formation of double-strand breaks.
Richter, Otto; Moenickes, Sylvia; Suhling, Frank
2012-02-01
The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.
Application of the diffusion-convection equation to modeling the infection by histoplasma-capsulatum
NASA Astrophysics Data System (ADS)
Jaime, Sergio A.
2013-05-01
Using computer algebra, the respiratory infection of the histoplasma capsulatum fungus was modeled and analyzed; the effects of the infection could also be described as a change in the lungs capacity to expand (associated with its elastic modulus). A further analysis to the immune system was also done in order to describe and model the way the body can handle those kinds of infections once they are hosted in the body. Using those models we can describe the behavior of the respiratory infection and then how to reduce or control its effects. As an investigation in the medical field, we need to test the models obtained and compare the results with the real infection behavior. The models where made based on the diffusive-convective equation; giving some initial and boundary conditions, we can get to the results obtained, which can describe how the infection is spreading and with a previous study of the immune system, the infection control done by the body can also be modeled.
Jia, Jingfei; Kim, Hyun K; Hielscher, Andreas H
2015-12-01
It is well known that radiative transfer equation (RTE) provides more accurate tomographic results than its diffusion approximation (DA). However, RTE-based tomographic reconstruction codes have limited applicability in practice due to their high computational cost. In this article, we propose a new efficient method for solving the RTE forward problem with multiple light sources in an all-at-once manner instead of solving it for each source separately. To this end, we introduce here a novel linear solver called block biconjugate gradient stabilized method (block BiCGStab) that makes full use of the shared information between different right hand sides to accelerate solution convergence. Two parallelized block BiCGStab methods are proposed for additional acceleration under limited threads situation. We evaluate the performance of this algorithm with numerical simulation studies involving the Delta-Eddington approximation to the scattering phase function. The results show that the single threading block RTE solver proposed here reduces computation time by a factor of 1.5~3 as compared to the traditional sequential solution method and the parallel block solver by a factor of 1.5 as compared to the traditional parallel sequential method. This block linear solver is, moreover, independent of discretization schemes and preconditioners used; thus further acceleration and higher accuracy can be expected when combined with other existing discretization schemes or preconditioners.
Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation
NASA Astrophysics Data System (ADS)
Ye, Zhuan; Xu, Xiaojing
2016-06-01
In this paper, we study the 2D Boussinesq equations with fractional Laplacian dissipation. In particular, we prove the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The main ingredient of the proof is the utilization of the Hölder estimates for advection fractional-diffusion equations as well as Littlewood-Paley technique.
NASA Astrophysics Data System (ADS)
Hsuen, Hsiao-Kuo
The performance equations for cathodes of polymer electrolyte fuel cells (PEFCs) that describe the dependence of cathode potential on current density are developed. Formulation of the performance equations starts from the reduction of a one-dimensional model that considers, in detail, the potential losses pertinent to the limitations of electron conduction, oxygen diffusion, proton migration, and the oxygen reduction reaction. In particular, non-uniform accumulation of liquid water in the gas diffuser, which partially blocks the gas channels and imposes a greater resistance for oxygen transport, is taken into account. Reduction of the one-dimensional model is implemented by approximating the oxygen concentration profile in the catalyst layer with a parabolic polynomial or a piecewise parabolic one determined by the occurrence of oxygen depletion. The final forms of the equations are obtained by applying the method of weighted residuals over the catalyst layer. The weighting function is selected in such a way that the weighted residuals can be analytically integrated. Potential losses caused by the various limiting processes can be quantitatively estimated by the performance equations. Thus, they provide a convenient diagnostic tool for the cathode performance. Computational results reveal that the performance equations agree well with the original one-dimensional model over an extensive range of parameter values. This indicates that the present performance equations can be used as a substitute for the one-dimensional model to provide quantitatively correct predictions for the cathode performance of PEFCs.
Waves, advection, and cloud patterns on Venus
NASA Technical Reports Server (NTRS)
Schinder, Paul J.; Gierasch, Peter J.; Leroy, Stephen S.; Smith, Michael D.
1990-01-01
The stable layers adjacent to the nearly neutral layer within the Venus clouds are found to be capable of supporting vertically trapped, horizontally propagating waves with horizontal wavelengths of about 10 km and speeds of a few meters per second relative to the mean wind in the neutral layer. These waves may possibly be excited by turbulence within the neutral layer. Here, the properties of the waves, and the patterns which they might produce within the visible clouds if excited near the subsolar point are examined. The patterns can be in agreement with many features in images. The waves are capable of transferring momentum latitudinally to help maintain the general atmospheric spin, but at present we are not able to evaluate wave amplitudes. We also examine an alternative possibility that the cloud patterns are produced by advection and shearing by the mean zonal and meridional flow of blobs formed near the equator. It is concluded that advection and shearing by the mean flow is the most likely explanation for the general pattern of small scale striations.
Parallel algorithms for semi-lagrangian advection
NASA Astrophysics Data System (ADS)
Malevsky, A. V.; Thomas, S. J.
1997-08-01
Numerical time step limitations associated with the explicit treatment of advection-dominated problems in computational fluid dynamics are often relaxed by employing Eulerian-Lagrangian methods. These are also known as semi-Lagrangian methods in the atmospheric sciences. Such methods involve backward time integration of a characteristic equation to find the departure point of a fluid particle arriving at a Eulerian grid point. The value of the advected field at the departure point is obtained by interpolation. Both the trajectory integration and repeated interpolation influence accuracy. We compare the accuracy and performance of interpolation schemes based on piecewise cubic polynomials and cubic B-splines in the context of a distributed memory, parallel computing environment. The computational cost and interprocessor communication requirements for both methods are reported. Spline interpolation has better conservation properties but requires the solution of a global linear system, initially appearing to hinder a distributed memory implementation. The proposed parallel algorithm for multidimensional spline interpolation has almost the same communication overhead as local piecewise polynomial interpolation. We also compare various techniques for tracking trajectories given different values for the Courant number. Large Courant numbers require a high-order ODE solver involving multiple interpolations of the velocity field.
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
NASA Technical Reports Server (NTRS)
Kennedy, Christopher A.; Carpenter, Mark H.
2001-01-01
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.
Chapter 19: The age of scarplike landforms from diffusion-equation analysis
Hanks, Thomas C.
2000-01-01
The purpose of this paper is to review developments in the quantitative modeling of fault-scarp geomorphology, principally those since 1980. These developments utilize diffusionequation mathematics, in several different forms, as the basic model of fault-scarp evolution. Because solutions to the general diffusion equation evolve with time, as we expect faultscarp morphology to evolve with time, the model solutions carry information about the age of the structure and thus its time of formation; hence the inclusion of this paper in this volume. The evolution of fault-scarp morphology holds a small but special place in the much larger class of problems in landform evolution. In general, landform evolution means the evolution of topography as a function of both space and time. It is the outcome of the competition among those tectonic processes that make topography, erosive processes that destroy topography, and depositional processes that redistribute topography. Deposition and erosion can always be coupled through conservation-of-mass relations, but in general deposition occurs at great distance from the source region of detritus. Moreover, erosion is an inherently rough process whereas deposition is inherently smooth, as is evident from even casual inspection of shaded-relief, digital-elevation maps (e.g., Thelin and Pike, 1990; Simpson and Anders, 1992) and the current fascination with fractal representations oferoding terrains (e.g., Huang and Turcotte, 1989; Newman and Turcotte, 1990). Nevertheless, large-scale landform-evolution modeling, now a computationally intensive, advanced numerical exercise, is generating ever more realistic landforms (e.g., Willgoose and others, 1991a,b; Kooi and Beaumont, 1994; Tucker and Slingerland, 1994), although many of the rate coefficients remain poorly prescribed
ben-Avraham, D; Fokas, A S
2001-07-01
A new transform method for solving boundary value problems for linear and integrable nonlinear partial differential equations recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q(xx)(x,y)+q(yy)(x,y)-4 beta(2)q(x,y)=0 in the triangular domain 0< or =x< or =L-y< or =L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A+A<==>A, in the segment (-L/2,L/2), with traps at the edges.
Sanchez, J.R.; Evans, J.W.
1999-01-01
Exact results are presented for the surface diffusion of small two-dimensional clusters, the constituent atoms of which are commensurate with a square lattice of adsorption sites. Cluster motion is due to the hopping of atoms along the cluster perimeter with various rates. We apply the formalism of Titulaer and Deutch [J. Chem. Phys. {bold 77}, 472 (1982)], which describes evolution in reciprocal space via a linear master equation with dimension equal to the number of cluster configurations. We focus on the regime of rapid hopping of atoms along straight close-packed edges, where certain subsets of configurations cycle rapidly between each other. Each such subset is treated as a single quasiconfiguration, thereby reducing the dimension of the evolution equation, simplifying the analysis, and elucidating limiting behavior. We also discuss the influence of concerted atom motions on the diffusion of tetramers and larger clusters. {copyright} {ital 1999} {ital The American Physical Society}
NASA Astrophysics Data System (ADS)
Jia, Junxiong; Peng, Jigen; Yang, Jiaqing
2017-04-01
In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.
NASA Astrophysics Data System (ADS)
Simon, Emanuel; Krauter, Philipp; Kienle, Alwin
2014-07-01
Transmittance and reflectance from spruce wood and bovine ligamentum nuchae as two different fibrous media are examined by time-of-flight spectroscopy for varying source detector separations and several orientations of the fibers in the sample. The anisotropic diffusion theory is used to obtain the absorption coefficient and the diffusion coefficients parallel and perpendicular to the fibers. The results are compared to those obtained with the isotropic diffusion theory. It is shown that for increasing source detector separations, the retrieved optical properties change as expected from Monte Carlo simulations performed in a previous study. This confirms that the anisotropic diffusion theory yields useful results for certain experimental conditions.
NASA Astrophysics Data System (ADS)
Jin, Shi; Lu, Hanqing
2017-04-01
In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro-macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.
NASA Astrophysics Data System (ADS)
Mascarenhas, Brendan S.; Helenbrook, Brian T.; Atkins, Harold L.
2010-05-01
An improved p-multigrid algorithm for discontinuous Galerkin (DG) discretizations of convection-diffusion problems is presented. The general p-multigrid algorithm for DG discretizations involves a restriction from the p=1 to p=0 discontinuous polynomial solution spaces. This restriction is problematic and has limited the efficiency of the p-multigrid method. For purely diffusive problems, Helenbrook and Atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p=1. It is shown that this method is not directly applicable to the convection-diffusion (CD) equation because it results in a central-difference discretization for the convective term. To remedy this, ideas from the streamwise upwind Petrov-Galerkin (SUPG) formulation are used to devise a transition from the discontinuous to continuous space at p=1 that yields an upwind discretization. The results show that the new method converges rapidly for all Peclet numbers.
NASA Astrophysics Data System (ADS)
Fujii, Hiroyuki; Okawa, Shinpei; Yamada, Yukio; Hoshi, Yoko
2014-11-01
Numerical modeling of light propagation in random media has been an important issue for biomedical imaging, including diffuse optical tomography (DOT). For high resolution DOT, accurate and fast computation of light propagation in biological tissue is desirable. This paper proposes a space-time hybrid model for numerical modeling based on the radiative transfer and diffusion equations (RTE and DE, respectively) in random media under refractive-index mismatching. In the proposed model, the RTE and DE regions are separated into space and time by using a crossover length and the time from the ballistic regime to the diffusive regime, ρDA 10 / μt‧ and tDA 20 / v μt‧ where μt‧ and v represent a reduced transport coefficient and light velocity, respectively. The present model succeeds in describing light propagation accurately and reduces computational load by a quarter compared with full computation of the RTE.
Zhao, Renjie; Evans, James W.; Oliveira, Tiago J.
2016-04-08
Here, a discrete version of deposition-diffusion equations appropriate for description of step flow on a vicinal surface is analyzed for a two-dimensional grid of adsorption sites representing the stepped surface and explicitly incorporating kinks along the step edges. Model energetics and kinetics appropriately account for binding of adatoms at steps and kinks, distinct terrace and edge diffusion rates, and possible additional barriers for attachment to steps. Analysis of adatom attachment fluxes as well as limiting values of adatom densities at step edges for nonuniform deposition scenarios allows determination of both permeability and kinetic coefficients. Behavior of these quantities is assessed as a function of key system parameters including kink density, step attachment barriers, and the step edge diffusion rate.
Zhao, Renjie; Evans, James W.; Oliveira, Tiago J.
2016-04-08
Here, a discrete version of deposition-diffusion equations appropriate for description of step flow on a vicinal surface is analyzed for a two-dimensional grid of adsorption sites representing the stepped surface and explicitly incorporating kinks along the step edges. Model energetics and kinetics appropriately account for binding of adatoms at steps and kinks, distinct terrace and edge diffusion rates, and possible additional barriers for attachment to steps. Analysis of adatom attachment fluxes as well as limiting values of adatom densities at step edges for nonuniform deposition scenarios allows determination of both permeability and kinetic coefficients. Behavior of these quantities is assessedmore » as a function of key system parameters including kink density, step attachment barriers, and the step edge diffusion rate.« less
Rocco, A; Ramírez-Piscina, L; Casademunt, J
2002-05-01
We study the dynamics of generic reaction-diffusion fronts, including pulses and chemical waves, in the presence of multiplicative noise. We discuss the connection between the reaction-diffusion Langevin-like field equations and the kinematic (eikonal) description in terms of a stochastic moving-boundary or sharp-interface approximation. We find that the effective noise is additive and we relate its strength to the noise parameters in the original field equations, to first order in noise strength, but including a partial resummation to all orders which captures the singular dependence on the microscopic cutoff associated with the spatial correlation of the noise. This dependence is essential for a quantitative and qualitative understanding of fluctuating fronts, affecting both scaling properties and nonuniversal quantities. Our results predict phenomena such as the shift of the transition point between the pushed and pulled regimes of front propagation, in terms of the noise parameters, and the corresponding transition to a non-Kardar-Parisi-Zhang universality class. We assess the quantitative validity of the results in several examples including equilibrium fluctuations and kinetic roughening. We also predict and observe a noise-induced pushed-pulled transition. The analytical predictions are successfully tested against rigorous results and show excellent agreement with numerical simulations of reaction-diffusion field equations with multiplicative noise.
NASA Astrophysics Data System (ADS)
Sanghi, T.; Aluru, N. R.
2013-03-01
In this work, we combine our earlier proposed empirical potential based quasi-continuum theory, (EQT) [A. V. Raghunathan, J. H. Park, and N. R. Aluru, J. Chem. Phys. 127, 174701 (2007), 10.1063/1.2793070], which is a coarse-grained multiscale framework to predict the static structure of confined fluids, with a phenomenological Langevin equation to simulate the dynamics of confined fluids in thermal equilibrium. An attractive feature of this approach is that all the input parameters to the Langevin equation (mean force profile of the confined fluid and the static friction coefficient) can be determined using the outputs of the EQT and the self-diffusivity data of the corresponding bulk fluid. The potential of mean force profile, which is a direct output from EQT is used to compute the mean force profile of the confined fluid. The density profile, which is also a direct output from EQT, along with the self-diffusivity data of the bulk fluid is used to determine the static friction coefficient of the confined fluid. We use this approach to compute the mean square displacement and survival probabilities of some important fluids such as carbon-dioxide, water, and Lennard-Jones argon confined inside slit pores. The predictions from the model are compared with those obtained using molecular dynamics simulations. This approach of combining EQT with a phenomenological Langevin equation provides a mathematically simple and computationally efficient means to study the impact of structural inhomogeneity on the self-diffusion dynamics of confined fluids.
NASA Astrophysics Data System (ADS)
Alonso-Vargas, G.
A computer program has been developed which uses a technique of synthetic acceleration by diffusion by analytical schemes. Both in the diffusion equation as in that of transport, analytical schemes were used which allowed a substantial time saving in the number of iterations required by source iteration method to obtain the K(sub e)ff. The program developed ASD (Synthetic Diffusion Acceleration) by diffusion was written in FORTRAN and can be executed on a personal computer with a hard disc and mathematical O-processor. The program is unlimited as to the number of regions and energy groups. The results obtained by the ASD program for K(sub e)ff is nearly completely concordant with those obtained by utilizing the ANISN-PC code for different analytical type problems in this work. The ASD program allowed obtention of an approximate solution of the neutron transport equation with a relatively low number of internal reiterations with good precision. One of its applications would be in the direct determinations of axial distribution neutronic flow in a fuel assembly as well as in the obtention of the effective multiplication factor.
NASA Astrophysics Data System (ADS)
Hormuth, David A., II; Weis, Jared A.; Barnes, Stephanie L.; Miga, Michael I.; Rericha, Erin C.; Quaranta, Vito; Yankeelov, Thomas E.
2015-07-01
Reaction-diffusion models have been widely used to model glioma growth. However, it has not been shown how accurately this model can predict future tumor status using model parameters (i.e., tumor cell diffusion and proliferation) estimated from quantitative in vivo imaging data. To this end, we used in silico studies to develop the methods needed to accurately estimate tumor specific reaction-diffusion model parameters, and then tested the accuracy with which these parameters can predict future growth. The analogous study was then performed in a murine model of glioma growth. The parameter estimation approach was tested using an in silico tumor ‘grown’ for ten days as dictated by the reaction-diffusion equation. Parameters were estimated from early time points and used to predict subsequent growth. Prediction accuracy was assessed at global (total volume and Dice value) and local (concordance correlation coefficient, CCC) levels. Guided by the in silico study, rats (n = 9) with C6 gliomas, imaged with diffusion weighted magnetic resonance imaging, were used to evaluate the model’s accuracy for predicting in vivo tumor growth. The in silico study resulted in low global (tumor volume error <8.8%, Dice >0.92) and local (CCC values >0.80) level errors for predictions up to six days into the future. The in vivo study showed higher global (tumor volume error >11.7%, Dice <0.81) and higher local (CCC <0.33) level errors over the same time period. The in silico study shows that model parameters can be accurately estimated and used to accurately predict future tumor growth at both the global and local scale. However, the poor predictive accuracy in the experimental study suggests the reaction-diffusion equation is an incomplete description of in vivo C6 glioma biology and may require further modeling of intra-tumor interactions including segmentation of (for example) proliferative and necrotic regions.
NASA Technical Reports Server (NTRS)
Zhou, YE; Vahala, George
1993-01-01
The advection of a passive scalar by incompressible turbulence is considered using recursive renormalization group procedures in the differential sub grid shell thickness limit. It is shown explicitly that the higher order nonlinearities induced by the recursive renormalization group procedure preserve Galilean invariance. Differential equations, valid for the entire resolvable wave number k range, are determined for the eddy viscosity and eddy diffusivity coefficients, and it is shown that higher order nonlinearities do not contribute as k goes to 0, but have an essential role as k goes to k(sub c) the cutoff wave number separating the resolvable scales from the sub grid scales. The recursive renormalization transport coefficients and the associated eddy Prandtl number are in good agreement with the k-dependent transport coefficients derived from closure theories and experiments.
Phase Segregation of Passive Advective Particles in an Active Medium
NASA Astrophysics Data System (ADS)
Das, Amit; Polley, Anirban; Rao, Madan
2016-02-01
Localized contractile configurations or asters spontaneously appear and disappear as emergent structures in the collective stochastic dynamics of active polar actomyosin filaments. Passive particles which (un)bind to the active filaments get advected into the asters, forming transient clusters. We study the phase segregation of such passive advective scalars in a medium of dynamic asters, as a function of the aster density and the ratio of the rates of aster remodeling to particle diffusion. The dynamics of coarsening shows a violation of Porod behavior; the growing domains have diffuse interfaces and low interfacial tension. The phase-segregated steady state shows strong macroscopic fluctuations characterized by multiscaling and intermittency, signifying rapid reorganization of macroscopic structures. We expect these unique nonequilibrium features to manifest in the actin-dependent molecular clustering at the cell surface.
NASA Astrophysics Data System (ADS)
Wang, I. T.
A general method for determining the effective transport wind speed, overlineu, in the Gaussian plume equation is discussed. Physical arguments are given for using the generalized overlineu instead of the often adopted release-level wind speed with the plume diffusion equation. Simple analytical expressions for overlineu applicable to low-level point releases and a wide range of atmospheric conditions are developed. A non-linear plume kinematic equation is derived using these expressions. Crosswind-integrated SF 6 concentration data from the 1983 PNL tracer experiment are used to evaluate the proposed analytical procedures along with the usual approach of using the release-level wind speed. Results of the evaluation are briefly discussed.
NASA Astrophysics Data System (ADS)
Mvogo, Alain; Tambue, Antoine; Ben-Bolie, Germain H.; Kofané, Timoléon C.
2016-10-01
We investigate localized wave solutions in a network of Hindmarsh-Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number k → 0) can be governed by the complex fractional Ginzburg-Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindmarsh-Rose models for relatively large network.
Bahşı, Ayşe Kurt; Yalçınbaş, Salih
2016-01-01
In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method.
Lu, Benzhuo; Zhou, Y C
2011-05-18
The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations.
Lin, Guoxing
2015-10-28
Inter-molecular multiple quantum coherence (iMQC) has important applications in NMR and MRI. However, the current theoretical methods still have some difficulties in analyzing the behavior of iMQC signal attenuation of pulsed field gradient diffusion experiments. In this paper, the iMQC diffusion experiments were analyzed by an effective phase shift diffusion equation (EPSDE) method, which is based on the idea that the accumulating phase shift (APS) can be viewed as the result of a diffusion process in virtual phase space (VPS) with effective diffusion coefficient K(2)(t) D (rad(2)/s) where K(t)=∫0 (t)γg(t')dt' is a wavenumber and D is the physical diffusion coefficient of the spin carrier in the real space. The term K(t(tot)) z1 needs to be added to the APS when K(t(tot)) is not zero. Most of the time, K(t(tot)) equals zero. However, in iMQC experiments, the condition K(t(tot)) equaling zero or being non-zero for each spin depends on the gradient pulse setting. The signal attenuations of these two types of iMQC, zero or non-zero K(t(tot)), were analyzed in detail for free and restricted diffusions, which shows that there are significant differences between these two types of iMQC. Particularly, if an apparent diffusion coefficient D(app) is used to analyze the signal attenuation, it equals nD for zero K(t(tot)) which agrees with current theoretical and experimental reports, while for non-zero K(t(tot)), it equals (2n - 1) D which agrees with experimental results from the literature; there are no similar theoretical results reported for comparison. The result that D(app) equals (2n - 1) D is important because the higher value of D(app) means that non-zero K(t(tot)) iMQC can potentially provide more contrast and measure slower diffusion rates than zero K(t(tot)) iMQC. The EPSDE method provides a new way to analyze iMQC diffusion experiments.
NASA Astrophysics Data System (ADS)
Qin, Zhuanping; Hou, Qiang; Zhao, Huijuan; Yang, Yanshuang; Zhou, Xiaoqing; Gao, Feng
2013-03-01
In this paper, frequency-domain endoscopic diffuse optical tomography image reconstruction algorithm based on dual-modulation-frequency and dual-points source diffuse equation is investigated for the reconstruction of the optical parameters including the absorption and reducing scattering coefficients. The forward problem is solved by the finite element method based on the frequency domain diffuse equation (FD-DE) for dual-points source approximation and multi-modulation-frequency. In the image reconstruction, a multi-modulation-frequency Newton-Raphson algorithm is applied to obtain the solution. To further improve the image accuracy and quality, a method based on the region of interest (ROI) is applied on the above procedures. The simulation is performed in the tubular model to verify the validity of the algorithm. Results show that the FD-DE with dual-points source approximate is more accuracy at shorter source-detector separation. The reconstruction with dual-modulation-frequency improves the image accuracy and quality compared to the results with single-modulation-frequency and triple-modulation-frequency method. The peak optical coefficients in ROI (ROI_max) are almost equivalent to the true optical coefficients with the relative error less than 6.67%. The full width at half maximum (FWHM) achieves 82% of the true radius. The contrast-to-noise ratio (CNR) and image coefficient(IC) is 5.678 and 26.962, respectively. Additionally, the results with the method based on ROI show that the ROI_max is equivalent to the true value. The FWHM can improve by 88% of the true radius. The CNR and IC is improved over 7.782 and 45.335, respectively.
Colmenares, Pedro J; López, Floralba; Olivares-Rivas, Wilmer
2009-12-01
We carried out a molecular-dynamics (MD) study of the self-diffusion tensor of a Lennard-Jones-type fluid, confined in a slit pore with attractive walls. We developed Bayesian equations, which modify the virtual layer sampling method proposed by Liu, Harder, and Berne (LHB) [P. Liu, E. Harder, and B. J. Berne, J. Phys. Chem. B 108, 6595 (2004)]. Additionally, we obtained an analytical solution for the corresponding nonhomogeneous Langevin equation. The expressions found for the mean-squared displacement in the layers contain naturally a modification due to the mean force in the transverse component in terms of the anisotropic diffusion constants and mean exit time. Instead of running a time consuming dual MD-Langevin simulation dynamics, as proposed by LHB, our expression was used to fit the MD data in the entire survival time interval not only for the parallel but also for the perpendicular direction. The only fitting parameter was the diffusion constant in each layer.
Chai, Zhenhua; Zhao, T S
2014-07-01
In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.
NASA Astrophysics Data System (ADS)
Alsabry, A.; Zybura, A.
2016-05-01
When the structure of reinforcement is in danger of chloride corrosion it is possible to prevent this disadvantageous phenomenon through exposing the cover to the influence of an electric field. The forces of an electric field considerably reduce chloride ions in pore liquid in concrete, which helps to rebuild a passive layer on the surface of the reinforcement and stops corrosion. The process of removing chlorides can be described with multi-component diffusion equations. However, an essential parameter of these equations, the diffusion coefficient, can be determined on the basis of an inverse task. Since the solution was achieved for one-dimension flow, the method applied can be confirmed by experimental results and the material parameters of the process can be determined theoretically. Some examples of numerical calculations of the effective electro-diffusion coefficient of chloride ions confirmed the usefulness of the theoretical solution for generalizing experimental results. Moreover, the calculation process of the numerical example provides some practical clues for future experimental research, which could be carried out in close connection with the theoretical solution.
Ougouag, Abderrafi Mohammed-El-Ami; Terry, William Knox
2002-04-01
The usual strategy for solving the neutron diffusion equation in two or three dimensions by nodal methods is to reduce the multidimensional partial differential equation to a set of ordinary differential equations (ODEs) in the separate spatial coordinates. This reduction is accomplished by “transverse integration” of the equation.1 For example, in three-dimensional Cartesian coordinates, the three-dimensional equation is first integrated over x and y to obtain an ODE in z, then over x and z to obtain an ODE in y, and finally over y and z to obtain an ODE in x. Then the ODEs are solved to obtain onedimensional solutions for the neutron fluxes averaged over the other two dimensions. These solutions are found in regions (“nodes”) small enough for the material properties and cross sections in them to be adequately represented by average values. Because the solution in each node is an exact analytical solution, the nodes can be much larger than the mesh elements used in finite-difference solutions. Then the solutions in the different nodes are coupled by applying interface conditions, ultimately fixing the solutions to the external boundary conditions.
Yang, Xuguang; Shi, Baochang; Chai, Zhenhua
2014-07-01
In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = 10(12)).
Laghaei, Rozita; Nasrabad, Afshin Eskandari; Eu, Byung Chan
2005-03-31
The shear viscosity formula derived by the density fluctuation theory in previous papers is computed for argon, krypton, and methane by using the self-diffusion coefficients derived in the modified free volume theory with the help of the generic van der Waals equation of state. In the temperature regime near or above the critical temperature, the density dependence of the shear viscosity can be accounted for by ab initio calculations with the self-diffusion coefficients provided by the modified free volume theory if the minimum (critical) free volume is set equal to the molecular volume and the volume overlap parameter (alpha) is taken about unity in the expression for the self-diffusion coefficient. In the subcritical temperature regime, if the density fluctuation range parameter is chosen appropriately at a temperature, then the resulting expression for the shear viscosity can well account for its density and temperature dependence over the ranges of density and temperature experimentally studied. In the sense that once the density fluctuation range is fixed at a temperature, the theory can account for the experimental data at other subcritical temperatures on the basis of the intermolecular force only; the theory is predictive even in the subcritical regime of temperature. Theory is successfully tested in comparison with experimental data for self-diffusion coefficients and shear viscosity for argon, krypton, and methane.
The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion
NASA Astrophysics Data System (ADS)
Bessaih, H.; Ferrario, B.
2017-02-01
In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order α in the nonlinear term and a β-fractional Laplacian; we consider the critical case α + β =5/4 and we assume 1/2 < β <5/4. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order α. We prove global well posedness when the initial velocity is in Hr and the initial temperature is in H r - β for r > max (2 β , β + 1). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions.
The arbitrary order mixed mimetic finite difference method for the diffusion equation
Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco
2016-05-01
Here, we propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux andmore » scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.« less
The arbitrary order mixed mimetic finite difference method for the diffusion equation
Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco
2016-05-01
Here, we propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux and scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.
Traytak, Sergey D
2014-06-14
The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.
NASA Astrophysics Data System (ADS)
Traytak, Sergey D.
2014-06-01
The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.
Traytak, Sergey D.
2014-06-14
The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.
Alqasemi, Umar; Salehi, Hassan S.; Zhu, Quing
2016-01-01
This paper reports a method of estimating an approximate closed-form solution to the light diffusion equation for any type of geometry involving Dirichlet’s boundary condition with known source location. It is based on estimating the optimum locations of multiple imaginary point sources to cancel the fluence at the extrapolated boundary by constrained optimization using a genetic algorithm. The mathematical derivation of the problem to approach the optimum solution for the direct-current type of diffuse optical systems is described in detail. Our method is first applied to slab geometry and compared with a truncated series solution. After that, it is applied to hemispherical geometry and compared with Monte Carlo simulation results. The method provides a fast and sufficiently accurate fluence distribution for optical reconstruction. PMID:26831771
NASA Technical Reports Server (NTRS)
Douglass, A.; Kawa, S. R.; Einaudi, Franco (Technical Monitor)
2000-01-01
Three dimensional chemistry and transport models (CTMs) contain a set of coupled continuity equations which describe the evolution of constituents such as ozone and other minor species which affect ozone. Both advection and photochemical processes contribute to constituent evolution, and a CTM provides a means to evaluate these contributions separately. Such evaluation is particularly useful when both terms are important to the modeled tendency. An example is the ozone tendency in the high latitude winter lower stratosphere, where advection tends to increase ozone, and catalytic processes involving chlorine radicals tend to decrease ozone. The Goddard three dimensional chemistry and transport model uses meteorological fields from the Goddard Earth Observing System Data Assimilation System, thus the modeled ozone evolution may reproduce the observed evolution and provide a test of the model representation of photochemical processes if the transport is shown to be modeled appropriately. We have investigated the model advection further using diabatic trajectory calculations. For long lived constituents such as N2O, the model field for a particular time on a potential temperature surface is compared with a field produced by calculating 15 day back trajectories for a fixed latitude longitude grid, and mapping model N2O at the terminus of the back trajectories onto the initial grid. This provides a quantitative means to evaluate two aspects of the CTM transport: one, the model horizontal gradient between middle latitudes and the polar vortex is compared with the gradient produced using the non-diffusive trajectory calculation; two, the model vertical advection, which is produced by the divergence of the horizontal winds, is compared with the vertical transport expected from diabatic cooling.
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Larsen, Edward
2013-06-17
The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.
NASA Astrophysics Data System (ADS)
Petrishcheva, E.; Abart, R.
2012-04-01
We address mathematical modeling and computer simulations of phase decomposition in a multicomponent system. As opposed to binary alloys with one common diffusion parameter, our main concern is phase decomposition in real geological systems under influence of strongly different interdiffusion coefficients, as it is frequently encountered in mineral solid solutions with coupled diffusion on different sub-lattices. Our goal is to explain deviations from equilibrium element partitioning which are often observed in nature, e.g., in a cooled ternary feldspar. To this end we first adopt the standard Cahn-Hilliard model to the multicomponent diffusion problem and account for arbitrary diffusion coefficients. This is done by using Onsager's approach such that flux of each component results from the combined action of chemical potentials of all components. In a second step the generalized Cahn-Hilliard equation is solved numerically using finite-elements approach. We introduce and investigate several decomposition scenarios that may produce systematic deviations from the equilibrium element partitioning. Both ideal solutions and ternary feldspar are considered. Typically, the slowest component is initially "frozen" and the decomposition effectively takes place only for two "fast" components. At this stage the deviations from the equilibrium element partitioning are indeed observed. These deviations may became "frozen" under conditions of cooling. The final equilibration of the system occurs on a considerably slower time scale. Therefore the system may indeed remain unaccomplished at the observation point. Our approach reveals the intrinsic reasons for the specific phase separation path and rigorously describes it by direct numerical solution of the generalized Cahn-Hilliard equation.
NASA Technical Reports Server (NTRS)
Zhang, Jun; Ge, Lixin; Kouatchou, Jules
2000-01-01
A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it Only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with the Gauss-Seidel type iterative method. This is compared with the known 19 point fourth order compact differenCe scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and the 19 point fourth order compact schemes.
Druskin, V.; Knizhnerman, L.
1994-12-31
The authors solve the Cauchy problem for an ODE system Au + {partial_derivative}u/{partial_derivative}t = 0, u{vert_bar}{sub t=0} = {var_phi}, where A is a square real nonnegative definite symmetric matrix of the order N, {var_phi} is a vector from R{sup N}. The stiffness matrix A is obtained due to semi-discretization of a parabolic equation or system with time-independent coefficients. The authors are particularly interested in large stiff 3-D problems for the scalar diffusion and vectorial Maxwell`s equations. First they consider an explicit method in which the solution on a whole time interval is projected on a Krylov subspace originated by A. Then they suggest another Krylov subspace with better approximating properties using powers of an implicit transition operator. These Krylov subspace methods generate optimal in a spectral sense polynomial approximations for the solution of the ODE, similar to CG for SLE.
NASA Astrophysics Data System (ADS)
Guymer, T. M.; Moore, A. S.; Morton, J.; Kline, J. L.; Allan, S.; Bazin, N.; Benstead, J.; Bentley, C.; Comley, A. J.; Cowan, J.; Flippo, K.; Garbett, W.; Hamilton, C.; Lanier, N. E.; Mussack, K.; Obrey, K.; Reed, L.; Schmidt, D. W.; Stevenson, R. M.; Taccetti, J. M.; Workman, J.
2015-04-01
A well diagnosed campaign of supersonic, diffusive radiation flow experiments has been fielded on the National Ignition Facility. These experiments have used the accurate measurements of delivered laser energy and foam density to enable an investigation into SESAME's tabulated equation-of-state values and CASSANDRA's predicted opacity values for the low-density C8H7Cl foam used throughout the campaign. We report that the results from initial simulations under-predicted the arrival time of the radiation wave through the foam by ≈22%. A simulation study was conducted that artificially scaled the equation-of-state and opacity with the intended aim of quantifying the systematic offsets in both CASSANDRA and SESAME. Two separate hypotheses which describe these errors have been tested using the entire ensemble of data, with one being supported by these data.
Guymer, T. M. Moore, A. S.; Morton, J.; Allan, S.; Bazin, N.; Benstead, J.; Bentley, C.; Comley, A. J.; Garbett, W.; Reed, L.; Stevenson, R. M.; Kline, J. L.; Cowan, J.; Flippo, K.; Hamilton, C.; Lanier, N. E.; Mussack, K.; Obrey, K.; Schmidt, D. W.; Taccetti, J. M.; and others
2015-04-15
A well diagnosed campaign of supersonic, diffusive radiation flow experiments has been fielded on the National Ignition Facility. These experiments have used the accurate measurements of delivered laser energy and foam density to enable an investigation into SESAME's tabulated equation-of-state values and CASSANDRA's predicted opacity values for the low-density C{sub 8}H{sub 7}Cl foam used throughout the campaign. We report that the results from initial simulations under-predicted the arrival time of the radiation wave through the foam by ≈22%. A simulation study was conducted that artificially scaled the equation-of-state and opacity with the intended aim of quantifying the systematic offsets in both CASSANDRA and SESAME. Two separate hypotheses which describe these errors have been tested using the entire ensemble of data, with one being supported by these data.
NASA Astrophysics Data System (ADS)
Barrera, Frederick; Ahmed, Elharith; Nash, Patrick; Sardar, Dhiraj
2011-03-01
The tracking of photons through turbid media (e.g. tissues) has been studied extensively from an experimental vantage point. These turbid media are difficult to characterize- since their components are exceedingly variegated- and thus present many challenges to clinicians who require models which precisely predict the location and time evolution of energy deposition. Furthermore, the interaction of the turbid media sample with the source of radiation typically involves many dynamic mechanisms (e.g. photothermal etc.) Using diffuse light transport, and an electromagnetic wave approach (e.g. Maxwell's equations), an analysis of thermal energy distribution in tissues is performed. Assuming a highly absorbing chromophore model of melanocytes in tissues, a comparison of the variation of thermal energy is determined for different collections of melanocyte spatial distributions. This work was funded by NIH/NIGMS MBRS-RISE GM60655.
NASA Astrophysics Data System (ADS)
Healy, R. W.
2015-12-01
Water movement through soils is often dominated by preferential flow processes such as fingering and macropore flow. Traditional models of flow in the unsaturated zone are based on the diffusion or Richards equation and assume that diffusive (surface-tension viscous) flow is the only flow process. These models are incapable of accurately simulating preferential flow. Several alternative approaches, including kinematic wave, transfer function, and water-content wave models, have been suggested for simulating water movement through preferential flow paths. The source-responsive model proposed by Nimmo (2010) and Nimmo and Mitchell (2013) is unique among such models in that water transfer from land surface to depth is controlled by the water-application rate at land surface. The source-responsive model has been coupled with a one-dimensional version of the Richards-equation based model of variably saturated flow, VS2DT. The new model, can simulate flow within the preferential (S) domain alone, within the diffuse (D) domain alone, or within both the S and D domains simultaneously. Water exchange between the two domains is treated as a first-order diffusive process, with the exchange coefficient being a function of soil-water content. The new model was used to simulate field and laboratory infiltration experiments described in the literature. Simulations were calibrated against measured soil water contents with the PEST parameter estimation package; values for hydraulic conductivity and 3 van Genuchten and 3 source-responsive parameters were optimized. Although exact matches between measured and simulated water contents were not obtained, the simulation results captured the salient characteristics of the published data sets, including features typical of preferential as well as diffusive flow. Results obtained from simulating flow simultaneously in both the S and D domain provided better matches to measured data than results obtained from simulating flow independently
Lu, Benzhuo; Zhou, Y.C.
2011-01-01
The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations. PMID:21575582
Daly, Edoardo; Porporato, Amilcare
2004-11-01
Similarity solutions of the shallow-water equation with a generalized resistance term are studied for open channel flows when both inertial and gravity forces are negligible. The resulting model encompasses various particular cases that appear, in addition to mathematical hydraulics, in diverse physical phenomena, such as gravity currents, creeping flows of Newtonian and non-Newtonian fluids, thin films, and nonlinear Fokker-Planck equations. Solutions of both source-type and dam-break problems are analyzed. Closed-form solutions are discussed, when possible, along with a qualitative study of two phase-plane formulations based on two different variable transformations.
Aerosol particles and the formation of advection fog
NASA Technical Reports Server (NTRS)
Hung, R. J.; Liaw, G. S.; Vaughan, O. H., Jr.
1979-01-01
A study of numerical simulation of the effects of concentration, particle size, mass of nuclei, and chemical composition on the dynamics of warm fog formation, particularly the formation of advection fog, is presented. This formation is associated with the aerosol particle characteristics, and both macrophysical and microphysical processes are considered. In the macrophysical model, the evolution of wind components, water vapor content, liquid water content, and potential temperature under the influences of vertical turbulent diffusion, turbulent momentum, and turbulent energy transfers are taken into account. In the microphysical model, the supersaturation effect is incorporated with the surface tension and hygroscopic material solution. It is shown that the aerosol particles with the higher number density, larger size nuclei, the heavier nuclei mass, and the higher ratio of the Van't Hoff factor to the molecular weight favor the formation of the lower visibility advection fogs with stronger vertical energy transfer during the nucleation and condensation time period.
A new iterated two-band diffusion equation: theory and its application.
Shih, Arthur Chun-Chieh; Liao, Hong-Yuan Mark; Lu, Chun-Shien
2003-01-01
In this paper, we propose an iterated two-band filtering method to solve the selective image smoothing problem. We prove that a discrete computation step in an iterated nonlinear diffusion-based filtering algorithm is equivalent to a sequence of operations, including decomposition, regularization, and then reconstruction, in the proposed two-band filtering scheme. To correctly separate the high frequency components from the low frequency ones in the decomposition process, we adopt a dyadic wavelet-based approximation scheme. In the regularization process, we use a diffusivity function as a guide to retain useful data and suppress noises. Finally, the signal of the next stage, which is a "smoother" version of the signal at the previous stage, can be computed by reconstructing the decomposed low frequency component and the regularized high frequency component. Based on the proposed scheme, the smoothing operation can be applied to the correct targets. Experimental results show that our new approach is really efficient in noise removing.
NASA Astrophysics Data System (ADS)
O'Riordan, E.; Shishkin, G. I.
2007-09-01
A priori parameter explicit bounds on the solution of singularly perturbed elliptic problems of convection-diffusion type are established. Regular exponential boundary layers can appear in the solution. These bounds on the solutions and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. By introducing extensions of the coefficients to a larger domain, artificial compatibility conditions are not imposed in the derivation of these decompositions.
NASA Astrophysics Data System (ADS)
Utama, Briandhika; Purqon, Acep
2016-08-01
Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.
NASA Astrophysics Data System (ADS)
Tiguercha, Djlalli; Bennis, Anne-claire; Ezersky, Alexander
2015-04-01
The elliptical motion in surface waves causes an oscillating motion of the sand grains leading to the formation of ripple patterns on the bottom. Investigation how the grains with different properties are distributed inside the ripples is a difficult task because of the segration of particle. The work of Fernandez et al. (2003) was extended from one-dimensional to two-dimensional case. A new numerical model, based on these non-linear diffusion equations, was developed to simulate the grain distribution inside the marine sand ripples. The one and two-dimensional models are validated on several test cases where segregation appears. Starting from an homogeneous mixture of grains, the two-dimensional simulations demonstrate different segregation patterns: a) formation of zones with high concentration of light and heavy particles, b) formation of «cat's eye» patterns, c) appearance of inverse Brazil nut effect. Comparisons of numerical results with the new set of field data and wave flume experiments show that the two-dimensional non-linear diffusion equations allow us to reproduce qualitatively experimental results on particles segregation.
NASA Astrophysics Data System (ADS)
Miyakawa, Erina; Fujii, Hiroyuki; Hattori, Kiyohito; Tatekura, Yuki; Kobayashi, Kazumichi; Watanabe, Masao
2016-12-01
Diffuse optical tomography (DOT), which is still under development, has a potential to enable non-invasive diagnoses of thyroid cancers in the human neck using the near-infrared light. This modality needs a photon migration model because scattered light is used. There are two types of photon migration models: the radiative transport equation (RTE) and diffusion equation (DE). The RTE can describe photon migration in the human neck with accuracy, while the DE enables an efficient calculation. For developing the accurate and efficient model of photon migration, it is crucial to investigate a condition where the DE holds in a scattering medium including a void region under the refractive-index mismatch at the void boundary because the human neck has a trachea (void region) and the refractive indices are different between the human neck and trachea. Hence, in this paper, we compare photon migration using the RTE with that using the DE in the medium. The numerical results show that the DE is valid under the refractive-index match at the void boundary even though the void region is near the source and detector positions. Under the refractive-index mismatch at the boundary, the numerical results using the DE disagree with those using the RTE when the void region is near the source and detector positions. This is probably because the anisotropy of the light scattering remains around the void boundary.
NASA Astrophysics Data System (ADS)
Bleibel, Johannes; Domínguez, Alvaro; Oettel, Martin
2016-06-01
We build on an existing approximation scheme to the Smoluchowski equation in order to derive a dynamic density functional theory (DDFT) including two-body hydrodynamic interactions. A generalized diffusion equation and a wavenumber-dependent diffusion coefficient D(k) are derived by linearization in the density fluctuations. The result is applied to a colloidal monolayer at a fluid interface, having bulk-like hydrodynamic interactions and/or interacting via long-ranged capillary forces. In these cases, D(k) shows characteristic singularities as k\\to 0 . The consequences of these singularities are studied by means of analytical perturbation theory, numerical solution of DDFT and simulations for an explicit example: the capillary collapse of a finite, disk-like distribution of particles. There is in general a good agreement between DDFT and simulations if the initial density distributions for the theoretical prediction correspond to the actual initial configurations of simulations, rather than to an average over them. Otherwise, discrepancies arise that are discussed in detail.
Shendeleva, Margarita L; Molloy, John A
2007-09-01
A spatially varying refractive index leads to the bending of photon paths in a medium, which complicates the Monte Carlo modeling of a photon random walk. We show that the process of photon diffusion in a turbid medium with varying refractive index and curved photon paths can be mapped to the diffusion process in a medium with straight photon paths and modified optical properties. Specifically, the diffusion coefficient, the absorption, and the refractive index of the second medium should differ from the corresponding properties of the first medium by the factor of the squared refractive index of the first medium. The specific intensity of light in the second medium will then be equal to the specific intensity in the first medium divided by the same factor, which also means that the photon density distributions in the two media will be identical. In a Monte Carlo simulation the scaling property suggests that two different algorithms can be used to obtain the photon density distribution, namely, the algorithm with curved photon paths and given optical properties and the algorithm with straight photon paths and modified optical properties.
Tarvainen, Tanja; Vauhkonen, Marko; Kolehmainen, Ville; Arridge, Simon R; Kaipio, Jari P
2005-10-21
In this paper, a coupled radiative transfer equation and diffusion approximation model is extended for light propagation in turbid medium with low-scattering and non-scattering regions. The light propagation is modelled with the radiative transfer equation in sub-domains in which the assumptions of the diffusion approximation are not valid. The diffusion approximation is used elsewhere in the domain. The two equations are coupled through their boundary conditions and they are solved simultaneously using the finite element method. The streamline diffusion modification is used to avoid the ray-effect problem in the finite element solution of the radiative transfer equation. The proposed method is tested with simulations. The results of the coupled model are compared with the finite element solutions of the radiative transfer equation and the diffusion approximation and with results of Monte Carlo simulation. The results show that the coupled model can be used to describe photon migration in turbid medium with low-scattering and non-scattering regions more accurately than the conventional diffusion model.
High-resolution two dimensional advective transport
Smith, P.E.; Larock, B.E.
1989-01-01
The paper describes a two-dimensional high-resolution scheme for advective transport that is based on a Eulerian-Lagrangian method with a flux limiter. The scheme is applied to the problem of pure-advection of a rotated Gaussian hill and shown to preserve the monotonicity property of the governing conservation law.
Evolution and advection of solar mesogranulation
NASA Technical Reports Server (NTRS)
Muller, Richard; Auffret, Herve; Roudier, Thierry; Vigneau, Jean; Simon, George W.; Frank, Zoe; Shine, Richard A.; Title, Alan M.
1992-01-01
A three-hour sequence of observations at the Pic du Midi observatory has been obtained which shows the evolution of solar mesogranules from appearance to disappearance with unprecedented clarity. It is seen that the supergranules, which are known to advect the granules with their convective motion, also advect the mesogranules to their boundaries. This process controls the evolution and disappearance of mesogranules.
It is well known that the fate and transport of contaminants in the subsurface are controlled by complex processes including advection, dispersion-diffusion, and chemical reactions. However, the interplay between the physical transport processes and chemical reactions, and their...
Advection-Induced Spectrotemporal Defects in a Free-Electron Laser
Bielawski, S.; Szwaj, C.; Bruni, C.; Garzella, D.; Orlandi, G.L.; Couprie, M.E.
2005-07-15
We evidence numerically and experimentally that advection can induce spectrotemporal defects in a system presenting a localized structure. Those defects in the spectrum are associated with the breakings induced by the drift of the localized solution. The results are based on simulations and experiments performed on the super-ACO free-electron laser. However, we show that this instability can be generalized using a real Ginzburg-Landau equation with (i) advection and (ii) a finite-size supercritical region.
The SMM model as a boundary value problem using the discrete diffusion equation.
Campbell, Joel
2007-12-01
A generalized single-step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.
The SMM Model as a Boundary Value Problem Using the Discrete Diffusion Equation
NASA Technical Reports Server (NTRS)
Campbell, Joel
2007-01-01
A generalized single step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.
Differential operator multiplication method for fractional differential equations
NASA Astrophysics Data System (ADS)
Tang, Shaoqiang; Ying, Yuping; Lian, Yanping; Lin, Stephen; Yang, Yibo; Wagner, Gregory J.; Liu, Wing Kam
2016-11-01
Fractional derivatives play a very important role in modeling physical phenomena involving long-range correlation effects. However, they raise challenges of computational cost and memory storage requirements when solved using current well developed numerical methods. In this paper, the differential operator multiplication method is proposed to address the issues by considering a reaction-advection-diffusion equation with a fractional derivative in time. The linear fractional differential equation is transformed into an integer order differential equation by the proposed method, which can fundamentally fix the aforementioned issues for select fractional differential equations. In such a transform, special attention should be paid to the initial conditions for the resulting differential equation of higher integer order. Through numerical experiments, we verify the proposed method for both fractional ordinary differential equations and partial differential equations.
Helical turbulent Prandtl number in the A model of passive vector advection
NASA Astrophysics Data System (ADS)
Hnatič, M.; Zalom, P.
2016-11-01
Using the field theoretic renormalization group technique in the two-loop approximation, turbulent Prandtl numbers are obtained in the general A model of passive vector advected by fully developed turbulent velocity field with violation of spatial parity introduced via the continuous parameter ρ ranging from ρ =0 (no violation of spatial parity) to |ρ |=1 (maximum violation of spatial parity). Values of A represent a continuously adjustable parameter which governs the interaction structure of the model. In nonhelical environments, we demonstrate that A is restricted to the interval -1.723 ≤A ≤2.800 (rounded to 3 decimal places) in the two-loop order of the field theoretic model. However, when ρ >0.749 (rounded to 3 decimal places), the restrictions may be removed, which means that presence of helicity exerts a stabilizing effect onto the possible stationary regimes of the system. Furthermore, three physically important cases A ∈{-1 ,0 ,1 } are shown to lie deep within the allowed interval of A for all values of ρ . For the model of the linearized Navier-Stokes equations (A =-1 ) up to date unknown helical values of the turbulent Prandtl number have been shown to equal 1 regardless of parity violation. Furthermore, we have shown that interaction parameter A exerts strong influence on advection-diffusion processes in turbulent environments with broken spatial parity. By varying A continuously, we explain high stability of the kinematic MHD model (A =1 ) against helical effects as a result of its proximity to the A =0.912 (rounded to 3 decimal places) case where helical effects are completely suppressed. Contrary, for the physically important A =0 model, we show that it lies deep within the interval of models where helical effects cause the turbulent Prandtl number to decrease with |ρ | . We thus identify internal structure of interactions given by the parameter A , and not the vector character of the admixture itself being the dominant factor influencing
Helical turbulent Prandtl number in the A model of passive vector advection.
Hnatič, M; Zalom, P
2016-11-01
Using the field theoretic renormalization group technique in the two-loop approximation, turbulent Prandtl numbers are obtained in the general A model of passive vector advected by fully developed turbulent velocity field with violation of spatial parity introduced via the continuous parameter ρ ranging from ρ=0 (no violation of spatial parity) to |ρ|=1 (maximum violation of spatial parity). Values of A represent a continuously adjustable parameter which governs the interaction structure of the model. In nonhelical environments, we demonstrate that A is restricted to the interval -1.723≤A≤2.800 (rounded to 3 decimal places) in the two-loop order of the field theoretic model. However, when ρ>0.749 (rounded to 3 decimal places), the restrictions may be removed, which means that presence of helicity exerts a stabilizing effect onto the possible stationary regimes of the system. Furthermore, three physically important cases A∈{-1,0,1} are shown to lie deep within the allowed interval of A for all values of ρ. For the model of the linearized Navier-Stokes equations (A=-1) up to date unknown helical values of the turbulent Prandtl number have been shown to equal 1 regardless of parity violation. Furthermore, we have shown that interaction parameter A exerts strong influence on advection-diffusion processes in turbulent environments with broken spatial parity. By varying A continuously, we explain high stability of the kinematic MHD model (A=1) against helical effects as a result of its proximity to the A=0.912 (rounded to 3 decimal places) case where helical effects are completely suppressed. Contrary, for the physically important A=0 model, we show that it lies deep within the interval of models where helical effects cause the turbulent Prandtl number to decrease with |ρ|. We thus identify internal structure of interactions given by the parameter A, and not the vector character of the admixture itself being the dominant factor influencing diffusion-advection
NASA Technical Reports Server (NTRS)
Peltier, Leonard Joel; Biringen, Sedat; Chait, Arnon
1990-01-01
Implicit techniques for calculating three-dimensional, time-dependent heat diffusion in a cube are tested with emphasis on storage efficiency, accuracy, and speed of calculation. For this purpose, a tensor product technique with both Chebyshev collocation and finite differences and a generalized conjugate gradient technique with finite differences are used in conjunction with Crank-Nicolson discretization. An Euler explicit finite difference calculation is performed for use as a benchmark. The implicit techniques are found to be competitive with the Euler explicit method in terms of storage efficiency and speed of calculation and offer advantages both in accuracy and stability. Mesh stretching in the finite difference calculations is shown to markedly improve the accuracy of the solution.
Quantifying the Effects of Noise on Diffuse Interface Models: Cahn-Hilliard-Cook equations
NASA Astrophysics Data System (ADS)
Pfeifer, Spencer; Ganapathysubramanian, Baskar
2015-03-01
We present an investigation into the dynamics of phase separation through numerical simulations of the Cahn-Hilliard-Cook (CHC) equation. This model is an extension of the well-known Cahn- Hilliard equation, perturbed by an additive white noise. Studies have shown that random fluctuations are critical for proper resolution of physical phenomena. This is especially true for phase critical systems. We explore the transient behavior of the solution space for varying levels of noise. This is enabled by our massively scalable finite element-based numerical framework. We briefly examine the interplay between noise level and discretization (spatial and temporal) in obtaining statistically consistent solutions. We show that the added noise accelerates progress towards phase separation, but retards dynamics throughout subsequent coarsening. We identify a scaling exponent relating morphology metrics with the level of noise. We observe a very clear scaling effect of finite domain size, which is observed to be offset by increasing levels of noise. Domain scaling reveals a clear microstructural asymmetry at various stages of the evolution for lower noise levels. In contrast, higher noise levels tend to produce more uniform morphologies.
Gong, Chunye; Bao, Weimin; Tang, Guojian; Jiang, Yuewen; Liu, Jie
2014-01-01
It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE) with iterative implicit finite difference method is O(M(x)M(y)N(2)). In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16-4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.