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.
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. PMID:27176431
NASA Astrophysics Data System (ADS)
Ancey, Christophe; Bohorquez, Patricio; Heyman, Joris
2016-04-01
The advection-diffusion equation arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Stochastic models can also be used to derive this equation, with the significant advantage that they provide information on the statistical properties of particle activity. Stochastic models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. We develop an approach based on birth-death Markov processes, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received little attention. We show that particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particle displacement and the dimensions of the measurement system.
Technology Transfer Automated Retrieval System (TEKTRAN)
This paper presents a formal exact solution of the linear advection-diffusion transport equation with constant coefficients for both transient and steady-state regimes. A classical mathematical substitution transforms the original advection-diffusion equation into an exclusively diffusive equation. ...
General solution of a fractional diffusion-advection equation for solar cosmic-ray transport
NASA Astrophysics Data System (ADS)
Rocca, M. C.; Plastino, A. R.; Plastino, A.; Ferri, G. L.; de Paoli, A.
2016-04-01
In this effort we exactly solve the fractional diffusion-advection equation for solar cosmic-ray transport and give its general solution in terms of hypergeometric distributions. Numerical analysis of this equation shows that its solutions resemble power-laws.
NASA Astrophysics Data System (ADS)
Huber, Markus; Tailleux, Remi; Ferreira, David; Kuhlbrodt, Till; Gregory, Jonathan
2015-04-01
The classic vertical advection-diffusion (VAD) balance is a central concept in studying the ocean heat budget, in particular in simple climate models (SCMs). Here we present a new framework to calibrate the parameters of the VAD equation to the vertical ocean heat balance of two fully-coupled climate models that is traceable to the models' circulation as well as to vertical mixing and diffusion processes. Based on temperature diagnostics, we derive an effective vertical velocity w∗ and turbulent diffusivity kν∗ for each individual physical process. In steady state, we find that the residual vertical velocity and diffusivity change sign in middepth, highlighting the different regional contributions of isopycnal and diapycnal diffusion in balancing the models' residual advection and vertical mixing. We quantify the impacts of the time evolution of the effective quantities under a transient 1% CO2 simulation and make the link to the parameters of currently employed SCMs.
NASA Astrophysics Data System (ADS)
Ancey, C.; Bohorquez, P.; Heyman, J.
2015-12-01
The advection-diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it, with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in rivers. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection-diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection-diffusion equation for particle activity. Yet while this derivation is formally valid in the continuum limit, it runs into difficulty in practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due
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.
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.
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 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.
NASA Astrophysics Data System (ADS)
Kordilla, Jannes; Pan, Wenxiao; Tartakovsky, Alexandre
2014-12-01
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.
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)
Appadu, A. R.
2016-06-01
An unconditionally positive definite scheme has been derived in [1] to approximate a linear advection-diffusion-reaction equation which models exponential travelling waves and the coefficients of advective, diffusive and reactive terms have been chosen as one. The scheme has been baptised as Unconditionally Positive Finite Difference (UPFD). In this work, we use the UPFD scheme to solve the advection-diffusion-reaction problem in [1] and we also extend our study to three other important regimes involved in this model. The temporal step size is varied while fixing the spatial step size. We compute some errors namely; L1 error, dispersion, dissipation errors. We also study the variation of the modulus of the exact amplification factor, modulus of amplification factor of the scheme and relative phase error, all vs the phase angle for the four different regimes.
NASA Astrophysics Data System (ADS)
Rubbab, Qammar; Mirza, Itrat Abbas; Qureshi, M. Zubair Akbar
2016-07-01
The time-fractional advection-diffusion equation with Caputo-Fabrizio fractional derivatives (fractional derivatives without singular kernel) is considered under the time-dependent emissions on the boundary and the first order chemical reaction. The non-dimensional problem is formulated by using suitable dimensionless variables and the fundamental solutions to the Dirichlet problem for the fractional advection-diffusion equation are determined using the integral transforms technique. The fundamental solutions for the ordinary advection-diffusion equation, fractional and ordinary diffusion equation are obtained as limiting cases of the previous model. Using Duhamel's principle, the analytical solutions to the Dirichlet problem with time-dependent boundary pulses have been obtained. The influence of the fractional parameter and of the drift parameter on the solute concentration in various spatial positions was analyzed by numerical calculations. It is found that the variation of the fractional parameter has a significant effect on the solute concentration, namely, the memory effects lead to the retardation of the mass transport.
NASA Astrophysics Data System (ADS)
Mudunuru, M. K.; Nakshatrala, K.
2012-12-01
Advection-Diffusion-Reaction (ADR) equations naturally arises in many physical phenomena, which include seepage of contaminants in heterogeneous porous media, transport of injected tracers due to the flow of oil in a petroleum reservoir, and degradation of a deformable solid due to diffusing chemical species. Vast literature exists on how to solve this equation in the cases when the medium is isotropic, velocity field being divergence free, and for advection-dominated problems. However, it is well know that many popular finite element formulations (e.g., the standard Galerkin formulation, stabilized methods, variational multi-scale methods, subgrid-scale methods, and primitive least-squares formulations) do not satisfy element-by-element mass/species balance and do not produce non-negative solutions on general computational grids. Various post-processing based methods were developed in order to recover some properties of computed numerical solutions. Most of these post-processing techniques are ad hoc, and are not variationally consistent. In this poster, we shall present a novel numerical methodology for ADR equations that satisfy discrete maximum principles, the non-negative constraint, and element-by-element mass/species balance. The methodology can handle general computational grids, no additional restrictions on time-step, and for heterogeneous anisotropic media. Several numerical results pertinent to advection-dominated ADR problems will be presented to illustrate the performance of the proposed numerical formulation.
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...
EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD FOR THE ADVECTION-DIFFUSION EQUATION
Many numerical methods use characteristic analysis to accommodate the advective component of transport. uch characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. eneralization of characteristic...
MAGNETIC ADVECTION DUE TO DIFFUSIVITY GRADIENTS
NASA Astrophysics Data System (ADS)
Zita, E. J.
2009-12-01
We derive and discuss an important source of advection of magnetic fields in plasmas, for a completely general case. Magnetic diffusivity is proportional to electrical resistivity: where the value this parameter is high, it is well known that magnetic fields can leak (or diffuse) rapidly into (or out) of the plasma. Magnetohydrodynamic lore has it that where gradients, or changes in space, of the value of the diffusivity are high, magnetic fields can have enhanced flow (or advection). We derive this phenomenon rigorously, compare our results to other work in the literature, and discuss its implications, especially for kinematic dynamos. As an extra mathematical bonus, we find that the magnetic advection due to diffusivity gradients can be expressed in terms of a diffusion equation within the induction equation, making its computational implementation especially simple.
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.
Adaptive domain decomposition methods for advection-diffusion problems
Carlenzoli, C.; Quarteroni, A.
1995-12-31
Domain decomposition methods can perform poorly on advection-diffusion equations if diffusion is dominated by advection. Indeed, the hyperpolic part of the equations could affect the behavior of iterative schemes among subdomains slowing down dramatically their rate of convergence. Taking into account the direction of the characteristic lines we introduce suitable adaptive algorithms which are stable with respect to the magnitude of the convective field in the equations and very effective on bear boundary value problems.
Spiral defect chaos in an advection-reaction-diffusion system
NASA Astrophysics Data System (ADS)
Affan, H.; Friedrich, R.
2014-06-01
This paper comprises numerical and theoretical studies of spatiotemporal patterns in advection-reaction-diffusion systems in which the chemical species interact with the hydrodynamic fluid. Due to the interplay between the two, we obtained the spiral defect chaos in the activator-inhibitor-type model. We formulated the generalized Swift-Hohenberg-type model for this system. Then the evolution of fractal boundaries due to the effect of the strong nonlinearity at the interface of the two chemical species is studied numerically. The purpose of the present paper is to point out that spiral defect chaos, observed in model equations of the extended Swift-Hohenberg equation for low Prandtl number convection, may actually be obtained also in certain advection-reaction-diffusion systems.
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.
NASA Astrophysics Data System (ADS)
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
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
Advection and diffusion in shoreline change prediction
NASA Astrophysics Data System (ADS)
Anderson, T. R.; Frazer, L. N.
2010-12-01
We added longshore advection and diffusion to the simple cross-shore rate calculation method, as used widely by the USGS and others, to model historic shorelines and to predict future shoreline positions; and applied this to Hawaiian Island beach data. Aerial photographs, sporadically taken throughout the past century, yield usable, albeit limited, historic shoreline data. These photographs provide excellent spatial coverage, but poor temporal resolution, of the shoreline. Due to the sparse historic shoreline data, and the many natural and anthropogenic events influencing coastlines, we constructed a simplistic shoreline change model that can identify long-term behavior of a beach. Our new, two-dimensional model combines the simple rate method to accommodate for cross-shore sediment transport with the classic Pelnard-Considère model for diffusion, as well as a longshore advection speed term. Inverse methods identify cross-shore rate, longshore advection speed, and longshore diffusivity down a sandy coastline. A spatial averaging technique then identifies shoreline segments where one parameter can reasonably account for the cross-shore and longshore transport rates in that area. This produces model results with spatial resolution more appropriate to the temporal spacing of the data. Because changes in historic data can be accounted for by varying degrees of cross-shore and longshore sediment transport - for example, beach erosion can equally be explained by sand moving either off-shore or laterally - we tested several different model scenarios on the data: allowing only cross-shore sediment movement, only longshore movement, and a combination of the two. We used statistical information criteria to determine both the optimal spatial resolution and best-fitting scenario. Finally, we employed a voting method predicting the relaxed shoreline position over time.
Advection, diffusion and delivery over a network
Heaton, Luke L.M.; López, Eduardo; Maini, Philip K.; Fricker, Mark D.; Jones, Nick S.
2014-01-01
Many biological, geophysical and technological systems involve the transport of resource over a network. In this paper we present an algorithm for calculating the exact concentration of resource at any point in space or time, given that the resource in the network is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. We consider the implications of advection, diffusion and delivery for simple models of glucose delivery through a vascular network, and conclude that in certain circumstances, increasing the volume of blood and the number of glucose transporters can actually decrease the total rate of glucose delivery. We also consider the case of empirically determined fungal networks, and analyze the distribution of resource that emerges as such networks grow over time. Fungal growth involves the expansion of fluid filled vessels, which necessarily involves the movement of fluid. In three empirically determined fungal networks we found that the minimum currents consistent with the observed growth would effectively transport resource throughout the network over the time-scale of growth. This suggests that in foraging fungi, the active transport mechanisms observed in the growing tips may not be required for long range transport. PMID:23005783
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...
Advective and diffusive cosmic ray transport in galactic haloes
NASA Astrophysics Data System (ADS)
Heesen, Volker; Dettmar, Ralf-Jürgen; Krause, Marita; Beck, Rainer; Stein, Yelena
2016-05-01
We present 1D cosmic ray transport models, numerically solving equations of pure advection and diffusion for the electrons and calculating synchrotron emission spectra. We find that for exponential halo magnetic field distributions advection leads to approximately exponential radio continuum intensity profiles, whereas diffusion leads to profiles that can be better approximated by a Gaussian function. Accordingly, the vertical radio spectral profiles for advection are approximately linear, whereas for diffusion they are of `parabolic' shape. We compare our models with deep Australia Telescope Compact Array observations of two edge-on galaxies, NGC 7090 and 7462, at λλ 22 and 6 cm. Our result is that the cosmic ray transport in NGC 7090 is advection dominated with V=150^{+80}_{-30} km s^{-1}, and that the one in NGC 7462 is diffusion dominated with D=3.0± 1.0 × 10^{28}E_GeV^{0.5} cm^2 s^{-1}. NGC 7090 has both a thin and thick radio disc with respective magnetic field scale heights of hB1 = 0.8 ± 0.1 kpc and hB2 = 4.7 ± 1.0 kpc. NGC 7462 has only a thick radio disc with hB2 = 3.8 ± 1.0 kpc. In both galaxies, the magnetic field scale heights are significantly smaller than what estimates from energy equipartition would suggest. A non-negligible fraction of cosmic ray electrons can escape from NGC 7090, so that this galaxy is not an electron calorimeter.
Spectral approximation to advection-diffusion problems by the fictitious interface method
NASA Astrophysics Data System (ADS)
Frati, A.; Pasquarelli, F.; Quarteroni, A.
1993-08-01
The algorithmic aspects of the 'fictitious interface' method used in numerical approximations of convection-dominated flows are discussed. The solution algorithm presented alternates the advection-equation solution with that of the advection-diffusion equation within complementary subdomains. For the problems presently considered, spatial discretization is obtained by the spectral collocation method via Legendre-Gaussian modes. Attention is given to the the fictitious interface method's application to the Burgers equation.
Advection, diffusion, and delivery over a network
NASA Astrophysics Data System (ADS)
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.
A fully implicit method for 3D quasi-steady state magnetic advection-diffusion.
Siefert, Christopher; Robinson, Allen Conrad
2009-09-01
We describe the implementation of a prototype fully implicit method for solving three-dimensional quasi-steady state magnetic advection-diffusion problems. This method allows us to solve the magnetic advection diffusion equations in an Eulerian frame with a fixed, user-prescribed velocity field. We have verified the correctness of method and implementation on two standard verification problems, the Solberg-White magnetic shear problem and the Perry-Jones-White rotating cylinder problem.
Chaotic advection, diffusion, and reactions in open flows
Tel, Tamas; Karolyi, Gyoergy; Pentek, Aron; Scheuring, Istvan; Toroczkai, Zoltan; Grebogi, Celso; Kadtke, James
2000-03-01
We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.
Shadowing and the role of small diffusivity in the chaotic advection of scalars
NASA Technical Reports Server (NTRS)
Klapper, I.
1992-01-01
Using techniques from shadowing theory, the solution of the scalar advection-diffusion equation is studied. It is shown that, under certain circumstances, the effect of small scalar diffusivity is to smooth the zero-diffusivity solution by averaging local fine-scaled structure against a Gaussian. The method of study depends on shadowing and thus fails for nonuniformly stretching systems, its failure suggesting the ways in which the effects of asymptotically small molecular diffusion can become nonlocal in chaotic fluid flows.
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.
Multirate Runge-Kutta schemes for advection equations
NASA Astrophysics Data System (ADS)
Schlegel, Martin; Knoth, Oswald; Arnold, Martin; Wolke, Ralf
2009-04-01
Explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena. The wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step. Multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps, resulting in a moderate to substantial reduction of the computational cost, depending on the scenario to simulate [S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput. 41 (1983) 321-336; H. Tang, G. Warnecke, A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and pace grids, SIAM J. Sci. Comput. 26 (4) (2005) 1415-1431; E. Constantinescu, A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, SIAM J. Sci. Comput. 33 (3) (2007) 239-278]. In air pollution modeling the advection part is usually integrated explicitly in time, where the time step is constrained by a locally varying Courant-Friedrichs-Lewy (CFL) number. Multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction. Therefore it is of major interest to apply multirate schemes to the advection equation. We introduce a generic recursive multirate Runge-Kutta scheme that can be easily adapted to an arbitrary number of refinement levels. It preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit Runge-Kutta methods as base method.
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.
Anomalous diffusion of a tracer advected by wave turbulence
NASA Astrophysics Data System (ADS)
Balk, Alexander M.
2001-02-01
We consider the advection of a passive tracer when the velocity field is a superposition of random waves. Green's function for the turbulent transport (turbulent diffusion and turbulent drift) is derived. This Green's function is shown to imply sub-diffusive or super-diffusive behavior of the tracer. For the analysis we introduce the statistical near-identity transformation. The results are confirmed by numerical simulations.
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...
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.
Super-diffusion versus competitive advection: a simulation
NASA Astrophysics Data System (ADS)
Del Moro, D.; Giannattasio, F.; Berrilli, F.; Consolini, G.; Lepreti, F.; Gošić, M.
2015-04-01
Context. Magnetic element tracking is often used to study the transport and diffusion of the magnetic field on the solar photosphere. From the analysis of the displacement spectrum of these tracers, it has recently been agreed that a regime of super-diffusivity dominates the solar surface. Quite habitually this result is discussed in the framework of fully developed turbulence. Aims: However, the debate whether the super-diffusivity is generated by a turbulent dispersion process, by the advection due to the convective pattern, or even by another process is still open, as is the question of the amount of diffusivity at the scales relevant to the local dynamo process. Methods: To understand how such peculiar diffusion in the solar atmosphere takes place, we compared the results from two different data sets (ground-based and space-borne) and developed a simulation of passive tracers advection by the deformation of a Voronoi network. Results: The displacement spectra of the magnetic elements obtained by the data sets are consistent in retrieving a super-diffusive regime for the solar photosphere, but the simulation also shows a super-diffusive displacement spectrum: its competitive advection process can reproduce the signature of super-diffusion. Conclusions: Therefore, it is not necessary to hypothesize a totally developed turbulence regime to explain the motion of the magnetic elements on the solar surface.
NASA Astrophysics Data System (ADS)
Hamdi, Adel
2009-11-01
This paper deals with the identification of a point source (localization of its position and recovering the history of its time-varying intensity function) that constitutes the right-hand side of the first equation in a system of two coupled 1D linear transport equations. Assuming that the source intensity function vanishes before reaching the final control time, we prove the identifiability of the sought point source from recording the state relative to the second coupled transport equation at two observation points framing the source region. Note that at least one of the two observation points should be strategic. We establish an identification method that uses these records to identify the source position as the root of a continuous and strictly monotonic function. Whereas the source intensity function is recovered using a recursive formula without any need of an iterative process. Some numerical experiments on a variant of the surface water pollution BOD-OD coupled model are presented.
Multiple anisotropic collisions for advection-diffusion Lattice Boltzmann schemes
NASA Astrophysics Data System (ADS)
Ginzburg, Irina
2013-01-01
This paper develops a symmetrized framework for the analysis of the anisotropic advection-diffusion Lattice Boltzmann schemes. Two main approaches build the anisotropic diffusion coefficients either from the anisotropic anti-symmetric collision matrix or from the anisotropic symmetric equilibrium distribution. We combine and extend existing approaches for all commonly used velocity sets, prescribe most general equilibrium and build the diffusion and numerical-diffusion forms, then derive and compare solvability conditions, examine available anisotropy and stable velocity magnitudes in the presence of advection. Besides the deterioration of accuracy, the numerical diffusion dictates the stable velocity range. Three techniques are proposed for its elimination: (i) velocity-dependent relaxation entries; (ii) their combination with the coordinate-link equilibrium correction; and (iii) equilibrium correction for all links. Two first techniques are also available for the minimal (coordinate) velocity sets. Even then, the two-relaxation-times model with the isotropic rates often gains in effective stability and accuracy. The key point is that the symmetric collision mode does not modify the modeled diffusion tensor but it controls the effective accuracy and stability, via eigenvalue combinations of the opposite parity eigenmodes. We propose to reduce the eigenvalue spectrum by properly combining different anisotropic collision elements. The stability role of the symmetric, multiple-relaxation-times component, is further investigated with the exact von Neumann stability analysis developed in diffusion-dominant limit.
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.
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.
CDF Solutions of Advection-Reaction equations with uncertain parameters (Invited)
NASA Astrophysics Data System (ADS)
Boso, F.; Tartakovsky, D. M.
2013-12-01
Flow and transport models are affected by parametric uncertainty. Quantitative forecasting of such processes in natural porous media are especially prone to uncertainty because of the inaccessibility and multi-scale nature of the subsurface. We consider a reduced-complexity stochastic transport system which takes into account advection and nonlinear reactions in advection-reaction equations (AREs) with uncertain (random) velocity and reaction parameters. We derive a deterministic equation that governs the evolution of cumulative distribution function (CDF) of a solution of the underlying ARE. Although requiring closure, this differential equation benefits from uniquely defined boundary and initial conditions and can be solved with classic techniques. Here we analyze the accuracy and robustness of the large-eddy-diffusivity closure by comparison with Monte Carlo simulations for different correlation structures and parameters.
A balancing domain decomposition method by constraints for advection-diffusion problems
Tu, Xuemin; Li, Jing
2008-12-10
The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advection-diffusion equations. A pre-conditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. In the preconditioning step of each iteration, a partially sub-assembled finite element problem is solved. A convergence rate estimate for the GMRES iteration is established, under the condition that the diameters of subdomains are small enough. It is independent of the number of subdomains and grows only slowly with the subdomain problem size. Numerical experiments for several two-dimensional advection-diffusion problems illustrate the fast convergence of the proposed algorithm.
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.
NASA Astrophysics Data System (ADS)
Darbandi, Masoud; Ghafourizadeh, Majid
2015-12-01
In this work, we derive a few new advective flux approximation expressions, apply them in a hybrid finite-volume-element (FVE) formulation, and solve the turbulent reacting flow governing equations in the cylindrical frame. To derive these advective-kinetic-based expressions, we benefit from the advantages of a physical influence scheme (PIS) basically, extend it to the cylindrical frame suitably, and approximate the required advective flux terms at the cell faces more accurately. The present numerical scheme not only respects the physics of flow correctly but also resolves the pressure-velocity coupling problem automatically. We also suggest a bi-implicit algorithm to solve the set of coupled turbulent reacting flow governing equations, in which the turbulence and chemistry governing equations are solved simultaneously. To evaluate the accuracy of new derived FVE-PIS expressions, we compare the current solutions with other available numerical solutions and experimental data. The comparisons show that the new derived expressions provide some more advantages over the past numerical approaches in solving turbulent diffusion flame in the cylindrical frame. Indeed, the current method and formulations can be used to solve and analyze the turbulent diffusion flames in the cylindrical coordinates very reliably.
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.
Multinomial diffusion equation
NASA Astrophysics Data System (ADS)
Balter, Ariel; Tartakovsky, Alexandre M.
2011-06-01
We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N→∞, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
Multinomial diffusion equation
Balter, Ariel I.; Tartakovsky, Alexandre M.
2011-06-24
We describe a new, microscopic model for diffusion that captures diffusion induced uctuations at scales where the concept of concentration gives way to discrete par- ticles. We show that in the limit as the number of particles N ! 1, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
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
Exact PDF equations and closure approximations for advective-reactive transport
Venturi, D.; Tartakovsky, Daniel M.; Tartakovsky, Alexandre M.; Karniadakis, George E.
2013-06-01
Mathematical models of advection–reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection–reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty.
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.
Multinomial Diffusion Equation
Balter, Ariel I.; Tartakovsky, Alexandre M.
2011-06-01
We have developed a novel stochastic, space/time discrete representation of particle diffusion (e.g. Brownian motion) based on discrete probability distributions. We show that in the limit of both very small time step and large concentration, our description is equivalent to the space/time continuous stochastic diffusion equation. Being discrete in both time and space, our model can be used as an extremely accurate, efficient, and stable stochastic finite-difference diffusion algorithm when concentrations are so small that computationally expensive particle-based methods are usually needed. Through numerical simulations, we show that our method can generate realizations that capture the statistical properties of particle simulations. While our method converges converges to both the correct ensemble mean and ensemble variance very quickly with decreasing time step, but for small concentration, the stochastic diffusion PDE does not, even for very small time steps.
Numerical Modeling of Deep Mantle Convection: Advection and Diffusion Schemes for Marker Methods
NASA Astrophysics Data System (ADS)
Mulyukova, Elvira; Dabrowski, Marcin; Steinberger, Bernhard
2013-04-01
Thermal and chemical evolution of Earth's deep mantle can be studied by modeling vigorous convection in a chemically heterogeneous fluid. Numerical modeling of such a system poses several computational challenges. Dominance of heat advection over the diffusive heat transport, and a negligible amount of chemical diffusion results in sharp gradients of thermal and chemical fields. The exponential dependence of the viscosity of mantle materials on temperature also leads to high gradients of the velocity field. The accuracy of many numerical advection schemes degrades quickly with increasing gradient of the solution, while the computational effort, in terms of the scheme complexity and required resolution, grows. Additional numerical challenges arise due to a large range of length-scales characteristic of a thermochemical convection system with highly variable viscosity. To examplify, the thickness of the stem of a rising thermal plume may be a few percent of the mantle thickness. An even thinner filament of an anomalous material that is entrained by that plume may consitute less than a tenth of a percent of the mantle thickness. We have developed a two-dimensional FEM code to model thermochemical convection in a hollow cylinder domain, with a depth- and temperature-dependent viscosity representative of the mantle (Steinberger and Calderwood, 2006). We use marker-in-cell method for advection of chemical and thermal fields. The main advantage of perfoming advection using markers is absence of numerical diffusion during the advection step, as opposed to the more diffusive field-methods. However, in the common implementation of the marker-methods, the solution of the momentum and energy equations takes place on a computational grid, and nodes do not generally coincide with the positions of the markers. Transferring velocity-, temperature-, and chemistry- information between nodes and markers introduces errors inherent to inter- and extrapolation. In the numerical scheme
Fractional Advective-Dispersive Equation as a Model of Solute Transport in Porous Media
Technology Transfer Automated Retrieval System (TEKTRAN)
Understanding and modeling transport of solutes in porous media is a critical issue in the environmental protection. The common model is the advective-dispersive equation (ADE) describing the superposition of the advective transport and the Brownian motion in water-filled pore space. Deviations from...
Multiscale numerical methods for passive advection-diffusion in incompressible turbulent flow fields
NASA Astrophysics Data System (ADS)
Lee, Yoonsang; Engquist, Bjorn
2016-07-01
We propose a seamless multiscale method which approximates the macroscopic behavior of the passive advection-diffusion equations with steady incompressible velocity fields with multi-spatial scales. The method uses decompositions of the velocity fields in the Fourier space, which are similar to the decomposition in large eddy simulations. It also uses a hierarchy of local domains with different resolutions as in multigrid methods. The effective diffusivity from finer scale is used for the next coarser level computation and this process is repeated up to the coarsest scale of interest. The grids are only in local domains whose sizes decrease depending on the resolution level so that the overall computational complexity increases linearly as the number of different resolution grids increases. The method captures interactions between finer and coarser scales but has to sacrifice some of interactions between different scales. The proposed method is numerically tested with 2D examples including a successful approximation to a continuous spectrum flow.
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...
NASA Astrophysics Data System (ADS)
Karimi, S.; Nakshatrala, K. B.
2014-12-01
Advection-Diffusion-Reaction (ADR) equations play a crucial role in simulating numerous geo- physical phenomena. It is well-known that the solution to these equations exhibit disparate spatial and temporal scales. These mathematical scales occur due to relative dominance of either advec- tion, diffusion, or reaction processes. Hence, in a careful simulation, one has to choose appropriate time-integrators, time-steps, and numerical formulations for spatial discretization. Multi-time-step coupling methods allow specific choice of integration methods (either temporal or spatial) in dif- ferent regions of the spatial domain. In recent years, most of the attempts to design monolithic multi-time-step frameworks favored second-order transient systems in structural dynamics. In this presentation, we will introduce monolithic multi-time-step computational frameworks for ADR equations. These methods are based on the theory of differential/algebraic equations. We shall also provide an overview of results from stability analysis, study of drift from compatibility con- straints, and analysis of influence of perturbations. Several benchmark problems will be utilized to demonstrate the theoretical findings and features of the proposed frameworks. Finally, application of the proposed methods to fast bimolecular reactive systems will be shown.
Bad behavior of Godunov mixed methods for strongly anisotropic advection-dispersion equations
NASA Astrophysics Data System (ADS)
Mazzia, Annamaria; Manzini, Gianmarco; Putti, Mario
2011-09-01
We study the performance of Godunov mixed methods, which combine a mixed-hybrid finite element solver and a Godunov-like shock-capturing solver, for the numerical treatment of the advection-dispersion equation with strong anisotropic tensor coefficients. It turns out that a mesh locking phenomenon may cause ill-conditioning and reduce the accuracy of the numerical approximation especially on coarse meshes. This problem may be partially alleviated by substituting the mixed-hybrid finite element solver used in the discretization of the dispersive (diffusive) term with a linear Galerkin finite element solver, which does not display such a strong ill conditioning. To illustrate the different mechanisms that come into play, we investigate the spectral properties of such numerical discretizations when applied to a strongly anisotropic diffusive term on a small regular mesh. A thorough comparison of the stiffness matrix eigenvalues reveals that the accuracy loss of the Godunov mixed method is a structural feature of the mixed-hybrid method. In fact, the varied response of the two methods is due to the different way the smallest and largest eigenvalues of the dispersion (diffusion) tensor influence the diagonal and off-diagonal terms of the final stiffness matrix. One and two dimensional test cases support our findings.
A Novel Electrical Model for Advection-Diffusion-Based Molecular Communication in Nanonetworks.
Azadi, Mehdi; Abouei, Jamshid
2016-04-01
In this paper, we propose an end-to-end electrical model to characterize the communication between two nanomachines via advection-diffusion motion along the conventional pipe medium. For this modeling, we consider three modules consisting of transmitter, advection-diffusion propagation and receiver. The modulation scheme and releasing molecules through the conventional pipe medium from the transmitter nanomachine is represented in the transmitter module. The advection-diffusion propagation of molecules along the flow-induced path is shown in advection-diffusion propagation module, and the demodulation scheme of bounded particles at the receiver nanomachine is characterized in the receiver module. Our objective is to find an electrical model of each module under the zero initial condition which enables us to represent the electrical circuit related to each module. The transmitter and the signal propagation models are built on the basis of the molecular advection-diffusion physics, whereas the receiver model is interpreted by stemming from the theory of the ligand-receptor binding chemical process. In addition, we employ the transfer function of modules to derive the normalized gain and the delay of each module. Supported by numerical results, we analyze the effect of physical parameters of the pipe medium on the total normalized gain and delay of molecular communications. PMID:27046879
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.
Kim, Hong; Benson, Craig H
2004-07-01
The relative contributions of four mechanisms of oxygen transport in multilayer composite (MLC) caps placed over oxygen-consuming mine waste were evaluated using numerical and analytical methods. MLC caps are defined here as caps consisting of earthen and geosynthetic (polymeric) components where a composite barrier layer consisting of a geomembrane (1-2 mm thick polymeric sheet) overlying a clay layer is the primary barrier to transport. The transport mechanisms that were considered are gas-phase advective transport, gas-phase diffusive transport, liquid-phase advective transport via infiltrating precipitation and liquid-phase diffusive transport. A numerical model was developed to simulate gas-phase advective-diffusive transport of oxygen through a multilayer cap containing seven layers. This model was also used to simulate oxygen diffusion in the liquid phase. An approximate analytical method was used to compute the advective flux of oxygen in the liquid phase. The numerical model was verified for limiting cases using an analytical solution. Comparisons were also made between model predictions and field data for earthen caps reported by others. Results of the analysis show that the dominant mechanism for oxygen transport through MLC caps is gas-phase diffusion. For the cases that were considered, the gas-phase diffusive flux typically comprises at least 99% of the total oxygen flux. Thus, designers of MLC caps should focus on design elements and features that will limit diffusion of gas-phase oxygen. PMID:15145567
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.
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.
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.
Magnetic flux and heat losses by diffusive, advective, and Nernst effects in MagLIF-like plasma
NASA Astrophysics Data System (ADS)
Velikovich, A. L.; Giuliani, J. L.; Zalesak, S. T.
2014-12-01
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 ωeτe effective diffusion coefficients determining the losses of heat and magnetic flux are both shown to decrease with ωeτe 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.
Parashar, R.; Cushman, J.H.
2008-06-20
Microbial motility is often characterized by 'run and tumble' behavior which consists of bacteria making sequences of runs followed by tumbles (random changes in direction). As a superset of Brownian motion, Levy motion seems to describe such a motility pattern. The Eulerian (Fokker-Planck) equation describing these motions is similar to the classical advection-diffusion equation except that the order of highest derivative is fractional, {alpha} element of (0, 2]. The Lagrangian equation, driven by a Levy measure with drift, is stochastic and employed to numerically explore the dynamics of microbes in a flow cell with sticky boundaries. The Eulerian equation is used to non-dimensionalize parameters. The amount of sorbed time on the boundaries is modeled as a random variable that can vary over a wide range of values. Salient features of first passage time are studied with respect to scaled parameters.
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. PMID:26063527
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...
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
NASA Astrophysics Data System (ADS)
Pelosi, A.; Schumer, R.; Parker, G.; Ferguson, R. I.
2016-03-01
Tracer pebbles are often used to study bed load transport processes in gravel bed rivers. Models have been proposed for their downstream dispersion, and also for vertical dispersion, but not for the combined effects of downstream and vertical movement. Here we use the Exner-Based Master Equation to characterize the transient coevolution of streamwise and vertical advection-diffusion of tracer pebbles under equilibrium transport conditions (no net aggradation or degradation). The coevolution of streamwise and vertical dispersion gives rise to behavior that can differ markedly from that associated with purely streamwise processes with no vertical exchange. One example is streamwise advective slowdown. Particles that are advected downward into zones where the probability of reentrainment becomes asymptotically small are essentially trapped and can no longer participate in streamwise advection. As a result, the mean streamwise velocity of the tracer plume declines in time. Qualitative and quantitative comparisons with two field experiments show encouraging agreement despite the simplified boundary conditions in the model.
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.
Fractional-calculus diffusion equation
2010-01-01
Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677
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.
NASA Astrophysics Data System (ADS)
Dean, A. M.; Benson, D. A.; Major, E.
2010-12-01
By adding a fractional-in-time term to the traditional advection dispersion equation, a model is able to simulate a late-time heavy-tailed contaminant breakthrough curve. This heavy-tailed breakthrough curve is observed in data collected during a conservative tracer “push-pull” test at the Macrodispersion Experiment (MADE) site. A time fractional advection dispersion equation (fADE) is able to predict power law tailing of conservative solutes by accounting for solutes transferring between the mobile and relatively immobile phases. Solutes can become trapped in a low permeability zone where the transport is controlled by diffusion instead of advection. It has been observed that the late-time heavy-tailed breakthrough curve may follow a power law due to the movement into these low flow zones. By solving the time fADE in a particle tracking program (SLIM-FAST) the model accounts for mass transfer between various phases and produces the same power law tail as observed in field data. For the implementation of the time fADE, in SLIM-FAST, the particles move based on a random-walk motion but have the ability to transition into a relatively immobile phase after (exponentially) random mobile times. Following a period in the immobile phase, the particle re-enters the mobile phase to be moved by advection and Fickian dispersion. To test the fADE approach, a recent single-well push-pull tracer test at the MADE site is reproduced using a groundwater flow code (ParFlow) and a particle tracking code (SLIM-FAST) using various immobile residence-time distributions.
A deterministic Lagrangian particle separation-based method for advective-diffusion problems
NASA Astrophysics Data System (ADS)
Wong, Ken T. M.; Lee, Joseph H. W.; Choi, K. W.
2008-12-01
A simple and robust Lagrangian particle scheme is proposed to solve the advective-diffusion transport problem. The scheme is based on relative diffusion concepts and simulates diffusion by regulating particle separation. This new approach generates a deterministic result and requires far less number of particles than the random walk method. For the advection process, particles are simply moved according to their velocity. The general scheme is mass conservative and is free from numerical diffusion. It can be applied to a wide variety of advective-diffusion problems, but is particularly suited for ecological and water quality modelling when definition of particle attributes (e.g., cell status for modelling algal blooms or red tides) is a necessity. The basic derivation, numerical stability and practical implementation of the NEighborhood Separation Technique (NEST) are presented. The accuracy of the method is demonstrated through a series of test cases which embrace realistic features of coastal environmental transport problems. Two field application examples on the tidal flushing of a fish farm and the dynamics of vertically migrating marine algae are also presented.
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.
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
Ibiş, Birol; Bayram, Mustafa
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
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.
NASA Technical Reports Server (NTRS)
Wang, Xiao-Yen; Chow, Chuen-Yen; Chang, Sin-Chung
1995-01-01
The existing 2-D alpha-mu scheme and alpha-epsilon scheme based on the method of space-time conservation element and solution element, which were constructed for solving the linear 2-D unsteady advection-diffusion equation and unsteady advection equation, respectively, are tested. Also, the alpha-epsilon scheme is modified to become the V-E scheme for solving the nonlinear 2-D inviscid Burgers equation. Numerical solutions of six test problems are presented in comparison with their exact solutions or numerical solutions obtained by traditional finite-difference or finite-element methods. It is demonstrated that the 2-D alpha-mu, alpha-epsilon, and nu-epsilon schemes can be used to obtain numerical results which are more accurate than those based on some of the traditional methods but without using any artificial tuning in the computation. Similar to the previous 1-D test problems, the high accuracy and simplicity features of the space-time conservation element and solution element method have been revealed again in the present 2-D test results.
NASA Astrophysics Data System (ADS)
Moura, R. C.; Sherwin, S. J.; Peiró, J.
2016-02-01
This study addresses linear dispersion-diffusion analysis for the spectral/hp continuous Galerkin (CG) formulation in one dimension. First, numerical dispersion and diffusion curves are obtained for the advection-diffusion problem and the role of multiple eigencurves peculiar to spectral/hp methods is discussed. From the eigencurves' behaviour, we observe that CG might feature potentially undesirable non-smooth dispersion/diffusion characteristics for under-resolved simulations of problems strongly dominated by either convection or diffusion. Subsequently, the linear advection equation augmented with spectral vanishing viscosity (SVV) is analysed. Dispersion and diffusion characteristics of CG with SVV-based stabilization are verified to display similar non-smooth features in flow regions where convection is much stronger than dissipation or vice-versa, owing to a dependency of the standard SVV operator on a local Péclet number. First a modification is proposed to the traditional SVV scaling that enforces a globally constant Péclet number so as to avoid the previous issues. In addition, a new SVV kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher-order discretizations, as measured by the extent of the wavenumber range where numerical errors are negligible. The dissipation characteristics of CG with the SVV modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous Galerkin (DG) scheme. Nevertheless, for the kernel function proposed, the full upwind DG scheme is found to have a slightly higher resolution power for the same dissipation levels. These results show that improved CG-SVV characteristics can be pursued via different kernel functions with the aid of optimization algorithms.
A New 2D-Advection-Diffusion Model Simulating Trace Gas Distributions in the Lowermost Stratosphere
NASA Astrophysics Data System (ADS)
Hegglin, M. I.; Brunner, D.; Peter, T.; Wirth, V.; Fischer, H.; Hoor, P.
2004-12-01
Tracer distributions in the lowermost stratosphere are affected by both, transport (advective and non-advective) and in situ sources and sinks. They influence ozone photochemistry, radiative forcing, and heating budgets. In-situ measurements of long-lived species during eight measurement campaigns revealed relatively simple behavior of the tracers in the lowermost stratosphere when represented in an equivalent-latitude versus potential temperature framework. We here present a new 2D-advection-diffusion model that simulates the main transport pathways influencing the tracer distributions in the lowermost stratosphere. The model includes slow diabatic descent of aged stratospheric air and vertical and/or horizontal diffusion across the tropopause and within the lowermost stratosphere. The diffusion coefficients used in the model represent the combined effects of different processes with the potential of mixing tropospheric air into the lowermost stratosphere such as breaking Rossby and gravity waves, deep convection penetrating the tropopause, turbulent diffusion, radiatively driven upwelling etc. They were specified by matching model simulations to observed distributions of long-lived trace gases such as CO and N2O obtained during the project SPURT. The seasonally conducted campaigns allow us to study the seasonal dependency of the diffusion coefficients. Despite its simplicity the model yields a surprisingly good description of the small scale features of the measurements and in particular of the observed tracer gradients at the tropopause. The correlation coefficients between modeled and measured trace gas distributions were up to 0.95. Moreover, mixing across isentropes appears to be more important than mixing across surfaces of constant equivalent latitude (or PV). With the aid of the model, the distribution of the fraction of tropospheric air in the lowermost stratosphere can be determined.
NASA Technical Reports Server (NTRS)
Leonard, B. P.
1988-01-01
A fresh approach is taken to the embarrassingly difficult problem of adequately modeling simple pure advection. An explicit conservative control-volume formation makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ULTIMATE conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three critical test profiles: an isolated sine-squared wave, a discontinuous step, and a semi-ellipse. The goal, of course, is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservation schemes of any order of accuracy. Second-order schemes are unsatisfactory, showing steepening and clipping typical of currently popular so-called high resolution shock-capturing of TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. Little is to be gained above ULTIMATE fifth-order upwinding which gives results close to the ultimate for which one might hope.
NASA Astrophysics Data System (ADS)
Dvoretskaya, Olga A.; Kondratenko, Peter S.
2009-04-01
We study the transport of impurity particles on a comb structure in the presence of advection. The main body concentration and asymptotic concentration distributions are obtained. Seven different transport regimes occur on the comb structure with finite teeth: classical diffusion, advection, quasidiffusion, subdiffusion, slow classical diffusion, and two kinds of slow advection. Quasidiffusion deserves special attention. It is characterized by a linear growth of the mean-square displacement. However, quasidiffusion is an anomalous transport regime. We established that a change in transport regimes in time leads to a change in regimes in space. Concentration tails have a cascade structure, namely, consisting of several parts.
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.
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.
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.
Modeling of advection-diffusion-reaction processes using transport dissipative particle dynamics
NASA Astrophysics Data System (ADS)
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-11-01
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. In particular, 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 Lagrangian particles. To validate the proposed tDPD model and the boundary conditions, three benchmark simulations of one-dimensional diffusion with different boundary conditions are performed, and the results show excellent agreement with the theoretical solutions. Also, two-dimensional simulations of ADR systems are performed and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, an application of tDPD to the spatio-temporal dynamics of blood coagulation involving twenty-five reacting species is performed to demonstrate the promising biological applications of the tDPD model. Supported by the DOE Center on Mathematics for Mesoscopic Modeling of Materials (CM4) and an INCITE grant.
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
NASA Astrophysics Data System (ADS)
Fan, Niannian; Singh, Arvind; Guala, Michele; Foufoula-Georgiou, Efi; Wu, Baosheng
2016-04-01
Bed load transport is a highly stochastic, multiscale process, where particle advection and diffusion regimes are governed by the dynamics of each sediment grain during its motion and resting states. Having a quantitative understanding of the macroscale behavior emerging from the microscale interactions is important for proper model selection in the absence of individual grain-scale observations. Here we develop a semimechanistic sediment transport model based on individual particle dynamics, which incorporates the episodic movement (steps separated by rests) of sediment particles and study their macroscale behavior. By incorporating different types of probability distribution functions (PDFs) of particle resting times Tr, under the assumption of thin-tailed PDF of particle velocities, we study the emergent behavior of particle advection and diffusion regimes across a wide range of spatial and temporal scales. For exponential PDFs of resting times Tr, we observe normal advection and diffusion at long time scales. For a power-law PDF of resting times (i.e., f>(Tr>)˜Tr-ν), the tail thickness parameter ν is observed to affect the advection regimes (both sub and normal advective), and the diffusion regimes (both subdiffusive and superdiffusive). By comparing our semimechanistic model with two random walk models in the literature, we further suggest that in order to reproduce accurately the emerging diffusive regimes, the resting time model has to be coupled with a particle motion model able to produce finite particle velocities during steps, as the episodic model discussed here.
Riemann equation for prime number diffusion
NASA Astrophysics Data System (ADS)
Chen, Wen; Liang, Yingjie
2015-05-01
This study makes the first attempt to propose the Riemann diffusion equation to describe in a manner of partial differential equation and interpret in physics of diffusion the classical Riemann method for prime number distribution. The analytical solution of this equation is the well-known Riemann representation. The diffusion coefficient is dependent on natural number, a kind of position-dependent diffusivity diffusion. We find that the diffusion coefficient of the Riemann diffusion equation is nearly a straight line having a slope 0.99734 in the double-logarithmic axis. Consequently, an approximate solution of the Riemann diffusion equation is obtained, which agrees well with the Riemann representation in predicting the prime number distribution. Moreover, we interpret the scale-free property of prime number distribution via a power law function with 1.0169 the scale-free exponent in respect to logarithmic transform of the natural number, and then the fractal characteristic of prime number distribution is disclosed.
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems.
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. PMID:26156459
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
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.
Elton, A.B.H.
1990-09-24
A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.
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)
Moiseev, N. Ya.; Silant'eva, I. Yu.
2008-07-01
An approach to the construction of second-and higher order accurate difference schemes in time and space is described for solving the linear one-and multidimensional advection equations with constant coefficients by the Godunov method with antidiffusion. The differential approximations for schemes of up to the fifth order are constructed and written. For multidimensional advection equations with constant coefficients, it is shown that Godunov schemes with splitting over spatial variables are preferable, since they have a smaller truncation error than schemes without splitting. The high resolution and efficiency of the difference schemes are demonstrated using test computations.
Approximate Solutions Of Equations Of Steady Diffusion
NASA Technical Reports Server (NTRS)
Edmonds, Larry D.
1992-01-01
Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.
Evaluation of realtime spray drift using RTDrift Gaussian advection-diffusion model.
Lebeau, Frédéric; Verstraete, Arnaud; Schiffers, Bruno; Destain, Marie-France
2009-01-01
A spray drift model was developed to deliver real time information to the pesticide applicator. The sprayer is equipped with sensors to deliver real time measurement of operational parameters as spray pressure, boom height, horizontal boom movements and geolocalization. The spray droplet size spectrum as a function of pressure was characterized using PDI measurements. Wind speed and direction were measured using a sprayer mounted 2-D ultrasonic anemometer. For each successive boom position, a diffusion-advection Gaussian tilting plume model is used to compute the spray drift deposits downwind. Drift is computed independently for each droplet classes and each nozzle based on the operating parameters. Field trials were performed on a test plot in various wind conditions. The ground drift was measured for different drift distances using fluorimetry analysis. Results show that drift deposits are mainly affected by wind speed and direction what was correctly accounted for by the model. Short distance drift deposits values were overestimated by the model while long distance drift was underestimated. It appears that this most probably origins from embarked wind speed measurements and diffusion parameter. It is concluded that a treatment of embarked wind speed and diffusion measurement should be used to minimize these errors. PMID:20218507
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.; 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 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
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.
2015-06-26
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
NASA Astrophysics Data System (ADS)
Karra, Satish; Makedonska, Nataliia; Viswanathan, Hari S.; Painter, Scott L.; Hyman, Jeffrey D.
2015-10-01
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. These results also suggest that matrix diffusion may support reduced production over longer time frames.
The Riesz-Bessel Fractional Diffusion Equation
Anh, V.V. McVinish, R.
2004-05-15
This paper examines the properties of a fractional diffusion equation defined by the composition of the inverses of the Riesz potential and the Bessel potential. The first part determines the conditions under which the Green function of this equation is the transition probability density function of a Levy motion. This Levy motion is obtained by the subordination of Brownian motion, and the Levy representation of the subordinator is determined. The second part studies the semigroup formed by the Green function of the fractional diffusion equation. Applications of these results to certain evolution equations is considered. Some results on the numerical solution of the fractional diffusion equation are also provided.
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.
Technology Transfer Automated Retrieval System (TEKTRAN)
The classical model to describe solute transport in soil is based on the advective-dispersive equation where Fick’s law is used to explain dispersion. From the microscopic point of view this is equivalent to consider that the motion of the particles of solute may be simulated by the Brownian motion....
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
Fractional diffusion equations coupled by reaction terms
NASA Astrophysics Data System (ADS)
Lenzi, E. K.; Menechini Neto, R.; Tateishi, A. A.; Lenzi, M. K.; Ribeiro, H. V.
2016-09-01
We investigate the behavior for a set of fractional reaction-diffusion equations that extend the usual ones by the presence of spatial fractional derivatives of distributed order in the diffusive term. These equations are coupled via the reaction terms which may represent reversible or irreversible processes. For these equations, we find exact solutions and show that the spreading of the distributions is asymptotically governed by the same the long-tailed distribution. Furthermore, we observe that the coupling introduced by reaction terms creates an interplay between different diffusive regimes leading us to a rich class of behaviors related to anomalous diffusion.
Wang, H.; Man, S.; Ewing, R.E.; Qin, G.; Lyons, S.L.; Al-Lawatia, M.
1999-06-10
Many difficult problems arise in the numerical simulation of fluid flow processes within porous media in petroleum reservoir simulation and in subsurface contaminant transport and remediation. The authors develop a family of Eulerian-Lagrangian localized adjoint methods for the solution of the initial-boundary value problems for first-order advection-reaction equations on general multi-dimensional domains. Different tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes, which are full mass conservative, naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary conditions. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices, which can be efficiently solved by the conjugate gradient method in an optimal order number of iterations without any preconditioning needed. Numerical results are presented to compare the performance of the ELLAM schemes with many well studied and widely used methods, including the upwind finite difference method, the Galerkin and the Petrov-Galerkin finite element methods with backward-Euler or Crank-Nicolson temporal discretization, the streamline diffusion finite element methods, the monotonic upstream-centered scheme for conservation laws (MUSCL), and the Minmod scheme.
NASA Astrophysics Data System (ADS)
Guihéneuf, N.; Boisson, A.; Bour, O.; Le Borgne, T.; Marechal, J.; Nigon, B.; Wajiddudin, M.; Ahmed, S.
2013-12-01
The prediction of transport in weathered and fractured rocks is critical as it represents the primary control of contaminant transfer from the subsurface in many parts of the world. This is the case in Southern India, where the subsurface is composed mainly of weathered and fractured granite and where the overexploitation of the groundwater resource since the 70's has led to high water table depletion and strong groundwater quality deterioration. One key issue for modelling transport in such systems is to quantify the respective role of advective heterogeneities and matrix diffusion, which can both lead to strongly non Fickian transport properties. We investigate this question by analysing tracer test experiments performed under different flow configurations at a fractured granite experimental site located in Andhra Pradesh (India). We performed both convergent and push-pull tracer tests within the same fracture and at different scales. Three convergent tracer tests were performed with a solution of fluorescein for different pumping rate and for different distances between injection and pumping boreholes: 6, 30 and 41 meters. To evaluate diffusive process, we performed two long-duration push-pull tests (push time of 3 hours) with a solution of two conservative tracers of different diffusion coefficient (fluorescein and sodium chloride). We performed also six others push-pull tests with only fluorescein but for a variable push times of 14 min and 55 min with or without resting time of about 60 min. The late-time behaviour on the breakthrough curves (BTCs) obtained for all convergent tracer tests showed a power-law slope of -2. Two of them showed an inflexion in the BTCs suggesting the existence of two independent flow paths and thus a highly channelized flow. The long-duration push-pull tests showed similar late-time behaviour with a power-law slope of -2.2 for both tracers. The six others push-pull tests showed a variation of power-law exponent from -3 to -2
Advection and Diffusion of Substances in Biological Tissues With Complex Vascular Networks
Beard, Daniel A.; Bassingthwaighte, James B.
2010-01-01
For highly diffusive solutes the kinetics of blood–tissue exchange is only poorly represented by a model consisting of sets of independent parallel capillary–tissue units. We constructed a more realistic multicapillary network model conforming statistically to morphometric data. Flows through the tortuous paths in the network were calculated based on constant resistance per unit length throughout the network and the resulting advective intracapillary velocity field was used as a framework for describing the extravascular diffusion of a substance for which there is no barrier or permeability limitation. Simulated impulse responses from the system, analogous to tracer water outflow dilution curves, showed flow-limited behavior over a range of flows from about 2 to 5 ml min−1 g−1, as is observed for water in the heart in vivo. The present model serves as a reference standard against which to evaluate computationally simpler, less physically realistic models. The simulated outflow curves from the network model, like experimental water curves, were matched to outflow curves from the commonly used axially distributed models only by setting the capillary wall permeability–surface area (PS) to a value so artifactually low that it is incompatible with the experimental observations that transport is flow limited. However, simple axially distributed models with appropriately high PSs will fit water outflow dilution curves if axial diffusion coefficients are set at high enough values to account for enhanced dispersion due to the complex geometry of the capillary network. Without incorporating this enhanced dispersion, when applied to experimental curves over a range of flows, the simpler models give a false inference that there is recruitment of capillary surface area with increasing flow. Thus distributed models must account for diffusional as well as permeation processes to provide physiologically appropriate parameter estimates. PMID:10784090
NASA Astrophysics Data System (ADS)
Benson, D. A.; Zhang, Y.
2006-12-01
Conservative solute transport through natural media is typically "anomalous" or non-Fickian. The anomalous transport may be characterized by faster than linear growth of the centered second moment, or non-Gaussian leading or trailing edges of a plume emanating from a point source. These characteristics develop because of non-local dependence on either past (time) or far upstream (space) concentrations. Non-local equations developed to describe anomalous dispersion usually focus on constant transport parameters and/or independence of the transport on space dimension. These simplifications have been useful for fitting simple transport processes, such as laboratory column tests or 1-D projections of field data. However, they may be insufficient for real field settings, where direction-dependent depositional processes and nonstationary heterogeneity can occur. We develop a generalized, multi-dimensional, spatiotemporal fractional advection- dispersion equation (fADE) with variable parameters to characterize regional-scale anomalous dispersion processes including trapping in immobile zones and/or super-Fickian rapid transport. A Lagrangian numerical model of the space-time fractional transport equation is developed in which solute particles can disperse in both space and time, depending on the medium heterogeneity properties, such as the connectivity and statistical distributions of high versus low-permeability deposits. In the generalized fADE, the range of the order of fractional time derivative is (0 2], representing a wide range of possible trapping behavior. The extension of the order to the range (1 2] is novel to transport theory. We apply the numerical model in 1-D and 2-D to the MADE site tritium plumes, and results indicate that this method can capture the main behaviors of realistic plumes, including local variations of spreading, direction-dependent scaling rates, and arbitrary rapid transport along preferential flow paths. Since the governing equation
Xiaoyi Li; Hai Huang; Paul Meakin
2008-09-01
The nonlinear coupling of fluid flow, reactive chemical transport and pore structure changes due to mineral precipitation (or dissolution) in porous media play a key role in a wide variety of processes of scientific interest and practical importance. Significant examples include the evolution of fracture apertures in the subsurface, acid fracturing stimulation for enhanced oil recovery and immobilizations of radionuclides and heavy metals in contaminated groundwater. We have developed a pore-scale simulation technique for modeling coupled reactive flow and structure evolution in porous media and fracture apertures. Advection, diffusion, and mineral precipitation resulting in changes in pore geometries are treated simultaneously by solving fully coupled fluid momentum and reactive solute transport equations. In this model, the reaction-induced evolution of solid grain surfaces is captured using a level set method. A sub-grid representation of the interface, based on the level set approach, is used instead of pixel representations of the interface often used in cellular-automata and most lattice-Boltzmann methods. The model is validated against analytical solutions for simplified geometries. Precipitation processes were simulated under various flow conditions and reaction rates, and the resulting pore geometry changes are discussed. Quantitative relationships between permeability and porosity under various flow conditions and reaction rates are reported.
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
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
NASA Astrophysics Data System (ADS)
Maghrebi, M.; Jankovic, I.; Rabideau, A. J.; Allen-King, R. M.; Weissmann, G. S.
2011-12-01
Effects of three key transport mechanisms (advection, diffusion and sorption) on transport and contaminant tailing of chlorinated solvents have been investigated using a numerical model. Thousands of model simulations have been conducted for various combinations of transport parameters that govern three key mechanisms in order to quantify tailing and relative importance of each mechanism. Hydraulic conductivity model contains a single inclusion of constant conductivity K embedded in a homogeneous anisotropic background of conductivity Kh,Kv. The inclusion is shaped as an oblate ellipsoid and subject to uniform flow. The background represents "average" conductivity of a heterogeneous formation while inclusion is used to represent geologic units that are more or less conductive than the background. The ratio of long to short semi-axis of the inclusion (a/b) models the ratio of horizontal to vertical integral scales (Ih/Iv) of different geologic units, where integral scales can be obtained, for example, using indicator variograms. The flow solution for present problem is obtained analytically as a closed form solution with exact expressions for Darcy velocity valid both inside and outside the inclusion. Sorption is modeled as an equilibrium process governed by a linear isotherm. The effects on transport and tailing are accounted for using retardation factors. Sorption heterogeneity is created by allowing different values of retardation factor for the interior (Ri) and the exterior of the inclusion (Rb). Diffusive displacements have been added to retarded advective displacements using random walk method. Peclet number, defined as Pe=U Ih/D (U is the groundwater velocity, D is the molecular diffusion coefficient for chlorinated solvents), is used to quantify the diffusion process. Very large numbers of particles (hundreds of thousands) have been tracked using very small time steps (as small as a/10,000) to provide sufficient resolution to breakthrough curves and to
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.
Experimental study of advective-diffusive gaseous CO2 transport through porous media
NASA Astrophysics Data System (ADS)
Basirat, Farzad; Sharma, Prabhakar; Niemi, Auli; Fagerlund, Fritjof
2014-05-01
Leakage of gaseous CO2 into the shallow subsurface system is one of the main concerns associated with geologic storage resources. A better understanding of CO2 leakage in the shallow subsurface plays an important role for developing leakage monitoring programs. CO2 may reach the unsaturated zone by different leak mechanisms such as exsolution from CO2 supersaturated water and continuous bubbling or gas flow along a leakage path. In the unsaturated zone, the CO2 is heavier than air and may accumulate below the ground surface and move laterally. We developed a small-scale experiment setup to study the possible gaseous CO2 transport mechanisms with different controlled conditions. In this study, the experiment setup was applied to measure CO2 distributions in time and space through homogenous dry sand in which the CO2 concentrations through the domain were measured by sensitive gas sensors. The preliminary analysis of the result suggests that the transport and distribution of gaseous CO2 is spatially and temporally sensitive for the selected experimental conditions of gas flow rate and porous media. To better understand the advection and diffusion processes through the unsaturated zone, the experimental results are coupled with the dusty gas model (DGM) of Mason et al. (1967). The dusty gas model's constitutive relationships are integrated into a numerical model for multicomponent gas mixture flow and transport in porous media. The DGM considers interactions between all gaseous species and Knudsen diffusion which is important in fine grained soils. Results from the applied model were consistent with the experimental breakthrough curves obtained in this study.
Riemann equation for prime number diffusion.
Chen, Wen; Liang, Yingjie
2015-05-01
This study makes the first attempt to propose the Riemann diffusion equation to describe in a manner of partial differential equation and interpret in physics of diffusion the classical Riemann method for prime number distribution. The analytical solution of this equation is the well-known Riemann representation. The diffusion coefficient is dependent on natural number, a kind of position-dependent diffusivity diffusion. We find that the diffusion coefficient of the Riemann diffusion equation is nearly a straight line having a slope 0.99734 in the double-logarithmic axis. Consequently, an approximate solution of the Riemann diffusion equation is obtained, which agrees well with the Riemann representation in predicting the prime number distribution. Moreover, we interpret the scale-free property of prime number distribution via a power law function with 1.0169 the scale-free exponent in respect to logarithmic transform of the natural number, and then the fractal characteristic of prime number distribution is disclosed. PMID:26026319
Advective, Diffusive and Eruptive Leakage of CO2 and Brine within Fault Zone
NASA Astrophysics Data System (ADS)
Jung, N. H.; Han, W. S.
2014-12-01
This study investigated a natural analogue for CO2 leakage near the 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 to 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 overtly 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 the low-k fault scenario engenders development of CO2 anticlinal trap within the shallow aquifers (Entrada and Navajo), concentrating high CO2 fluxes (~1,273 g m-2 d-1) within the northern footwall of the LGW fault similar to the field. Moreover, eruptive CO2 leakage at a well
Lyapunov Spectra in Diffusion Replicator Equation
NASA Astrophysics Data System (ADS)
Orihashi, Kenji; Aizawa, Yoji
2008-11-01
Statistical Properties of the turbulence in the diffusion replicator equation of three species are numerically studied. The maximal Lyapunov exponent and Lyapunov dimension are derived precisely. Further, these characteristics obey some characteristic scaling laws.
Discrete Fractional Diffusion Equation of Chaotic Order
NASA Astrophysics Data System (ADS)
Wu, Guo-Cheng; Baleanu, Dumitru; Xie, He-Ping; Zeng, Sheng-Da
Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases.
Shadid, John Nicolas; Bochev, Pavel Blagoveston; Gunzburger, Max Donald
2003-09-01
Implicit time integration coupled with SUPG discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows. Recently, it has been suggested that for small Courant numbers these terms may dominate the streamline diffusion term, ostensibly causing destabilization of the SUPG method. While consistent with a straightforward finite element stability analysis, this contention is not supported by computational experiments and contradicts earlier Von-Neumann stability analyses of the semidiscrete SUPG equations. This prompts us to re-examine finite element stability of the fully discrete SUPG equations. A careful analysis of the additional terms reveals that, regardless of the time step size, they are always dominated by the consistent mass matrix. Consequently, SUPG cannot be destabilized for small Courant numbers. Numerical results that illustrate our conclusions are reported.
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.
Webb, S.W.
1996-05-01
Two models for gas-phase diffusion and advection in porous media, the Advective-Dispersive Model (ADM) and the Dusty-Gas Model (DGM), are reviewed. The ADM, which is more widely used, is based on a linear addition of advection calculated by Darcy`s Law and ordinary diffusion using Fick`s Law. Knudsen diffusion is often included through the use of a Klinkenberg factor for advection, while the effect of a porous medium on the diffusion process is through a porosity-tortuosity-gas saturation multiplier. Another, more comprehensive approach for gas-phase transport in porous media has been formulated by Evans and Mason, and is referred to as the Dusty- Gas Model (DGM). This model applies the kinetic theory of gases to the gaseous components and the porous media (or ``dust``) to develop an approach for combined transport due to ordinary and Knudsen diffusion and advection including porous medium effects. While these two models both consider advection and diffusion, the formulations are considerably different, especially for ordinary diffusion. The various components of flow (advection and diffusion) are compared for both models. Results from these two models are compared to isothermal experimental data for He-Ar gas diffusion in a low-permeability graphite. Air-water vapor comparisons have also been performed, although data are not available, for the low-permeability graphite system used for the helium-argon data. Radial and linear air-water heat pipes involving heat, advection, capillary transport, and diffusion under nonisothermal conditions have also been considered.
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.
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.
Solves the Multigroup Neutron Diffusion Equation
1995-06-23
GNOMER is a program which solves the multigroup neutron diffusion equation in 1D, 2D and 3D cartesian geometry. The program is designed to calculate the global core power distributions (with thermohydraulic feedbacks), as well as power distribution and homogenized cross sections over a fuel assembly.
Transformed Fourier and Fick equations for the control of heat and mass diffusion
NASA Astrophysics Data System (ADS)
Guenneau, S.; Petiteau, D.; Zerrad, M.; Amra, C.; Puvirajesinghe, T.
2015-05-01
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.
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.
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.
NASA Astrophysics Data System (ADS)
Vikas, Kumar; K. Gupta, R.; Ram, Jiwari
2014-03-01
In this paper, the variable-coefficient diffusion—advection (DA) equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended (G'/G)-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.
Healy, R.W.; Russell, T.F.
1992-01-01
A finite-volume Eulerian-Lagrangian local adjoint method for solution of the advection-dispersion equation is developed and discussed. The method is mass conservative and can solve advection-dominated ground-water solute-transport problems accurately and efficiently. An integrated finite-difference approach is used in the method. A key component of the method is that the integral representing the mass-storage term is evaluated numerically at the current time level. Integration points, and the mass associated with these points, are then forward tracked up to the next time level. The number of integration points required to reach a specified level of accuracy is problem dependent and increases as the sharpness of the simulated solute front increases. Integration points are generally equally spaced within each grid cell. For problems involving variable coefficients it has been found to be advantageous to include additional integration points at strategic locations in each well. These locations are determined by backtracking. Forward tracking of boundary fluxes by the method alleviates problems that are encountered in the backtracking approaches of most characteristic methods. A test problem is used to illustrate that the new method offers substantial advantages over other numerical methods for a wide range of problems.
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
Technology Transfer Automated Retrieval System (TEKTRAN)
It has been reported that this model cannot take into account several important features of solute movement through soil. Recently, a new model has been suggested that results in a solute transport equation with fractional spatial derivatives, or FADE. We have assembled a database on published solu...
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.
Production Density Diffusion Equation Propagation and Production
NASA Astrophysics Data System (ADS)
Shirai, Kenji; Amano, Yoshinori
When we call the production flow to transition elements in the next step in the process of product manufactured one, the production flow is considered to be displaced in the direction of the unit production density. Density and production, as captured from different perspectives, also said production costs per unit of production. However, it is assumed that contributed to the production cost of manufacturing 100 percent. They may not correspond to the physical propagation conditions after each step of the production density, the equations governing the manufacturing process, which is intended to be represented by a single diffusion equation. We can also apply the concept of energy levels in statistical mechanics, production density function, in other words, in statistical mechanics “place” that if you use the world of manufacturing and production term. If the free energy in this production (potential) that are consuming the substance is nothing but the entropy production. That is, productivity is defined as the entropy production has to be. Normally, when we increase the number of production units, the product nears completion at year-end number of units completed and will aim to be delivered to the contractor from the turnover order. However, if you stop at any number of units, that will increase production density over time. Thus, the diffusion does not proceed from that would be irreversible. In other words, the congestion will occur in production. This fact and to report the results of analysis based on real data.
Advective and diapycnal diffusive oceanic flux in Tenerife - La Gomera Channel
NASA Astrophysics Data System (ADS)
Marrero-Díaz, A.; Rodriguez-Santana, A.; Hernández-Arencibia, M.; Machín, F.; García-Weil, L.
2012-04-01
During the year 2008, using the commercial passenger ship Volcán de Tauce of the Naviera Armas company several months, it was possible to obtain vertical profiles of temperature from expandable bathythermograph probes in eight stations across the Tenerife - La Gomera channel. With these data of temperature we have been estimated vertical sections of potential density and geostrophic transport with high spatial and temporal resolution (5 nm between stations, and one- two months between cruises). The seasonal variability obtained for the geostrophic transport in this channel shows important differences with others Canary Islands channels. From potential density and geostrophic velocity data we estimated the vertical diffusion coefficients and diapycnal diffusive fluxes, using a parameterization that depends of Richardson gradient number. In the center of the channel and close to La Gomera Island, we found higher values for these diffusive fluxes. Convergence and divergence of these fluxes requires further study so that we can draw conclusions about its impact on the distribution of nutrients in the study area and its impact in marine ecosystems. This work is being used in research projects TRAMIC and PROMECA.
Embry, Irucka; Roland, Victor; Agbaje, Oluropo; Watson, Valetta; Martin, Marquan; Painter, Roger; Byl, Tom; Sharpe, Lonnie
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
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.
Concentration through large advection
NASA Astrophysics Data System (ADS)
Aleja, D.; López-Gómez, J.
2014-11-01
In this paper we extend the elegant results of Chen, Lam and Lou [6, Section 2], where a concentration phenomenon was established as the advection blows up, to a general class of adventive-diffusive generalized logistic equations of degenerate type. Our improvements are really sharp as we allow the carrying capacity of the species to vanish in some subdomain with non-empty interior. The main technical devices used in the derivation of the concentration phenomenon are Proposition 3.2 of Cano-Casanova and López-Gómez [5], Theorem 2.4 of Amann and López-Gómez [1] and the classical Harnack inequality. By the relevance of these results in spatial ecology, complete technical details seem imperative, because the proof of Theorem 2.2 of [6] contains some gaps originated by an “optimistic” use of Proposition 3.2 of [5]. Some of the general assumptions of [6] are substantially relaxed.
Generalized diffusion equation and analytical expressions to neutron scattering experiments
NASA Astrophysics Data System (ADS)
Fa, Kwok Sau
2014-12-01
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. Analytical expressions related to neutron scattering experiments are presented and analyzed, which can be used to describe, for instance, biological systems.
Solution spectrum of nonlinear diffusion equations
Ulmer, W.
1992-08-01
The stationary version of the nonlinear diffusion equation -{partial_derivative}c/{partial_derivative}t+D{Delta}c=A{sub 1}c-A{sub 2}c{sup 2} can be solved with the ansatz c={summation}{sub p=1}{sup {infinity}} A{sub p}(cosh kx){sup -p}, inducing a band structure with regard to the ratio {lambda}{sub 1}/{lambda}{sub 2}. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c={summation}{sub p,q=1}{sup infinity}A{sub pa}(cosh kx){sup -p}[(cosh {alpha}t){sup -q-1} sinh {alpha}t+b(cosh {alpha}t){sup -q}] represents a solution spectrum of the time-dependent nonlinear equations, and the stationary version can be found from the asymptotic behaviour of the expansion. The solutions can be associated with reactive processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheriods. 33 refs., 1 tab.
NASA Astrophysics Data System (ADS)
Kawamura, Akira; Jinno, Kenji; Berndtsson, Ronny; Furukawa, Takashi
1997-12-01
There is a need to improve rainfall forecasting capabilities for small ungaged urban catchments to reduce flooding hazards and pollution release. For this purpose, information is required on small-scale and short-term convective cell behavior. We use a two-dimensional stochastic advection-diffusion model to parameterize the space-time rainfall intensity from convective rainfall. The rainfall intensity resulting from different separable components of the rain cell, such as apparent turbulent diffusion and development/decay of rainfall intensity, is quantified for 10 observed and, for southern Sweden, representative high-intensity rainfall events. This is done following a Lagrangian approach. It is shown the used model was able to respond to rapid changes in observed rainfall intensity in both space and time, thus giving a small average root-mean-square error for all 10 events (0.06 mm min -1). When dividing the total rainfall intensity into apparent turbulent diffusion and development/decay terms, respectively, it was shown that Dy, center and γcenter contribute approximately equally to the observed rainfall intensity. The Dx, center is usually only half the value of Dy, center , thus indicating less intensity contribution from this term and that the general elliptical shape of rain cells are elongated in the direction of movement. The observations indicate that the cumulus stage represents half and the dissipating stage half of the total cell development, respectively. The results can be used as first choice of parameter values when modeling rain cell movement over ungaged areas and the presented methodology can be used to study the effects of different cell components on total rainfall intensity.
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.
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. PMID:27375818
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.
Integro-differential diffusion equation and neutron scattering experiment
NASA Astrophysics Data System (ADS)
Sau Fa, Kwok
2015-02-01
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. Analytical expression for the intermediate scattering function is obtained and applied to ribonucleic acid (RNA) hydration water data from torula yeast. The model can capture the dynamics of hydrogen atoms in RNA hydration water, including the long-relaxation times.
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
Technology Transfer Automated Retrieval System (TEKTRAN)
Understanding and modeling transport of solutes in porous media is a critical issue in the environmental protection. Contaminants from various industrial and agricultural sources can travel in soil and ground water and eventually affect human and animal health. The parabolic advective-dispersive equ...
Uniqueness in inverse boundary value problems for fractional diffusion equations
NASA Astrophysics Data System (ADS)
Li, Zhiyuan; Imanuvilov, Oleg Yu; Yamamoto, Masahiro
2016-01-01
We consider an inverse boundary value problem for diffusion equations with multiple fractional time derivatives. We prove the uniqueness in determining the number of fractional time-derivative terms, the orders of the derivatives and spatially varying coefficients.
Multivariate Padé Approximations For Solving Nonlinear Diffusion Equations
NASA Astrophysics Data System (ADS)
Turut, V.
2015-11-01
In this paper, multivariate Padé approximation is applied to power series solutions of nonlinear diffusion equations. As it is seen from tables, multivariate Padé approximation (MPA) gives reliable solutions and numerical results.
The generalized diffusion-convection equation
NASA Technical Reports Server (NTRS)
Jones, Frank C.
1990-01-01
Starting from the Boltzmann equation, a transport equation is derived for energetic particles in a moving magnetized plasma in which the scattering centers that keep the particles quasi-isotropic are moving with a velocity that is not necessarily the same as that of the plasma. The scattering is characterized by three very loose constraints: (1) there is a rest frame for each scatterer in which the particles scatter elastically; (2) in this frame the scattering will not disturb an isotropic distribution; and (3) the momentum transfer in an average collision may be described by a tensor operating on the particles original momentum. Since the strength of the scattering is not specified, the derivation should be as valid for plasma microturbulence as for hard-sphere scattering. The results show clearly which phenomena are responsible for tying the particles to the plasma in the transport equation.
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.
Ages estimated from a diffusion equation model for scarp degradation
Colman, Steven M.; Watson, K.E.N.
1983-01-01
The diffusion equation derived from the continuity equation for hillslopes is applied to scarp erosion in unconsolidated materials. Solutions to this equation allow direct calculation of the product of the rate coefficient and the age of the scarp from measurements of scarp morphology. Where the rate coefficient can be estimated or can be derived from scarps of known age, this method allows direct calculation of unknown ages of scarps.
Wave and pseudo-diffusion equations from squeezed states
NASA Technical Reports Server (NTRS)
Daboul, Jamil
1993-01-01
We show that the probability distributions P(sub n)(q,p;y) := the absolute value squared of (n(p,q;y), which are obtained from squeezed states, obey an interesting partial differential equation, to which we give two intuitive interpretations: as a wave equation in one space dimension; and as a pseudo-diffusion equation. We also study the corresponding Wehrl entropies S(sub n)(y), and we show that they have minima at zero squeezing, y = 0.
Multilevel methods for transport equations in diffusive regimes
NASA Technical Reports Server (NTRS)
Manteuffel, Thomas A.; Ressel, Klaus
1993-01-01
We consider the numerical solution of the single-group, steady state, isotropic transport equation. An analysis by means of the moment equations shows that a discrete ordinate S(sub N) discretization in direction (angle) with a least squares finite element discretization in space does not behave properly in the diffusion limit. A scaling of the S(sub N) equations is introduced so that the least squares discretization has the correct diffusion limit. For the resulting discrete system a full multigrid algorithm was developed.
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.
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.
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.
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)
Jang, Juhi; Li, Fengyan; Qiu, Jing-Mei; Xiong, Tao
2015-01-01
In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the porous media equation, the advection-diffusion equation, and the viscous Burgers' equation. Our approach is based on the micro-macro reformulation of the kinetic equation which involves a natural decomposition of the equation to the equilibrium and non-equilibrium parts. To achieve high order accuracy and uniform stability as well as to capture the correct asymptotic limit, two new ingredients are employed in the proposed methods: discontinuous Galerkin (DG) spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta scheme in time equipped with a properly chosen implicit-explicit strategy. Formal asymptotic analysis shows that the proposed scheme in the limit of ε → 0 is a consistent high order discretization for the limiting equation. Numerical results are presented to demonstrate the stability and high order accuracy of the proposed schemes together with their performance in the limit. Our methods are also tested for the continuous-velocity one-group transport equation in slab geometry and for several examples with spatially varying parameters.
Efficient stochastic Galerkin methods for random diffusion equations
Xiu Dongbin Shen Jie
2009-02-01
We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projection. Although the resulting system of diffusion equations are coupled, we show that one can construct fast numerical methods to solve them in a decoupled fashion. The methods are based on separation of the diagonal terms and off-diagonal terms in the matrix of the Galerkin system. We examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters. Numerical examples are provided, for both steady and unsteady random diffusions, to support the analysis.
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
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
NASA Astrophysics Data System (ADS)
Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin
2013-02-01
In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized ( L 1) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.
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.
Identifiability for the pointwise source detection in Fisher’s reaction-diffusion equation
NASA Astrophysics Data System (ADS)
Ben Belgacem, Faker
2012-06-01
We are interested in the detection of a pointwise source in a class of semi-linear advection-diffusion-reaction equations of Fisher type. The source is determined by its location, which may be steady or unsteady, and its time-dependent intensity. Observations recorded at a couple of points are the available data. One observing station is located upstream of the source and the other downstream. This is a severely ill-posed nonlinear inverse problem. In this paper, we pursue an identifiability result. The process we follow has been developed earlier for the linear model and may be sharpened to operate for the semi-linear equation. It is based on the uniqueness for a parabolic (semi-linear) sideways problem, which is obtained by a suitable unique continuation theorem. We state a maximum principle that turns out to be necessary for our proof. The identifiability is finally obtained for a stationary or a moving source. Many applications may be found in biology, chemical physiology or environmental science. The problem we deal with is the detection of pointwise organic pollution sources in rivers and channels. The basic equation to consider is the one-dimensional biochemical oxygen demand equation, with a nonlinear power growth inhibitor and/or the Michaelis-Menten reaction coefficient.
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.
Healy, R.W.; Russell, T.F.
1998-01-01
We extend the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) for solution of the advection-dispersion equation to two dimensions. The method can conserve mass globally and is not limited by restrictions on the size of the grid Peclet or Courant number. Therefore, it is well suited for solution of advection-dominated ground-water solute transport problems. In test problem comparisons with standard finite differences, FVELLAM is able to attain accurate solutions on much coarser space and time grids. On fine grids, the accuracy of the two methods is comparable. A critical aspect of FVELLAM (and all other ELLAMs) is evaluation of the mass storage integral from the preceding time level. In FVELLAM this may be accomplished with either a forward or backtracking approach. The forward tracking approach conserves mass globally and is the preferred approach. The backtracking approach is less computationally intensive, but not globally mass conservative. Boundary terms are systematically represented as integrals in space and time which are evaluated by a common integration scheme in conjunction with forward tracking through time. Unlike the one-dimensional case, local mass conservation cannot be guaranteed, so slight oscillations in concentration can develop, particularly in the vicinity of inflow or outflow boundaries. Published by Elsevier Science Ltd.
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).
Green's Function Nodal Algorithm for the Diffusion Equation.
1989-12-04
Version 00 GRENADE is a coarse-mesh program designed for neutronic flux and power calculations in nuclear reactors. It solves the static diffusion equation for neutrons in multidimensional problems, assuming Cartesian Geometry. The program yields flux and power distributions and the effective neutron multiplication factor .
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)
NASA Astrophysics Data System (ADS)
Vikas, V.; Wang, Z. J.; Fox, R. O.
2013-09-01
Population balance equations with advection and diffusion terms can be solved using quadrature-based moment methods. Recently, high-order realizable finite-volume schemes with appropriate realizability criteria have been derived for the advection term. However, hitherto no work has been reported with respect to realizability problems for the diffusion term. The current work focuses on developing high-order realizable finite-volume schemes for diffusion. The pitfalls of existing finite-volume schemes for the diffusion term based on the reconstruction of moments are discussed, and it is shown that realizability can be guaranteed only with the 2nd-order scheme and that the realizability criterion for the 2nd-order scheme is the same as the stability criterion. However, realizability of moments cannot be guaranteed when higher-order moment-based reconstruction schemes are used. To overcome this problem, realizable high-order finite-volume schemes based on the reconstruction of weights and abscissas are proposed and suitable realizability criteria are derived. The realizable schemes can achieve higher than 2nd-order accuracy for problems with smoothly varying abscissas. In the worst-case scenario of highly nonlinear abscissas, the realizable schemes are 2nd-order accurate but have lower error magnitudes compared to existing schemes. The results obtained using the realizable high-order schemes are shown to be consistent with those obtained using the 2nd-order moment-based reconstruction scheme.
Methods for diffusive relaxation in the Pn equation
Hauck, Cory D; Mcclarren, Ryan G; Lowrie, Robert B
2008-01-01
We present recent progress in the development of two substantially different approaches for simulating the so-called of P{sub N} equations. These are linear hyperbolic systems of PDEs that are used to model particle transport in a material medium, that in highly collisional regimes, are accurately approximated by a simple diffusion equation. This limit is based on a balance between function values and gradients of certain variables in the P{sub N} system. Conventional reconstruction methods based on upwinding approximate such gradients with an error that is dependent on the size of the computational mesh. Thus in order to capture the diffusion limit, a given mesh must resolve the dynamics of the continuum equation at the level of the mean-free-path, which tends to zero in the diffusion limit. The two methods analyzed here produce accurate solutions in both collisional and non-collisional regimes; in particular, they do not require resolution of the mean-free-path in order to properly capture the diffusion limit. The first method is a straight-forward application of the discrete Galerkin (DG) methodology, which uses additional variables in each computational cell to capture the balance between function values and gradients, which are computed locally. The second method uses a temporal splitting of the fast and slow dynamics in the P{sub N} system to derive so-called regularized equations for which the diffusion limit is built-in. We focus specifically on the P{sub N} equations for one-dimensional, slab geometries. Preliminary results for several benchmark problems are presented which highlight the advantages and disadvantages of each method. Further improvements and extensions are also discussed.
Kinetic equations for diffusion in the presence of entropic barriers.
Reguera, D; Rubí, J M
2001-12-01
We use the mesoscopic nonequilibrium thermodynamics theory to derive the general kinetic equation of a system in the presence of potential barriers. The result is applied to a description of the evolution of systems whose dynamics is influenced by entropic barriers. We analyze in detail the case of diffusion in a domain of irregular geometry in which the presence of the boundaries induces an entropy barrier when approaching the exact dynamics by a coarsening of the description. The corresponding kinetic equation, named the Fick-Jacobs equation, is obtained, and its validity is generalized through the formulation of a scaling law for the diffusion coefficient which depends on the shape of the boundaries. The method we propose can be useful to analyze the dynamics of systems at the nanoscale where the presence of entropy barriers is a common feature. PMID:11736170
Differencing the diffusion equation on unstructured meshes in 2-D
Palmer, T.S.
1994-10-24
During the last few years, there has been an increased effort to devise robust transport differencings for unstructured meshes, specifically arbitrarily connected grids of polygons. Adams has investigated unstructured mesh discretization techniques for the even- and odd-parity forms of the transport equation, and for the more traditional first-order form. Conversely, development of unstructured mesh diffusion methods has been lacking. While Morel, Kershaw, Shestakov and others have done a great deal of work on diffusion schemes for logically-rectangular grids, to the author`s knowledge there has been no work on discretizations of the diffusion equation on unstructured meshes of polygons. In this paper, the authors introduce a point-centered diffusion differencing for two-dimensional unstructured meshes. They have designed the method to have the following attractive properties: (1) the scheme is equivalent to the standard five-point point-centered scheme on an orthogonal mesh; (2) the method preserves the homogeneous linear solution; (3) the method gives second-order accuracy; (4) they have strict conservation within the control volume surrounding each point; and (5) the numerical solution converges to the exact result as the mesh is refined, regardless of the smoothness of the mesh. A potential disadvantage of the method is that the diffusion matrix is asymmetric, in general.
Geometric Correction for Diffusive Expansion of Steady Neutron Transport Equation
NASA Astrophysics Data System (ADS)
Wu, Lei; Guo, Yan
2015-06-01
We revisit the diffusive limit of a steady neutron transport equation in a two-dimensional unit disk with one-speed velocity. A classical theorem by Bensoussan et al. (Publ Res Inst Math Sci 15(1):53-157, 1979) states that its solution can be approximated in L ∞ by the leading order interior solution plus the Knudsen layer in the diffusive limit. In this paper, we construct a counterexample to this result via a different boundary layer expansion with geometric correction.
Langevin equation with fluctuating diffusivity: A two-state model
NASA Astrophysics Data System (ADS)
Miyaguchi, Tomoshige; Akimoto, Takuma; Yamamoto, Eiji
2016-07-01
Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.
Langevin equation with fluctuating diffusivity: A two-state model.
Miyaguchi, Tomoshige; Akimoto, Takuma; Yamamoto, Eiji
2016-07-01
Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool. PMID:27575079
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.
Reaction diffusion equation with spatio-temporal delay
NASA Astrophysics Data System (ADS)
Zhao, Zhihong; Rong, Erhua
2014-07-01
We investigate reaction-diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction-diffusion equation with spatio-temporal delay. Applying this theory to Lotka-Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem's steady-state solution.
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.
Laser speckle contrast imaging is sensitive to advective flux
NASA Astrophysics Data System (ADS)
Khaksari, Kosar; Kirkpatrick, Sean J.
2016-07-01
Unlike laser Doppler flowmetry, there has yet to be presented a clear description of the physical variables that laser speckle contrast imaging (LSCI) is sensitive to. Herein, we present a theoretical basis for demonstrating that LSCI is sensitive to total flux and, in particular, the summation of diffusive flux and advective flux. We view LSCI from the perspective of mass transport and briefly derive the diffusion with drift equation in terms of an LSCI experiment. This equation reveals the relative sensitivity of LSCI to both diffusive flux and advective flux and, thereby, to both concentration and the ordered velocity of the scattering particles. We demonstrate this dependence through a short series of flow experiments that yield relationships between the calculated speckle contrast and the concentration of the scatterers (manifesting as changes in scattering coefficient), between speckle contrast and the velocity of the scattering fluid, and ultimately between speckle contrast and advective flux. Finally, we argue that the diffusion with drift equation can be used to support both Lorentzian and Gaussian correlation models that relate observed contrast to the movement of the scattering particles and that a weighted linear combination of these two models is likely the most appropriate model for relating speckle contrast to particle motion.
Persistence exponent of the diffusion equation in ε dimensions
NASA Astrophysics Data System (ADS)
Hilhorst, H. J.
2000-03-01
We consider the d-dimensional diffusion equation ∂ tφ( x,t)= Δφ( x,t) with random initial condition, and observe that, when appropriately scaled, φ(0, t) is Gaussian and Markovian in the limit d→0. This leads via the Majumdar-Sire perturbation theory to a small d expansion for the persistence exponent θ( d). We find θ(d)= {1}/{4}d-0.12065…d 3/2+⋯
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...
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 on the order of the reaction radius of a reacting pair of molecules. PMID:26871190
Reaction rates for a generalized reaction-diffusion master equation
NASA Astrophysics Data System (ADS)
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.
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.
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.
Compact finite volume methods for the diffusion equation
NASA Technical Reports Server (NTRS)
Rose, Milton E.
1989-01-01
The paper describes an approach to treating initial-boundary-value problems by finite volume methods in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second-order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and ADI methods.
Analysis of a mixed space-time diffusion equation
NASA Astrophysics Data System (ADS)
Momoniat, Ebrahim
2015-06-01
An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank-Nicolson scheme. The Crank-Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.
Compact finite volume methods for the diffusion equation
NASA Technical Reports Server (NTRS)
Rose, Milton E.
1989-01-01
An approach to treating initial-boundary value problems by finite volume methods is described, in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and alternating direction implicit (ADI) methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.
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)
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, Kβ(t)D radβ/sα, 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(-γβgβδβDf1 tα), a stretched exponential function (SEF) attenuation, while from the fractional derivative model the attenuation is Eα,1(-γβgβδβDf2 tα), 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 ∫0τ Kβ (t)dtα ] . The results obtained in this study are in good agreement with the results in literature. Several expressions that describe signal
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
Numerical simulation of life cycles of advection warm fog
NASA Technical Reports Server (NTRS)
Hung, R. J.; Vaughan, O. H.
1977-01-01
The formation, development and dissipation of advection warm fog is investigated. The equations employed in the model include the equation of continuity, momentum and energy for the descriptions of density, wind component and potential temperature, respectively, together with two diffusion equations for the modification of water-vapor mixing ratio and liquid-water mixing ratios. A description of the vertical turbulent transfer of heat, moisture and momentum has been taken into consideration. The turbulent exchange coefficients adopted in the model are based on empirical flux-gradient relations.
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.
Mimetic discretization of two-dimensional magnetic diffusion equations
NASA Astrophysics Data System (ADS)
Lipnikov, Konstantin; Reynolds, James; Nelson, Eric
2013-08-01
In case of non-constant resistivity, cylindrical coordinates, and highly distorted polygonal meshes, a consistent discretization of the magnetic diffusion equations requires new discretization tools based on a discrete vector and tensor calculus. We developed a new discretization method using the mimetic finite difference framework. It is second-order accurate on arbitrary polygonal meshes and a consistent calculation of the Joule heating is intrinsic within it. The second-order convergence rates in L2 and L1 norms were verified with numerical experiments.
Bifurcations of diffusive soliton solutions to Kuznetsov's equation
NASA Astrophysics Data System (ADS)
Jordan, Pedro M.
2003-04-01
Exact traveling wave solutions are determined for Kuznetsov's equation, a nonlinear PDE of 3rd order which describes finite amplitude acoustic disturbances in thermoviscous Newtonian fluids. Specifically, it is shown that traveling wave solutions exist, and assume the form of diffusive solitons, if and only if the Mach number is less than or equal to a bifurcation value. It is also shown that the wave speed v is always supersonic, that Max[v] occurs at the bifurcation value of the Mach number, and that a shock develops as the Reynolds number tends to infinity. Finally, special cases and asymptotic results are listed, a relationship to Burgers' equation is noted, and 3-D bifurcation diagrams are given.
New variable separation solutions for the generalized nonlinear diffusion equations
NASA Astrophysics Data System (ADS)
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
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.
Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems
NASA Astrophysics Data System (ADS)
Colli, Pierluigi; Fukao, Takeshi
2016-05-01
An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved.
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
CONTAMINANT TRANSPORT IN SEDIMENT UNDER THE INFLUENCE OF ADVECTIVE FLUX
Chemical flux across the sediment/water interface is controlled by a combination of diffusive, dispersive and advective processes. The advective process is a function of submarine groundwater discharge and tidal effects. In areas where surface water interacts with groundwater, ...
Zhan, Wang; Jiang, Li; Loew, Murray; Yang, Yihong
2008-01-01
Diffusion is an important mechanism for molecular transport in living biological tissues. Diffusion magnetic resonance imaging (dMRI) provides a unique probe to examine microscopic structures of the tissues in vivo, but current dMRI techniques usually ignore the spatio-temporal evolution process of the diffusive medium. In the present study, we demonstrate the feasibility to reveal the spatio-temporal diffusion process inside the human brain based on a numerical solution of the diffusion equation. Normal human subjects were scanned with a diffusion tensor imaging (DTI) technique on a 3-Tesla MRI scanner, and the diffusion tensor in each voxel was calculated from the DTI data. The diffusion equation, a partial-derivative description of Fick’s Law for the diffusion process, was discretized into equivalent algebraic equations. A finite-difference method was employed to obtain the numerical solution of the diffusion equation with a Crank-Nicholson iteration scheme to enhance the numerical stability. By specifying boundary and initial conditions, the spatio-temporal evolution of the diffusion process inside the brain can be virtually reconstructed. Our results exhibit similar medium profiles and diffusion coefficients as those of light fluorescence dextrans measured in integrative optical imaging experiments. The proposed method highlights the feasibility to non-invasively estimate the macroscopic diffusive transport time for a molecule in a given region of the brain. PMID:18440744
Garges, J.A.; Baehr, A.L.
1998-01-01
The relative importance of advection and dispersion for both solute and vapor transport can be determined from type curves or concentration, flux, or cumulative flux. The dimensionless form of the type curves provides a means to directly evaluate the importance of mass transport by advection relative to that of mass transport by diffusion and dispersion. Type curves based on an analytical solution to the advection-dispersion equation are plotted in terms of dimensionless time and Peclet number. Flux and cumulative flux type curves provide additional rationale for transport regime determination in addition to the traditional concentration type curves. The extension of type curves to include vapor transport with phase partitioning in the unsaturated zone is a new development. Type curves for negative Peclet numbers also are presented. A negative Peclet number characterizes a problem in which one direction of flow is toward the contamination source, and thereby diffusion and advection can act in opposite directions. Examples are the diffusion of solutes away from the downgradient edge of a pump-and-treat capture zone, the upward diffusion of vapors through the unsaturated zone with recharge, and the diffusion of solutes through a low hydraulic conductivity cutoff wall with an inward advective gradient.
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.
From baking a cake to solving the diffusion equation
NASA Astrophysics Data System (ADS)
Olszewski, Edward A.
2006-06-01
We explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. We restrict our consideration to the génoise and obtain a semiempirical relation for the baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The relation, which is based on the diffusion equation, has three parameters whose values are estimated from data obtained by baking cakes in cylindrical pans of various diameters. The relation takes into account the evaporation of moisture at the top surface of the cake, which is the dominant factor affecting the baking time of a cake.
Generalized Landauer equation: absorption-controlled diffusion processes.
Godoy, S; García-Colín, L S; Micenmacher, V
1999-05-01
The exact expression of the one-dimensional Boltzmann multiple-scattering coefficients, for the passage of particles through a slab of a given material, is obtained in terms of the single-scattering cross section of the material, including absorption. The remarkable feature of the result is that for multiple scattering in a metal, free from absorption, one recovers the well-known Landauer result for conduction electrons. In the case of particles, such as neutrons, moving through a weak absorbing media, Landuer's formula is modified due to the absorption cross section. For photons, in a strong absorbing media, one recovers the Lambert-Beer equation. In this latter case one may therefore speak of absorption-controlled diffusive processes. PMID:11969603
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.
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
A New Contraction Family for Porous Medium and Fast Diffusion Equations
NASA Astrophysics Data System (ADS)
Chmaycem, G.; Jazar, M.; Monneau, R.
2016-08-01
In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation.
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
Wang, Lei; Shi, Baochang; Chai, Zhenhua
2015-10-01
In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368
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.
NASA Astrophysics Data System (ADS)
Akimoto, Takuma; Yamamoto, Eiji
2016-06-01
We consider the Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously over time, in order to study fluctuations of time-averaged observables in temporally heterogeneous diffusion processes. We find that the time-averaged mean-square displacement (TMSD) can be represented by the occupation time of a state in the asymptotic limit of the measurement time and hence occupation time statistics is a powerful tool for calculating the TMSD in the model. We show that the TMSD increases linearly with time (normal diffusion) but the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges, i.e., intrinsic nonequilibrium processes. Thus, we find that temporally heterogeneous environments provide anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficients obtained by single-particle-tracking trajectories in experiments.
Rotationally symmetric solutions of the Landau-Lifshitz and diffusion equations
Mayergoyz, I. D.; Bertotti, G.; Serpico, C.
2000-05-01
The problem of isotropic conducting ferromagnetic film subject to in-plane circular polarized magnetic fields is discussed. This problem requires simultaneous solution of diffusion and Landau-Lifshitz equations. It is observed that the mathematical formulation of the problem is invariant with respect to rotations in the film plane. By exploiting this invariance, the rotationally symmetric solutions of the Landau-Lifshitz equation coupled with the diffusion equation are obtained and examined. (c) 2000 American Institute of Physics.
NASA Astrophysics Data System (ADS)
Frank, T. D.
2008-02-01
We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.
Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix
Schulze-Halberg, Axel
2012-10-15
We construct a Darboux transformation for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix. Our transformation is based on the two-dimensional supersymmetry formalism for the Schroedinger equation. The transformed Fokker-Planck equation and its solutions are obtained in explicit form.
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.
Application of The Full-Sweep AOR Iteration Concept for Space-Fractional Diffusion Equation
NASA Astrophysics Data System (ADS)
Sunarto, A.; Sulaiman, J.; Saudi, A.
2016-04-01
The aim of this paper is to investigate the effectiveness of the Full-Sweep AOR Iterative method by using Full-Sweep Caputo’s approximation equation to solve space-fractional diffusion equations. The governing space-fractional diffusion equations were discretized by using Full-Sweep Caputo’s implicit finite difference scheme to generate a system of linear equations. Then, the Full-Sweep AOR iterative method is applied to solve the generated linear system To examine the application of FSAOR method two numerical tests are conducted to show that the FSAOR method is superior to the FSSOR and FSGS methods.
Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations
NASA Astrophysics Data System (ADS)
Chakraverty, S.; Smita, Tapaswini
2014-12-01
The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 < α <= 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.
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 comparison of implicit numerical methods for solving the transient spherical diffusion equation
NASA Technical Reports Server (NTRS)
Curry, D. M.
1977-01-01
Comparative numerical temperature results obtained by using two implicit finite difference procedures for the solution of the transient diffusion equation in spherical coordinates are presented. The validity and accuracy of these solutions are demonstrated by comparison with exact analytical solutions.
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. PMID:17552680
Fractional Diffusion Equation, Quantum Subdynamics and EINSTEIN'S Theory of Brownian Motion
NASA Astrophysics Data System (ADS)
Abe, Sumiyoshi
The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.
Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations
Zhou Tao; Tang Tao
2010-11-01
In a recent work by Xiu and Shen [D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys. 228 (2009) 266-281], the Galerkin methods are used to solve stochastic diffusion equations in random media, where some properties for the coefficient matrix of the resulting system are provided. They also posed an open question on the properties of the coefficient matrix. In this work, we will provide some results related to the open question.
Vázquez, J. L.
2010-01-01
The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy–Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations. PMID:20823259
NASA Astrophysics Data System (ADS)
Horowitz, Jordan M.
2015-07-01
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.
Advection of methane in the hydrate zone: model, analysis and examples
NASA Astrophysics Data System (ADS)
Peszynska, Malgorzata; Showalter, Ralph E.; Webster, Justin T.
2015-12-01
A two-phase two-component model is formulated for the advective-diffusive transport of methane in liquid phase through sediment with the accompanying formation and dissolution of methane hydrate. This free-boundary problem has a unique generalized solution in $L^1$; the proof combines analysis of the stationary semilinear elliptic Dirichlet problem with the nonlinear semigroup theory in Banach space for an m-accretive multi-valued operator. Additional estimates of maximum principle type are obtained, and these permit appropriate maximal extensions of the phase-change relations. An example with pure advection indicates the limitations of these estimates and of the model developed here. We also consider and analyze the coupled pressure equation that determines the advective flux in the transport model.
Antidiffusive velocities for multipass donor cell advection
Margolin, L.; Smolarkiewicz, P.K.
1999-01-01
Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successfully more accurate solutions to the advection equation. In this paper, the authors show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. The authors discuss the implementation of the antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme.
Lattice fractional diffusion equation in terms of a Riesz-Caputo difference
NASA Astrophysics Data System (ADS)
Wu, Guo-Cheng; Baleanu, Dumitru; Deng, Zhen-Guo; Zeng, Sheng-Da
2015-11-01
A fractional difference is defined by the use of the right and the left Caputo fractional differences. The definition is a two-sided operator of Riesz type and introduces back and forward memory effects in space difference. Then, a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains. A lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders. The result shows that the Riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media.
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.
Efficient mass transport by optical advection
NASA Astrophysics Data System (ADS)
Kajorndejnukul, Veerachart; Sukhov, Sergey; Dogariu, Aristide
2015-10-01
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.
NASA Astrophysics Data System (ADS)
Valeau, Vincent; Sakout, Anas; Li, Feng; Picaut, Judicael
2002-11-01
Over the last years, some publications [e.g., Picaut, Simon, and Hardy, J. Acoust. Soc. Am. 106, 2638-2645 (1999)] showed that the acoustic energy density in closed or semiclosed spaces is the solution of a diffusion equation. This approach allows the nonuniform repartition of energy, and is especially relevant in room acoustics for complex spaces or long rooms. In this work, the 3-D diffusion equation is solved directly by using a finite element solver, for a set of long rooms and absorbing rooms. The stationary equation is first solved. A constant-power acoustic source is modelized by setting appropriate boundary conditions. The time-dependent problem is also solved to simulate the sound decay, with an impulse source defined in a subregion with relevant initial conditions. Results concerning sound attenuation and reverberation times match satisfactorily with other theoretical and numerical models. An application is also given for two coupled rooms.
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.
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.
Fractional diffusion equation for an n -dimensional correlated Lévy walk
NASA Astrophysics Data System (ADS)
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.
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.
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. PMID:27575074
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.
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.
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.
Solution of phase space diffusion equations using interacting trajectory ensembles
NASA Astrophysics Data System (ADS)
Donoso, Arnaldo; Martens, Craig C.
2002-06-01
In this paper, we present a new method for simulating the evolution of the phase space distribution function describing a system coupled to a Markovian thermal bath. The approach is based on the propagation of ensembles of trajectories. Instead of incorporating environmental perturbations as stochastic forces, however, the present method includes these effects by additional deterministic interactions between the ensemble members. The general formalism is developed and tested on model systems describing one-dimensional diffusion, relaxation of a coherently excited harmonic oscillator coupled to a thermal bath, and activated barrier crossing in a bistable potential. Excellent agreement with exact results or approximate theories is obtained in all cases. The method provides an entirely deterministic trajectory-based approach to the solution of condensed phase dynamics and chemical reactions.
Multigrid techniques for the numerical solution of the diffusion equation
NASA Technical Reports Server (NTRS)
Phillips, R. E.; Schmidt, F. W.
1984-01-01
An accurate numerical solution of diffusion problems containing large local gradients can be obtained with a significant reduction in computational time by using a multigrid computational scheme. The spatial domain is covered with sets of uniform square grids of different sizes. The finer grid patterns overlap the coarse grid patterns. The finite-difference expressions for each grid pattern are solved independently by iterative techniques. Two interpolation methods were used to establish the values of the potential function on the fine grid boundaries with information obtained from the coarse grid solution. The accuracy and computational requirements for solving a test problem by a simple multigrid and a multilevel-multigrid method were compared. The multilevel-multigrid method combined with a Taylor series interpolation scheme was found to be best.
NASA Astrophysics Data System (ADS)
Rhew, Jung-Hoon; Lundstrom, Mark S.
2002-11-01
We develop a drift-diffusion equation that describes ballistic transport in a nanoscale metal-oxide-semiconductor field effect transistor (MOSFET). We treat injection from different contacts separately, and describe each injection with a set of extended McKelvey one-flux equations [Phys. Rev. 123, 51 (1961); 125, 1570 (1962)] that include hierarchy closure approximations appropriate for high-field ballistic transport and degenerate carrier statistics. We then reexpress the extended one-flux equations in a drift-diffusion form with a properly defined Einstein relationship. The results obtained for a nanoscale MOSFET show excellent agreement with the solution of the ballistic Boltzmann transport equation with no fitting parameters. These results show that a macroscopic transport model based on the moments of the Boltzmann transport equation can describe ballistic transport.
NASA Astrophysics Data System (ADS)
Polyanin, Andrei D.; Zhurov, Alexei I.
2014-03-01
We propose a new method for constructing exact solutions to nonlinear delay reaction-diffusion equations of the form ut=kuxx+F(u,w), where u=u(x,t),w=u(x,t-τ), and τ is the delay time. The method is based on searching for solutions in the form u=∑n=1Nξn(x)ηn(t), where the functions ξn(x) and ηn(t) are determined from additional functional constraints (which are difference or functional equations) and the original delay partial differential equation. All of the equations considered contain one or two arbitrary functions of a single argument. We describe a considerable number of new exact generalized separable solutions and a few more complex solutions representing a nonlinear superposition of generalized separable and traveling wave solutions. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for nonlinear delay PDEs. The results are extended to a wide class of nonlinear partial differential-difference equations involving arbitrary linear differential operators of any order with respect to the independent variables x and t (in particular, this class includes the nonlinear delay Klein-Gordon equation) as well as to some partial functional differential equations with time-varying delay.
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.
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.
Simulate-HEX - The multi-group diffusion equation in hexagonal-z geometry
Lindahl, S. O.
2013-07-01
The multigroup diffusion equation is solved for the hexagonal-z geometry by dividing each hexagon into 6 triangles. In each triangle, the Fourier solution of the wave equation is approximated by 8 plane waves to describe the intra-nodal flux accurately. In the end an efficient Finite Difference like equation is obtained. The coefficients of this equation depend on the flux solution itself and they are updated once per power/void iteration. A numerical example demonstrates the high accuracy of the method. (authors)
Conservation Laws of a Family of Reaction-Diffusion-Convection Equations
NASA Astrophysics Data System (ADS)
Bruzón, M. S.; Gandarias, M. L.; de la Rosa, R.
Ibragimov introduced the concept of nonlinear self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi-self-adjoint equations. Gandarias defined the concept of weak self-adjoint. In this paper, we found a class of nonlinear self-adjoint nonlinear reaction-diffusion-convection equations which are neither self-adjoint nor quasi-self-adjoint nor weak self-adjoint. From a general theorem on conservation laws proved by Ibragimov we obtain conservation laws for these equations.
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.
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. PMID:26407013
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.
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. PMID:27504247
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.
A moving mesh finite difference method for equilibrium radiation diffusion equations
NASA Astrophysics Data System (ADS)
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.
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.
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
NASA Astrophysics Data System (ADS)
Nefedov, N. N.; Ni, Minkang
2015-12-01
A singularly perturbed boundary value problem for a second-order ordinary differential equation known in applications as a stationary reaction-diffusion equation is studied. A new class of problems is considered, namely, problems with nonlinearity having discontinuities localized in some domains, which leads to the formation of sharp transition layers in these domains. The existence of solutions with an internal transition layer is proved, and their asymptotic expansion is constructed.
Monotone waves for non-monotone and non-local monostable reaction-diffusion equations
NASA Astrophysics Data System (ADS)
Trofimchuk, Elena; Pinto, Manuel; Trofimchuk, Sergei
2016-07-01
We propose a new approach for proving existence of monotone wavefronts in non-monotone and non-local monostable diffusive equations. This allows to extend recent results established for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (modulo translations) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then each monotone wavefront is unique (modulo translations) within the class of all non-constant traveling waves. Our approach is based on the construction of suitable fundamental solutions for linear integral-differential equations. We consider two alternative scenarios: in the first one, the fundamental solution is negative (typically holds for the Mackey-Glass diffusive equations) while in the second one, the fundamental solution is non-negative (typically holds for the KPP-Fisher diffusive equations).
Axial expansion methods for solution of the multi-dimensional neutron diffusion equation
Beaklini Filho, J.F.
1984-01-01
The feasibility and practical implementation of axial expansion methods for the solution of the multi-dimensional multigroup neutron diffusion (MGD) equations is investigated. The theoretical examination which is applicable to the general MGD equations in arbitrary geometry includes the derivation of a new weak (reduced) form of the MGD equations by expanding the axial component of the neutron flux in a series of known trial functions and utilizing the Galerkin weighting. A general two-group albedo boundary condition is included in the weak form as a natural boundary condition. The application of different types of trial functions is presented.
NASA Astrophysics Data System (ADS)
Tóth, Gábor; Keppens, Rony
2012-07-01
The Versatile Advection Code (VAC) is a freely available general hydrodynamic and magnetohydrodynamic simulation software that works in 1, 2 or 3 dimensions on Cartesian and logically Cartesian grids. VAC runs on any Unix/Linux system with a Fortran 90 (or 77) compiler and Perl interpreter. VAC can run on parallel machines using either the Message Passing Interface (MPI) library or a High Performance Fortran (HPF) compiler.
NASA Astrophysics Data System (ADS)
Simon, Emanuel; Foschum, Florian; Kienle, Alwin
2013-06-01
Time-resolved diffuse optical spectroscopy measurements of phantoms at small source-detector separations yield good results for the retrieved coefficients of reduced scattering and absorption when a hybrid Green's function of the radiative transfer equation for semi-infinite media is used.
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.
A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation.
Wang, Kangle; Liu, Sanyang
2016-01-01
In this paper, a new Sumudu transform iterative method is established and successfully applied to find the approximate analytical solutions for time-fractional Cauchy reaction-diffusion equations. The approach is easy to implement and understand. The numerical results show that the proposed method is very simple and efficient. PMID:27386314
NASA Astrophysics Data System (ADS)
Wang, Zhi-Cheng; Bu, Zhen-Hui
2016-04-01
This paper is concerned with nonplanar traveling fronts in reaction-diffusion equations with combustion nonlinearity and degenerate Fisher-KPP nonlinearity. Our study contains two parts: in the first part we establish the existence of traveling fronts of pyramidal shape in R3, and in the second part we establish the existence and stability of V-shaped traveling fronts in R2.
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.
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.
Breakdown of the reaction-diffusion master equation with nonelementary rates
NASA Astrophysics Data System (ADS)
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.
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.
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.
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
Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations
NASA Astrophysics Data System (ADS)
Collier, N.; Knepley, M.
2015-12-01
The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).
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).
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. PMID:23208221
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.
Antidiffusive velocities for multipass donor cell advection
Margolin, L.G. ); Smolarkiewicz, P.K. )
1989-12-01
Smolarkiewicz describes an iterative process for approximating the advection equation. Basically, he uses a donor cell approximation to correct for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times leading to successively more accurate solutions to the advection equation. In this report, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done dimension to illustrate the method. The analysis is then repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications. 6 refs., 5 figs., 4 tabs.
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
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.
A modified multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation
NASA Astrophysics Data System (ADS)
Huang, Rongzong; Wu, Huiying
2014-10-01
A modified lattice Boltzmann model with multiple relaxation times (MRT) for the convection-diffusion equation (CDE) is proposed. By modifying the relaxation matrix, as well as choosing the corresponding equilibrium distribution function properly, the present model can recover the CDE with anisotropic diffusion coefficient with no deviation term even when the velocity vector varies generally with space or time through the Chapman-Enskog analysis. This model is firstly validated by simulating the diffusion of a Gaussian hill, which demonstrates it can handle the anisotropic diffusion problem correctly. Then it is adopted to calculate the longitudinal dispersion coefficient of the Taylo-Aris dispersion. Numerical results show that the present model can further reduce the numerical error under the condition of non-zero velocity vector, especially when the dimensionless relaxation time is relatively large.
Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions
NASA Astrophysics Data System (ADS)
Bagchi, Partha; Das, Arpan; Sengupta, Srikumar; Srivastava, Ajit M.
2015-09-01
Reaction-diffusion equations with suitable boundary conditions have special propagating solutions which very closely resemble the moving interfaces in a first-order transition. We show that the dynamics of the chiral order parameter for the chiral symmetry breaking transition in heavy-ion collisions, with dissipative dynamics, is governed by one such equation; specifically, the Newell-Whitehead equation. Furthermore, required boundary conditions are automatically satisfied due to the geometry of the collision. The chiral transition is, therefore, completed by a propagating interface, exactly as for a first-order transition, even though the transition actually is a crossover for relativistic heavy-ion collisions. The same thing also happens when we consider the initial confinement-deconfinement transition with the Polyakov loop order parameter. The resulting equation, again with dissipative dynamics, can then be identified with the reaction-diffusion equation known as the FitzHugh-Nagumo equation which is used in population genetics. Observational constraints imply that the entire phase conversion cannot be achieved by such slow moving fronts, and some alternate faster dynamics needs also to be invoked; for example, involving fluctuations. We discuss the implications of these results for heavy-ion collisions. We also discuss possible extensions for the case of the early universe.
Investigation of acoustically coupled enclosures using a diffusion-equation model.
Xiang, Ning; Jing, Yun; Bockman, Alexander C
2009-09-01
Recent application of coupled-room systems in performing arts spaces has prompted active research on sound fields in these complex geometries. This paper applies a diffusion-equation model to the study of acoustics in coupled-rooms. Acoustical measurements are conducted on a scale-model of two coupled-rooms. Using the diffusion model and the experimental results the current work conducts in-depth investigations on sound pressure level distributions, providing further evidence supporting the valid application of the diffusion-equation model. Analysis of the results within the Bayesian framework allows for quantification of the double-slope characteristics of sound-energy decays obtained from the diffusion-equation numerical modeling and the experimental measurements. In particular, Bayesian decay analysis confirms sound-energy flux modeling predictions that time-dependent sound-energy flows in coupled-room systems experience feedback in the form of energy flow-direction change across the aperture connecting the two rooms in cases where the dependent room is more reverberant than the source room. PMID:19739732
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.
NASA Astrophysics Data System (ADS)
Huang, Rongzong; Wu, Huiying
2015-03-01
A lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) with divergence-free velocity field is proposed, and the Chapman-Enskog analysis shows that the CDE can be recovered correctly. In the present model, the convection term is treated as a source term in the lattice Boltzmann equation (LBE) rather than being directly recovered by LBE; thus the CDE is intrinsically solved as a pure diffusion equation with a corresponding source term. To avoid the adoption of a nonlocal finite-difference scheme for computing the convection term, a local scheme is developed based on the Chapman-Enskog analysis. Most importantly, by properly specifying the discrete source term in the moment space, the local scheme can reach the same order (ɛ2) at which the CDE is recovered by a LB model. Numerical tests, including a one-dimensional periodic problem, diffusion of a Gaussian hill, diffusion of a rectangular pulse, and natural convection in a square cavity, are carried out to verify the present model. Numerical results are satisfactorily consistent with analytical solutions or previous numerical results, and show higher accuracy due to the correct recovery of CDE.
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.
Extremal equilibria for reaction-diffusion equations in bounded domains and applications
NASA Astrophysics Data System (ADS)
Rodríguez-Bernal, Aníbal; Vidal-López, Alejandro
We show the existence of two special equilibria, the extremal ones, for a wide class of reaction-diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation.
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.
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
Turing-Hopf bifurcation in the reaction-diffusion equations and its applications
NASA Astrophysics Data System (ADS)
Song, Yongli; Zhang, Tonghua; Peng, Yahong
2016-04-01
In this paper, we consider the Turing-Hopf bifurcation arising from the reaction-diffusion equations. It is a degenerate case and where the characteristic equation has a pair of simple purely imaginary roots and a simple zero root. First, the normal form theory for partial differential equations (PDEs) with delays developed by Faria is adopted to this degenerate case so that it can be easily applied to Turing-Hopf bifurcation. Then, we present a rigorous procedure for calculating the normal form associated with the Turing-Hopf bifurcation of PDEs. We show that the reduced dynamics associated with Turing-Hopf bifurcation is exactly the dynamics of codimension-two ordinary differential equations (ODE), which implies the ODE techniques can be employed to classify the reduced dynamics by the unfolding parameters. Finally, we apply our theoretical results to an autocatalysis model governed by reaction-diffusion equations; for such model, the dynamics in the neighbourhood of this bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations; and then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found.
NASA Astrophysics Data System (ADS)
Maassen, Jesse; Lundstrom, Mark
2016-03-01
Understanding ballistic phonon transport effects in transient thermoreflectance experiments and explaining the observed deviations from classical theory remains a challenge. Diffusion equations are simple and computationally efficient but are widely believed to break down when the characteristic length scale is similar or less than the phonon mean-free-path. Building on our prior work, we demonstrate how well-known diffusion equations, namely, the hyperbolic heat equation and the Cattaneo equation, can be used to model ballistic phonon effects in frequency-dependent periodic steady-state thermal transport. Our analytical solutions are found to compare excellently to rigorous numerical results of the phonon Boltzmann transport equation. The correct physical boundary conditions can be different from those traditionally used and are paramount for accurately capturing ballistic effects. To illustrate the technique, we consider a simple model problem using two different, commonly used heating conditions. We demonstrate how this framework can easily handle detailed material properties, by considering the case of bulk silicon using a full phonon dispersion and mean-free-path distribution. This physically transparent approach provides clear insights into the nonequilibrium physics of quasi-ballistic phonon transport and its impact on thermal transport properties.
On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains
NASA Astrophysics Data System (ADS)
Cantrell, Robert Stephen; Cosner, Chris
We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.
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.
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.
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.
Dynamical diffusion and renormalization group equation for the Fermi velocity in doped graphene
NASA Astrophysics Data System (ADS)
Ardenghi, J. S.; Bechthold, P.; Jasen, P.; Gonzalez, E.; Juan, A.
2014-11-01
The aim of this work is to study the electron transport in graphene with impurities by introducing a generalization of linear response theory for linear dispersion relations and spinor wave functions. Current response and density response functions are derived and computed in the Boltzmann limit showing that in the former case a minimum conductivity appears in the no-disorder limit. In turn, from the generalization of both functions, an exact relation can be obtained that relates both. Combining this result with the relation given by the continuity equation it is possible to obtain general functional behavior of the diffusion pole. Finally, a dynamical diffusion is computed in the quasistatic limit using the definition of relaxation function. A lower cutoff must be introduced to regularize infrared divergences which allow us to obtain a full renormalization group equation for the Fermi velocity, which is solved up to order O(ℏ2).
Modeling Heat Conduction and Radiation Transport with the Diffusion Equation in NIF ALE-AMR
Fisher, A C; Bailey, D S; Kaiser, T B; Gunney, B N; Masters, N D; Koniges, A E; Eder, D C; Anderson, R W
2009-10-06
The ALE-AMR code developed for NIF is a multi-material hydro-code that models target assembly fragmentation in the aftermath of a shot. The combination of ALE (Arbitrary Lagrangian Eulerian) hydro with AMR (Adaptive Mesh Refinement) allows the code to model a wide range of physical conditions and spatial scales. The large range of temperatures encountered in the NIF target chamber can lead to significant fluxes of energy due to thermal conduction and radiative transport. These physical effects can be modeled approximately with the aid of the diffusion equation. We present a novel method for the solution of the diffusion equation on a composite mesh in order to capture these physical effects.
Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation.
Wang, Yin; Zhang, Jun
2010-10-15
We present a sixth order explicit compact finite difference scheme to solve the three dimensional (3D) convection diffusion equation. We first use multiscale multigrid method to solve the linear systems arising from a 19-point fourth order discretization scheme to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equation with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact scheme (SOC), compared with the previously published fourth order compact scheme (FOC). PMID:21151737
A Widder's Type Theorem for the Heat Equation with Nonlocal Diffusion
NASA Astrophysics Data System (ADS)
Barrios, Begoña; Peral, Ireneo; Soria, Fernando; Valdinoci, Enrico
2014-08-01
The main goal of this work is to prove that every non-negative strong solution u( x, t) to the problem can be written as where and This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [
Global stability of travelling wave fronts for non-local diffusion equations with delay
NASA Astrophysics Data System (ADS)
Wang, X.; Lv, G.
2014-04-01
This paper is concerned with the global stability of travelling wave fronts for non-local diffusion equations with delay. We prove that the non-critical travelling wave fronts are globally exponentially stable under perturbations in some exponentially weighted L^\\infty-spaces. Moreover, we obtain the decay rates of \\sup_{x\\in{R}}\\vert u(x,t)-\\varphi(x+ct)\\vert using weighted energy estimates.
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.
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.
NASA Astrophysics Data System (ADS)
Moroney, Timothy; Yang, Qianqian
2013-08-01
We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space-fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Grünwald finite difference formulas to approximate the two-sided (i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobian-free Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space-fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.
A study of turbulent transport of an advective nature in a fluid plasma
NASA Astrophysics Data System (ADS)
Min, Byunghoon; An, Chan-Yong; Kim, Chang-Bae
2014-08-01
The advective nature of the electrostatic turbulent flux of plasma energy in Fourier space is studied numerically in a nearly adiabatic state. Such a state is represented by the Hasegawa-Mima equation, which is driven by a noise that may model the destabilization due to the phase mismatch of the plasma density and the electric potential. The noise is assumed to be Gaussian and not to be invariant under reflection along a direction ŝ. The flux density induced by such noise is found to be anisotropic: While it is random along ŝ, it is not along the perpendicular direction ŝ ⊥, and the flux is not diffusive. The renormalized response may be approximated as advective, with the velocity being proportional to ( kρ s )2, in the Fourier space.
Multi-moment advection scheme in three dimension for Vlasov simulations of magnetized plasma
Minoshima, Takashi; Matsumoto, Yosuke; Amano, Takanobu
2013-03-01
We present an extension of the multi-moment advection scheme [T. Minoshima, Y. Matsumoto, T. Amano, Multi-moment advection scheme for Vlasov simulations, Journal of Computational Physics 230 (2011) 6800–6823] to the three-dimensional case, for full electromagnetic Vlasov simulations of magnetized plasma. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, and advances them on the basis of their governing equations. Similar to the two-dimensional scheme, the three-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile with little numerical dispersion or diffusion. This is a very important property for Vlasov simulations of magnetized plasma. We apply the scheme to electromagnetic Vlasov simulations. Propagation of linear waves and nonlinear evolution of the electron temperature anisotropy instability are successfully simulated with a good accuracy of the energy conservation.
Comments on the Diffusive Behavior of Two Upwind Schemes
NASA Technical Reports Server (NTRS)
Wood, William A.; Kleb, William L.
1998-01-01
The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and locally one-dimensional finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2.5 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a speedup of 29 over finite volume.
Diffusion Characteristics of Upwind Schemes on Unstructured Triangulations
NASA Technical Reports Server (NTRS)
Wood, William A.; Kleb, William L.
1998-01-01
The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and dimensionally-split finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the dimensionally-split finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2-3 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a 20-25 speedup over finite volume.
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...
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.
The dilution wave in polymer crystallization is described by Fisher's reaction-diffusion equation
NASA Astrophysics Data System (ADS)
Higgs, Paul G.; Ungar, Goran
2001-04-01
Monodisperse long-chain alkanes such as C198H398 form lamellar crystals in both extended- and folded-chain forms. Folded-chain crystals are in a meta-stable equilibrium with polymer solution at a concentration CF. The crystal growth rate is virtually zero at this point, due to the self-poisoning phenomenon. If extended-chain crystallization is initiated from this state, a wave of crystallization proceeds through the solution, termed the dilution wave. The solution concentration falls as the wave passes, until a value CE is reached that is in equilibrium with the extended-chain crystal phase. We write down a reaction-diffusion equation to describe the dilution wave, and show that this is equivalent to Fisher's equation, which has previously been used to describe many other traveling wave phenomena. Numerical solutions of the equation are used to show examples of the wave shape.
The Hirota Method for Reaction-Diffusion Equations with Three Distinct Roots
Tanoglu, Gamze; Pashaev, Oktay
2004-10-04
The Hirota Method, with modified background is applied to construct exact analytical solutions of nonlinear reaction-diffusion equations of two types. The first equation has only nonlinear reaction part, while the second one has in addition the nonlinear transport term. For both cases, the reaction part has the form of the third order polynomial with three distinct roots. We found analytic one-soliton solutions and the relationships between three simple roots and the wave speed of the soliton. For the first case, if one of the roots is the mean value of other two roots, the soliton is static. We show that the restriction on three distinct roots to obtain moving soliton is removed in the second case by, adding nonlinear transport term to the first equation.
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 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.
Hellander, Andreas; Lawson, Michael J; Drawert, Brian; Petzold, Linda
2015-01-01
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. PMID:26865735
NASA Astrophysics Data System (ADS)
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 were 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.
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
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.
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. PMID:22734771
NASA Astrophysics Data System (ADS)
Künzli, Pierre; Tsunematsu, Kae; Albuquerque, Paul; Falcone, Jean-Luc; Chopard, Bastien; Bonadonna, Costanza
2016-04-01
Volcanic ash transport and dispersal models typically describe particle motion via a turbulent velocity field. Particles are advected inside this field from the moment they leave the vent of the volcano until they deposit on the ground. Several techniques exist to simulate particles in an advection field such as finite difference Eulerian, Lagrangian-puff or pure Lagrangian techniques. In this paper, we present a new flexible simulation tool called TETRAS (TEphra TRAnsport Simulator) based on a hybrid Eulerian-Lagrangian model. This scheme offers the advantages of being numerically stable with no numerical diffusion and easily parallelizable. It also allows us to output particle atmospheric concentration or ground mass load at any given time. The model is validated using the advection-diffusion analytical equation. We also obtained a good agreement with field observations of the tephra deposit associated with the 2450 BP Pululagua (Ecuador) and the 1996 Ruapehu (New Zealand) eruptions. As this kind of model can lead to computationally intensive simulations, a parallelization on a distributed memory architecture was developed. A related performance model, taking into account load imbalance, is proposed and its accuracy tested.
NASA Astrophysics Data System (ADS)
Xu, Zhaoquan; Xiao, Dongmei
2016-01-01
A class of reaction diffusion equation with spatio-temporal delays is systematically investigated. When the reaction function of this equation is nonlinear without monotonicity, it is shown that there exists a spreading speed c* > 0 for this equation such that c* is linearly determinate and coincides with the minimal wave speed of traveling waves, and that this equation admits a unique traveling wave (up to translation) with speed c >c* and no traveling wave with c
A spatial SIS model in advective heterogeneous environments
NASA Astrophysics Data System (ADS)
Cui, Renhao; Lou, Yuan
2016-09-01
We study the effects of diffusion and advection for a susceptible-infected-susceptible epidemic reaction-diffusion model in heterogeneous environments. The definition of the basic reproduction number R0 is given. If R0 < 1, the unique disease-free equilibrium (DFE) is globally asymptotically stable. Asymptotic behaviors of R0 for advection rate and mobility of the infected individuals (denoted by dI) are established, and the existence of the endemic equilibrium when R0 > 1 is studied. The effects of diffusion and advection rates on the stability of the DFE are further investigated. Among other things, we find that if the habitat is a low-risk domain, there may exist one critical value for the advection rate, under which the DFE changes its stability at least twice as dI varies from zero to infinity, while the DFE is unstable for any dI when the advection rate is larger than the critical value. These results are in strong contrast with the case of no advection, where the DFE changes its stability at most once as dI varies from zero to infinity.
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.
Hybrid simplified spherical harmonics with diffusion equation for light propagation in tissues.
Chen, Xueli; Sun, Fangfang; Yang, Defu; Ren, Shenghan; Zhang, Qian; Liang, Jimin
2015-08-21
Aiming at the limitations of the simplified spherical harmonics approximation (SPN) and diffusion equation (DE) in describing the light propagation in tissues, a hybrid simplified spherical harmonics with diffusion equation (HSDE) based diffuse light transport model is proposed. In the HSDE model, the living body is first segmented into several major organs, and then the organs are divided into high scattering tissues and other tissues. DE and SPN are employed to describe the light propagation in these two kinds of tissues respectively, which are finally coupled using the established boundary coupling condition. The HSDE model makes full use of the advantages of SPN and DE, and abandons their disadvantages, so that it can provide a perfect balance between accuracy and computation time. Using the finite element method, the HSDE is solved for light flux density map on body surface. The accuracy and efficiency of the HSDE are validated with both regular geometries and digital mouse model based simulations. Corresponding results reveal that a comparable accuracy and much less computation time are achieved compared with the SPN model as well as a much better accuracy compared with the DE one. PMID:26237074
Complete numerical solution of the diffusion equation of random genetic drift.
Zhao, Lei; Yue, Xingye; Waxman, David
2013-08-01
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
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...
Retinal Image Enhancement Using Robust Inverse Diffusion Equation and Self-Similarity Filtering.
Wang, Lu; Liu, Guohua; 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
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.
NASA Astrophysics Data System (ADS)
Harko, T.; Mak, M. K.
2015-11-01
We consider quasi-stationary (travelling wave type) solutions to a general nonlinear reaction-convection-diffusion equation with arbitrary, autonomous coefficients. The second order nonlinear equation describing one dimensional travelling waves can be reduced to a first kind first order Abel equation. By using two integrability conditions for the Abel equation (the Chiellini lemma and the Lemke transformation), several classes of exact travelling wave solutions of the general reaction-convection-diffusion equation are obtained, corresponding to different functional relations imposed between the diffusion, convection and reaction functions. In particular, we obtain travelling wave solutions for two non-linear second order partial differential equations, representing generalizations of the standard diffusion equation and of the classical Fisher-Kolmogorov equation, to which they reduce for some limiting values of the model parameters. The models correspond to some specific, power law type choices of the reaction and convection functions, respectively. The travelling wave solutions of these two classes of differential equation are investigated in detail by using both numerical and semi-analytical methods.
An improved second moment method for solution of pure advection problems
NASA Astrophysics Data System (ADS)
Ghods, Abdolreza; Sobouti, Farhad; Arkani-Hamed, Jafar
2000-04-01
The second moment numerical method (SMM) of Egan and Mahoney [Numerical modeling of advection and diffusion of urban area source pollutant. Journal of Applied Meteorology 1972; 11: 312-322] is adapted to solve for the pure advection transport equation in a variety of flow fields. SMM eliminates numerical diffusion by employing a procedure that takes into account the first and second moments of mass distribution in each grid element. For pure translational flow fields, the method is conservative, positive definite and shape-preserving. In rotational and/or shear flows, the accuracy of SMM is significantly reduced. Two improvements are presented to make the SMM applicable to a wider range of flow problems. It is shown that the improved SMM (ISMM) is less diffusive and more shape-preserving than the SMM in rotational and/or deformational flows. The ISMM can also be used to solve for a color function in compressible flow fields. The computational efficiency of this method is compared with that of other methods and, for a given accuracy, it is shown that ISMM is a cost-effective, non-diffusive and shape-preserving method. Copyright
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.
A new nonlinear finite volume scheme preserving positivity for diffusion equations
NASA Astrophysics Data System (ADS)
Sheng, Zhiqiang; Yuan, Guangwei
2016-06-01
In this paper we present a new nonlinear finite volume scheme preserving positivity for diffusion equations. The main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided. Two nonnegative parameters are introduced to define a new nonlinear two-point flux, in which one point is the cell-center and the other is the midpoint of cell-edge. The final flux on the edge is obtained by the continuity of normal flux. Numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes.
Visualizations of sound energy across coupled rooms using a diffusion equation model.
Jing, Yun; Xiang, Ning
2008-12-01
Visualizations, based on a diffusion equation model, are presented for both steady-state and transient sound energy. For steady-state sound-pressure level distributions, animations created by scanning from the primary room to the secondary room reveal discontinuous transitions of sound energy caused by a location change from the wall area to the aperture area. Animations of time-dependent energy flow directions visualize the energy flows across coupled spaces. This study also reveals a "reversal" characteristic of energy flow directions which seems to be dependent on the size and location of the aperture. PMID:19206694
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.
Stability and Bifurcation in a Delayed Reaction-Diffusion Equation with Dirichlet Boundary Condition
NASA Astrophysics Data System (ADS)
Guo, Shangjiang; Ma, Li
2016-04-01
In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov-Schmidt reduction. The existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution is derived by analyzing the distribution of the eigenvalues. The direction of Hopf bifurcation and stability of the bifurcating periodic solution are also investigated by means of normal form theory and center manifold reduction. Moreover, we illustrate our general results by applications to the Nicholson's blowflies models with one- dimensional spatial domain.
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.
New high order iterative scheme in the solution of convection-diffusion equation
NASA Astrophysics Data System (ADS)
Ling, Sam Teek; Ali, Norhashidah Hj. Mohd.
2014-07-01
In this paper, a new fourth-order nine-point finite difference scheme based on the rotated grid combined with the traditional Successive Over Relaxation (SOR)-type iterative method is discussed in solving the two-dimensional convection-diffusion partial differential equation (pde) with variable coefficients. Numerical experiments are carried out to verify the high accuracy solution of the scheme. Comparisons with the exact solutions also show that the rotated scheme converges faster than the existing compact scheme of the same order.
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
NASA Astrophysics Data System (ADS)
Bakhtiyari-Ramezani, M.; Mahmoodi, J.; Alinejad, N.
2015-11-01
In the fusion devices, ions, H atoms, and H2 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 H2 molecules, and desorption of the recombined H2 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.
Analytic solutions of the time-dependent quasilinear diffusion equation with source and loss terms
Hassan, M.H.A. ); Hamza, E.A. )
1993-08-01
A simplified one-dimensional quasilinear diffusion equation describing the time evolution of collisionless ions in the presence of ion-cyclotron-resonance heating, sources, and losses is solved analytically for all harmonics of the ion cyclotron frequency. Simple time-dependent distribution functions which are initially Maxwellian and vanish at high energies are obtained and calculated numerically for the first four harmonics of resonance heating. It is found that the strongest ion tail of the resulting anisotropic distribution function is driven by heating at the second harmonic followed by heating at the fundamental frequency.
A Monte Carlo synthetic-acceleration method for solving the thermal radiation diffusion equation
Evans, Thomas M.; Mosher, Scott W.; Slattery, Stuart R.; Hamilton, Steven P.
2014-02-01
We present a novel synthetic-acceleration-based Monte Carlo method for solving the equilibrium thermal radiation diffusion equation in three spatial dimensions. The algorithm performance is compared against traditional solution techniques using a Marshak benchmark problem and a more complex multiple material problem. Our results show that our Monte Carlo method is an effective solver for sparse matrix systems. For solutions converged to the same tolerance, it performs competitively with deterministic methods including preconditioned conjugate gradient and GMRES. We also discuss various aspects of preconditioning the method and its general applicability to broader classes of problems.
The diffusive logistic equation with a free boundary and sign-changing coefficient
NASA Astrophysics Data System (ADS)
Wang, Mingxin
2015-02-01
This short paper concerns a diffusive logistic equation with a free boundary and sign-changing coefficient, which is formulated to study the spread of an invasive species, where the free boundary represents the expanding front. A spreading-vanishing dichotomy is derived, namely the species either successfully spreads to the right-half-space as time t → ∞ and survives (persists) in the new environment, or it fails to establish itself and will extinct in the long run. The sharp criteria for spreading and vanishing are also obtained. When spreading happens, we estimate the asymptotic spreading speed of the free boundary.
Burgers turbulence and passive random advection
NASA Astrophysics Data System (ADS)
Boldyrev, Stanislav Anatolievich
1999-10-01
, and the diffusivity is neglected. These considerations illustrate that even with simple statistics of the velocity field, the statistics of advected quantities are nontrivial due to nonlinear interactions of different spatial directions. The last Chapter 5 summarizes the results and discusses future directions of research.
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.
Diffusion equations over arbitrary triangulated surfaces for filtering and texture applications.
Wu, Chunlin; Deng, Jiansong; Chen, Falai
2008-01-01
In computer graphics, triangular mesh representations of surfaces have become very popular. Compared with parametric and implicit forms of surfaces, triangular mesh surfaces have many advantages, such as easy to render, convenient to store and the ability to model geometric objects with arbitrary topology. In this paper, we are interested in data processing over triangular mesh surfaces through PDEs (partial differential equations). We study several diffusion equations over triangular mesh surfaces, and present corresponding numerical schemes to solve them. Our methods work for triangular mesh surfaces with arbitrary geometry (the angles of each triangle are arbitrary) and topology (open meshes or closed meshes of arbitrary genus). Besides the flexibility, our methods are efficient due to the implicit/semi-implicit time discretization. We finally apply our methods to several filtering and texture applications such as image processing, texture generating and regularization of harmonic maps over triangular mesh surfaces. The results demonstrate the flexibility and effectiveness of our methods. PMID:18369272
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.
Using adaptive proper orthogonal decomposition to solve the reaction-diffusion equation
Singer, M A; Green, W H
2007-12-03
We introduce an adaptive POD method to reduce the computational cost of reacting flow simulations. The scheme is coupled with an operator-splitting algorithm to solve the reaction-diffusion equation. For the reaction sub-steps, locally valid basis vectors, obtained via POD and the method of snapshots, are used to project the minor species mass fractions onto a reduced dimensional space thereby decreasing the number of equations that govern combustion chemistry. The method is applied to a one-dimensional laminar premixed CH{sub 4}-air flame using GRImech 3.0; with errors less than 0:25%, a speed-up factor of 3:5 is observed. The speed-up results from fewer source term evaluations required to compute the Jacobian matrices.
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.
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...
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.
Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain
NASA Astrophysics Data System (ADS)
Ren, Xiaoxia; Xiang, Zhaoyin; Zhang, Zhifei
2016-04-01
We study the initial boundary value problem of two dimensional MHD equations without magnetic diffusion in a strip domain. It was proved that the MHD equations have a unique global strong solution around the equilibrium state ≤ft(0,{{\\mathbf{e}}1}\\right) for both the non-slip boundary condition and Navier slip boundary condition on the velocity.
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.
Localized axial Green's function method for the convection-diffusion equations in arbitrary domains
NASA Astrophysics Data System (ADS)
Lee, Wanho; Kim, Do Wan
2014-10-01
A localized axial Green's function method (LAGM) is proposed for the convection-diffusion equation. The axial Green's function method (AGM) enables us to calculate the numerical solution of a multi-dimensional problem using only one-dimensional Green's functions for the axially split differential operators. This AGM has been developed not only for the elliptic boundary value problems but also for the steady Stokes flows, however, this paper is concerned with the localization of the AGM. This localization of the method is needed for practical purpose when computing the axial Green's function, specifically for the convection-diffusion equation on a line segment that we call the local axial line. Although our focus is mainly on the convection-dominated cases in arbitrary domains, this method can solve other cases in a unified way. Numerical results show that, despite irregular types of discretization on an arbitrary domain, we can calculate the numerical solutions using the LAGM without loss of accuracy even in cases of large convection. In particular, it is also shown that randomly distributed axial lines are available in our LAGM and complicated domains are not a burden.
Diffuse fluorescence tomography based on the radiative transfer equation for small animal imaging
NASA Astrophysics Data System (ADS)
Wang, Yihan; Zhang, Limin; Zhao, Huijuan; Gao, Feng; Li, Jiao
2014-02-01
Diffuse florescence tomography (DFT) as a high-sensitivity optical molecular imaging tool, can be applied to in vivo visualize interior cellular and molecular events for small-animal disease model through quantitatively recovering biodistributions of specific molecular probes. In DFT, the radiative transfer equation (RTE) and its approximation, such as the diffuse equation (DE), have been used as the forward models. The RTE-based DFT methodology is more suitable for biological tissue having void-like regions and the near-source area as in the situations of small animal imaging. We present a RTE-based scheme for the steady state DFT, which combines the discrete solid angle method and the finite difference method to obtain numerical solutions of the 2D steady RTE, with the natural boundary condition and collimating light source model. The approach is validated using the forward data from the Monte Carlo simulation for its better performances in the spatial resolution and reconstruction fidelity compared to the DE-based scheme.
Converged accelerated finite difference scheme for the multigroup neutron diffusion equation
Terranova, N.; Mostacci, D.; Ganapol, B. D.
2013-07-01
Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration. (authors)
Jing, Yun; Xiang, Ning
2008-01-01
This paper proposes a modified boundary condition to improve the room-acoustic prediction accuracy of a diffusion equation model. Previous boundary conditions for the diffusion equation model have certain limitations which restrict its application to a certain number of room types. The boundary condition employing the Sabine absorption coefficient [V. Valeau et al., J. Acoust. Soc. Am. 119, 1504-1513 (2006)] cannot predict the sound field well when the absorption coefficient is high, while the boundary condition employing the Eyring absorption coefficient [Y. Jing and N. Xiang, J. Acoust. Soc. Am. 121, 3284-3287 (2007); A. Billon et al., Appl. Acoust. 69, (2008)] has a singularity whenever any surface material has an absorption coefficient of 1.0. The modified boundary condition is derived based on an analogy between sound propagation and light propagation. Simulated and experimental data are compared to verify the modified boundary condition in terms of room-acoustic parameter prediction. The results of this comparison suggest that the modified boundary condition is valid for a range of absorption coefficient values and successfully eliminates the singularity problem. PMID:18177146
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. PMID:19257112
Zeng, Y; Albertus, P; Klein, R; Chaturvedi, N; Kojic, A; Bazant, MZ; Christensen, J
2013-06-07
Mathematical models of batteries which make use of the intercalation of a species into a solid phase need to solve the corresponding mass transfer problem. Because solving this equation can significantly add to the computational cost of a model, various methods have been devised to reduce the computational time. In this paper we focus on a comparison of the formulation, accuracy, and order of the accuracy for two numerical methods of solving the spherical diffusion problem with a constant or non-constant diffusion coefficient: the finite volume method and the control volume method. Both methods provide perfect mass conservation and second order accuracy in mesh spacing, but the control volume method provides the surface concentration directly, has a higher accuracy for a given numbers of mesh points and can also be easily extended to variable mesh spacing. Variable mesh spacing can significantly reduce the number of points that are required to achieve a given degree of accuracy in the surface concentration (which is typically coupled to the other battery equations) by locating more points where the concentration gradients are highest. (C) 2013 The Electrochemical Society. All rights reserved.
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.
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.
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. PMID:24815259
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
NASA Astrophysics Data System (ADS)
Vassal, J.-P.; Orgéas, L.; Favier, D.; Auriault, J.-L.; Le Corre, S.
2008-01-01
Many analytical and numerical works have been devoted to the prediction of macroscopic effective transport properties in particulate media. Usually, structure and properties of macroscopic balance and constitutive equations are stated a priori. In this paper, the upscaling of the transient diffusion equations in concentrated particulate media with possible particle-particle interfacial barriers, highly conductive particles, poorly conductive matrix, and temperature-dependent physical properties is revisited using the homogenization method based on multiple scale asymptotic expansions. This method uses no a priori assumptions on the physics at the macroscale. For the considered physics and microstructures and depending on the order of magnitude of dimensionless Biot and Fourier numbers, it is shown that some situations cannot be homogenized. For other situations, three different macroscopic models are identified, depending on the quality of particle-particle contacts. They are one-phase media, following the standard heat equation and Fourier’s law. Calculations of the effective conductivity tensor and heat capacity are proved to be uncoupled. Linear and steady state continuous localization problems must be solved on representative elementary volumes to compute the effective conductivity tensors for the two first models. For the third model, i.e., for highly resistive contacts, the localization problem becomes simpler and discrete whatever the shape of particles. In paper II [Vassal , Phys. Rev. E 77, 011303 (2008)], diffusion through networks of slender, wavy, entangled, and oriented fibers is considered. Discrete localization problems can then be obtained for all models, as well as semianalytical or fully analytical expressions of the corresponding effective conductivity tensors.
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).
NASA Astrophysics Data System (ADS)
Ghosh, Atiyo; Leier, Andre; Marquez-Lago, Tatiana
2014-03-01
Spatial stochastic effects are prevalent in many biological systems spanning a variety of scales, from intracellular (e.g. gene expression) to ecological (plankton aggregation). The most common ways of simulating such systems involve drawing sample paths from either the Reaction Diffusion Master Equation (RDME) or the Smoluchowski Equation, using methods such as Gillespie's Simulation Algorithm, Green's Function Reaction Dynamics and Single Particle Tracking. The simulation times of such techniques scale with the number of simulated particles, leading to much computational expense when considering large systems. The Spatial Chemical Langevin Equation (SCLE) can be simulated with fixed time intervals, independent of the number of particles, and can thus provide significant computational savings. However, very little work has been done to investigate the behavior of the SCLE. In this talk we summarize our findings on comparing the SCLE to the well-studied RDME. We use both analytical and numerical procedures to show when one should expect the moments of the SCLE to be close to the RDME, and also when they should differ.
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.
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.
On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations
NASA Astrophysics Data System (ADS)
John, Volker; Novo, Julia
2012-02-01
Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank-Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge-Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.
Boundary conditions of the lattice Boltzmann method for convection-diffusion equations
NASA Astrophysics Data System (ADS)
Huang, Juntao; Yong, Wen-An
2015-11-01
In this paper, we employ an asymptotic analysis technique and construct two boundary schemes accompanying the lattice Boltzmann method for convection-diffusion equations with general Robin boundary conditions. One scheme is for straight boundaries, with the boundary points locating at any distance from the lattice nodes, and has second-order accuracy. The other is for curved boundaries, has only first-order accuracy and is much simpler than the existing schemes. Unlike those in the literature, our schemes involve only the current lattice node. Such a "single-node" boundary schemes are highly desirable for problems with complex geometries. The two schemes are validated numerically with a number of examples. The numerical results show the utility of the constructed schemes and very well support our theoretical predications.
An inverse time-dependent source problem for a time-fractional diffusion equation
NASA Astrophysics Data System (ADS)
Wei, T.; Li, X. L.; Li, Y. S.
2016-08-01
This paper is devoted to identifying a time-dependent source term in a multi-dimensional time-fractional diffusion equation from boundary Cauchy data. The existence and uniqueness of a strong solution for the corresponding direct problem with homogeneous Neumann boundary condition are firstly proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Then we use the Tikhonov regularization method to solve the inverse source problem and propose a conjugate gradient algorithm to find a good approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. This paper was supported by the NSF of China (11371181) and the Fundamental Research Funds for the Central Universities (lzujbky-2013-k02).
Finite difference discretization of semiconductor drift-diffusion equations for nanowire solar cells
NASA Astrophysics Data System (ADS)
Deinega, Alexei; John, Sajeev
2012-10-01
We introduce a finite difference discretization of semiconductor drift-diffusion equations using cylindrical partial waves. It can be applied to describe the photo-generated current in radial pn-junction nanowire solar cells. We demonstrate that the cylindrically symmetric (l=0) partial wave accurately describes the electronic response of a square lattice of silicon nanowires at normal incidence. We investigate the accuracy of our discretization scheme by using different mesh resolution along the radial direction r and compare with 3D (x, y, z) discretization. We consider both straight nanowires and nanowires with radius modulation along the vertical axis. The charge carrier generation profile inside each nanowire is calculated using an independent finite-difference time-domain simulation.
Non-rigid registration of breast surfaces using the laplace and diffusion equations
2010-01-01
A semi-automated, non-rigid breast surface registration method is presented that involves solving the Laplace or diffusion equations over undeformed and deformed breast surfaces. The resulting potential energy fields and isocontours are used to establish surface correspondence. This novel surface-based method, which does not require intensity images, anatomical landmarks, or fiducials, is compared to a gold standard of thin-plate spline (TPS) interpolation. Realistic finite element simulations of breast compression and further testing against a tissue-mimicking phantom demonstrate that this method is capable of registering surfaces experiencing 6 - 36 mm compression to within a mean error of 0.5 - 5.7 mm. PMID:20149261
NASA Astrophysics Data System (ADS)
Hosseini, Vahid Reza; Shivanian, Elyas; Chen, Wen
2016-05-01
The purpose of the current investigation is to determine numerical solution of time-fractional diffusion-wave equation with damping for Caputo's fractional derivative of order α (1 < α ≤ 2). A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason of choosing MLRPI approach is that it does not require any background integrations cells, instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape. The unconditional stability and convergence with order O (τ 6 - 2 α) are proved, where τ is time stepping. Also, several numerical experiments are illustrated to verify theoretical analysis.
Gałdzicki, Z; Miekisz, S
1984-04-01
The role of viscosity in coupling between chemical reaction (complex formation) and diffusion in membranes has been investigated. The Fick law was replaced by the momentum balance equation with the viscous term. The irreversible thermodynamics admits coupling of the chemical reaction rate with the gradient of velocity. The proposed model has shown the contrary effect of viscosity and confirmed the experimental results. The chemical reaction rate increases only above the limit value of viscosity. The parameter Q (degree of complex formation) was introduced to investigate coupling. Q equals to the ratio of the chemical contribution into the flux of the complex to the total flux of the substance transported. For different values of the parameters of the model the dependence of Q upon position inside the membrane has been numerically calculated. The assumptions of the model limit it to a specific case and they only roughly model the biological situation. PMID:6537360
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.
Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
Jamelot, Erell; Ciarlet, Patrick
2013-05-15
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.
Rapidity window dependences of higher order cumulants and diffusion master equation
NASA Astrophysics Data System (ADS)
Kitazawa, Masakiyo
2015-10-01
We study the rapidity window dependences of higher order cumulants of conserved charges observed in relativistic heavy ion collisions. The time evolution and the rapidity window dependence of the non-Gaussian fluctuations are described by the diffusion master equation. Analytic formulas for the time evolution of cumulants in a rapidity window are obtained for arbitrary initial conditions. We discuss that the rapidity window dependences of the non-Gaussian cumulants have characteristic structures reflecting the non-equilibrium property of fluctuations, which can be observed in relativistic heavy ion collisions with the present detectors. It is argued that various information on the thermal and transport properties of the hot medium can be revealed experimentally by the study of the rapidity window dependences, especially by the combined use, of the higher order cumulants. Formulas of higher order cumulants for a probability distribution composed of sub-probabilities, which are useful for various studies of non-Gaussian cumulants, are also presented.
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.
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
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
Competing computational approaches to reaction-diffusion equations in clusters of cells
NASA Astrophysics Data System (ADS)
Stella, Sabrina; Chignola, Roberto; Milotti, Edoardo
2014-03-01
We have developed a numerical model that simulates the growth of small avascular solid tumors. At its core lies a set of partial differential equations that describe diffusion processes as well as transport and reaction mechanisms of a selected number of nutrients. Although the model relies on a restricted subset of molecular pathways, it compares well with experiments, and its emergent properties have recently led us to uncover a metabolic scaling law that stresses the common mechanisms that drive tumor growth. Now we plan to expand the biochemical model at the basis of the simulator to extend its reach. However, the introduction of additional molecular pathways requires an extensive revision of the reaction-diffusion part of the C++ code to make it more modular and to boost performance. To this end, we developed a novel computational abstract model where the individual molecular species represent the basic computational building blocks. Using a simple two-dimensional toy model to benchmark the new code, we find that the new implementation produces a more modular code without affecting performance. Preliminary results also show that a factor 2 speedup can be achieved with OpenMP multithreading, and other very preliminary results indicate that at least an order-of-magnitude speedup can be obtained using an NVidia Fermi GPU with CUDA code.
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
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)
Lubuma, J. M.-S.; Mureithi, E.; Terefe, Y. A.
2011-11-01
The classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the model (a), with the disease-free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided.
NASA Astrophysics Data System (ADS)
Bologna, Mauro; Svenkeson, Adam; West, Bruce J.; Grigolini, Paolo
2015-07-01
Diffusion processes in heterogeneous media, and biological systems in particular, are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory, or a special combination of the two. Accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well. This problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time-dependent coefficients. We propose an iterative scheme for solving fractional differential equations with time-dependent coefficients that is based on a parametric expansion in the fractional index. We demonstrate how this method can be used to predict the long-time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources.
Olson, Gordon L.
2011-08-20
Highlights: {yields} An existing multigroup transport algorithm is extended to be second-order in time. {yields} A new algorithm is presented that does not require a grey acceleration solution. {yields} The two algorithms are tested with 2D, multi-material problems. {yields} The two algorithms have comparable computational requirements. - Abstract: An existing solution method for solving the multigroup radiation equations, linear multifrequency-grey acceleration, is here extended to be second order in time. This method works for simple diffusion and for flux-limited diffusion, with or without material conduction. A new method is developed that does not require the solution of an averaged grey transport equation. It is effective solving both the diffusion and P{sub 1} forms of the transport equation. Two dimensional, multi-material test problems are used to compare the solution methods.
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.
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.
A CNN-based approach to integrate the 3-D turbolent diffusion equation
NASA Astrophysics Data System (ADS)
Nunnari, G.
2003-04-01
The paper deals with the integration of the 3-D turbulent diffusion equation. This problem is relevant in several application fields including fluid dynamics, air/water pollution, volcanic ash emissions and industrial hazard assessment. As it is well known numerical solution of such a kind of equation is very time consuming even by using modern digital computers and this represents a short-coming for on-line applications. To overcome this drawback a Cellular Neural Network Approach is proposed in this paper. CNN's proposed by Chua and Yang in 1988 are massive parallel analog non-linear circuits with local interconnections between the computing elements that allow very fast distributed computations. Nowadays several producers of semiconductors such as SGS-Thomson are producing on chip CNN's so that their massive use for heavy computing applications is expected in the near future. In the paper the methodological background of the proposed approach will be outlined. Further some results both in terms of accuracy and computation time will be presented also in comparison with traditional three-dimensional computation schemes. Some results obtained to model 3-D pollution problems in the industrial area of Siracusa (Italy), characterised by a large concentration of petrol-chemical plants, will be presented.
NASA Astrophysics Data System (ADS)
Orihashi, Kenji; Aizawa, Yoji
2011-11-01
This paper presents detailed numerical results of the competitive diffusion Lotka-Volterra equation (May-Leonard type). First, we derive the global phase diagrams of attractors in the parameter space including the system size, where transition lines between simple attractors are clearly obtained in accordance with the results of linear stability analysis, but the transition borders become complex when multi-basin structures appear. The complex aspects of the transition borders are studied in the case when the system size decreases. Next, we show the statistical aspects of the turbulence with special attention to the onset of the supercritical Hopf bifurcation. Several characteristic quantities, such as correlation length, correlation time, Lyapunov spectra and Lyapunov dimension, are investigated in detail near the onset of turbulence. Our data show the critical scaling law near the onset only in the restricted parameter domain. However even when the critical indices are not determined accurately, it is shown that the empirical scaling relations are obtained in a wide parameter domain far from the onset point and those scaling indices satisfy several relations. These scaling relations are discussed in comparison with the result derived by the phase reduction method. Lastly, we make a conjecture about the stability of an ecosystem based on the bifurcation diagram: the ecosystem obeying the Lotka-Volterra equation in the case of May-Leonard type is stabilized more as the system size increases.
NASA Astrophysics Data System (ADS)
Cooper, Crystal Diane
A computer program was modified to model the dynamics of morphogen concentrations in a developing eye of a Xenopus laevis frog. The dynamics were modelled because it is believed that the behavior of the morphogen concentrations determine how the developing eye maps to the brain. The eye in the xenophus grows as a series of rings, and thus this is the model used. The basis for the simulation are experiments done by Sullivan et al. Following the experiment, aIl eye ring is 'split' in half, inverted, and then 'pasted' onto a donor half. The purpose of the program is to replicate and analyze the results that were found experimentally: a graft made on a north to south axis (dorsal to ventral) produces a change in vision along the east to west axis (anterior to posterior). Four modified Gierer-Meinhardt reaction- diffusion equations are used to simulate the operation. In the second part of the research, the program was further modified and a time series analysis was done on the results. It was found that the modified Gierer- Meinhardt equations demonstrated chaotic behavior under certain conditions. The dynamics included fixed points, limit cycles, transient chaos, intermittent chaos, and strange attractors. The creation and destruction of fractal torii was found.
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.
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.; Department of Chemistry, Howard Hughes Medical Institute, University of California San Diego, La Jolla, California 92093-0636
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
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
NASA Astrophysics Data System (ADS)
Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter
2016-05-01
We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a "simple" and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion.
Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter
2016-05-20
We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a "simple" and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion. PMID:27258868
NASA Astrophysics Data System (ADS)
Kurganov, Alexander; Tadmor, Eitan
2000-05-01
Central schemes may serve as universal finite-difference methods for solving nonlinear convection-diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax-Friedrichs scheme (P. D. Lax, 1954) is the forerunner for such central schemes. The central Nessyahu-Tadmor (NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these central schemes is of order O((Δx)2r/Δt). In the convective regime where Δt∼Δx, the improved resolution of the NT scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of central schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms. In this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O(Δx)2r-1)). In particular, our new central schemes maintain their high-resolution independent of O(1/Δt), and letting Δt ↓ 0, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these central schemes is the use of more precise information of the local propagation speeds. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these central schemes realize the (nonsmooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size. The semi-discrete central scheme is
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)
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}.
NASA Astrophysics Data System (ADS)
Benguria, R. D.; Depassier, M. C.
2016-04-01
We study the dynamics of the equation obtained by Schryer and Walker for the motion of domain walls. The reduced equation is a reaction diffusion equation for the angle between the applied field and the magnetization vector. If the hard-axis anisotropy Kd is much larger than the easy-axis anisotropy Ku, there is a range of applied fields where the dynamics does not select the Schryer-Walker solution. We give an analytic expression for the speed of the domain wall in this regime and the conditions for its existence.
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.
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 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.
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
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...
NASA Technical Reports Server (NTRS)
Collier, G.
1967-01-01
Computer program VARI-QUIR 3 provides Gauss-Seidel type of solution with inner and outer iterations for steady-state, multigroup, two-dimensional neutron diffusion equations. The program has no restrictions on any of the input parameters such as the number of groups, regions, or materials.
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)
Ragusa, Jean C.
2015-01-01
In this paper, we propose a piece-wise linear discontinuous (PWLD) finite element discretization of the diffusion equation for arbitrary polygonal meshes. It is based on the standard diffusion form and uses the symmetric interior penalty technique, which yields a symmetric positive definite linear system matrix. A preconditioned conjugate gradient algorithm is employed to solve the linear system. Piece-wise linear approximations also allow a straightforward implementation of local mesh adaptation by allowing unrefined cells to be interpreted as polygons with an increased number of vertices. Several test cases, taken from the literature on the discretization of the radiation diffusion equation, are presented: random, sinusoidal, Shestakov, and Z meshes are used. The last numerical example demonstrates the application of the PWLD discretization to adaptive mesh refinement.
NASA Astrophysics Data System (ADS)
Gorpas, Dimitris; Andersson-Engels, Stefan
2012-03-01
The solution of the forward problem in fluorescence molecular imaging is among the most important premises for the successful confrontation of the inverse reconstruction problem. To date, the most typical approach has been the application of the diffusion approximation as the forward model. This model is basically a first order angular approximation for the radiative transfer equation, and thus it presents certain limitations. The scope of this manuscript is to present the dual coupled radiative transfer equation and diffusion approximation model for the solution of the forward problem in fluorescence molecular imaging. The integro-differential equations of its weak formalism were solved via the finite elements method. Algorithmic blocks with cubature rules and analytical solutions of the multiple integrals have been constructed for the solution. Furthermore, specialized mapping matrices have been developed to assembly the finite elements matrix. As a radiative transfer equation based model, the integration over the angular discretization was implemented analytically, while quadrature rules were applied whenever required. Finally, this model was evaluated on numerous virtual phantoms and its relative accuracy, with respect to the radiative transfer equation, was over 95%, when the widely applied diffusion approximation presented almost 85% corresponding relative accuracy for the fluorescence emission.
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. PMID:25897139
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.
Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation
NASA Astrophysics Data System (ADS)
Strani, Marta
2015-12-01
We study the one-dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation Metastable behaviour appears when the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches its steady state in a very long time interval. A rigorous analysis is used to study such behaviour by means of the construction of a one-parameter family {{≤ft\\{{{U}\\varepsilon}≤ft(x;ξ \\right)\\right\\}}ξ} of approximate stationary solutions and of a linearisation of the original system around an element of this family. We obtain a system consisting of an ODE for the parameter ξ, describing the position of the interface coupled with a PDE for the perturbation v and defined as the difference v:=u-{{U}\\varepsilon} . The key of our analysis are the spectral properties of the linearised operator around an element of the family ≤ft\\{{{U}\\varepsilon}\\right\\} : the presence of a first eigenvalue, small with respect to ε, leads to metastable behaviour when \\varepsilon \\ll 1 .
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.
Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method
NASA Astrophysics Data System (ADS)
Long, Feixiao; Li, Fengyan; Intes, Xavier; Kotha, Shiva P.
2016-03-01
Optical tomography has a wide range of biomedical applications. Accurate prediction of photon transport in media is critical, as it directly affects the accuracy of the reconstructions. The radiative transfer equation (RTE) is the most accurate deterministic forward model, yet it has not been widely employed in practice due to the challenges in robust and efficient numerical implementations in high dimensions. Herein, we propose a method that combines the discrete ordinate method (DOM) with a streamline diffusion modified continuous Galerkin method to numerically solve RTE. Additionally, a phase function normalization technique was employed to dramatically reduce the instability of the DOM with fewer discrete angular points. To illustrate the accuracy and robustness of our method, the computed solutions to RTE were compared with Monte Carlo (MC) simulations when two types of sources (ideal pencil beam and Gaussian beam) and multiple optical properties were tested. Results show that with standard optical properties of human tissue, photon densities obtained using RTE are, on average, around 5% of those predicted by MC simulations in the entire/deeper region. These results suggest that this implementation of the finite element method-RTE is an accurate forward model for optical tomography in human tissues.
NASA Astrophysics Data System (ADS)
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.
Wetting properties of gas diffusion layers: Application of the Cassie-Baxter and Wenzel equations
NASA Astrophysics Data System (ADS)
Parry, Valérie; Berthomé, Grégory; Joud, Jean-Charles
2012-05-01
In this paper, the wetting behaviours of as received and aged commercial 10% PTFE loaded gas diffusion layer were studied using the Wilhelmy plate method with liquid water temperature ranging from 5 to 60 °C. Comparison were made with an untreated sample and a PTFE smooth plate. These experimental results, supported by chemical and morphological surface characterizations, were discussed in the frame of the Wenzel and Cassie-Baxter regimes. For each wetting regime, surface fraction of solid, PTFE and carbon fibres and/or roughness coefficient were estimated by solving a system of Cassie-Baxter and/or Wenzel equations. The transition to one wetting regime to the other is also commented. Finally, the effects of ageing and of water temperature were studied. Ageing was found to alter the wetting behaviour of the GDL through its chemical degradation. An erosion and the crazing of the PTFE coating and an oxidation of the carbon fibres were pointed out. The decrease of the water surface tension linked to an increase of its temperature is also shown to lead to a better wetting and to an increase of the solid surface fraction value. This effect is reinforced by GDL ageing.
Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method.
Long, Feixiao; Li, Fengyan; Intes, Xavier; Kotha, Shiva P
2016-03-01
Optical tomography has a wide range of biomedical applications. Accurate prediction of photon transport in media is critical, as it directly affects the accuracy of the reconstructions. The radiative transfer equation (RTE) is the most accurate deterministic forward model, yet it has not been widely employed in practice due to the challenges in robust and efficient numerical implementations in high dimensions. Herein, we propose a method that combines the discrete ordinate method (DOM) with a streamline diffusion modified continuous Galerkin method to numerically solve RTE. Additionally, a phase function normalization technique was employed to dramatically reduce the instability of the DOM with fewer discrete angular points. To illustrate the accuracy and robustness of our method, the computed solutions to RTE were compared with Monte Carlo (MC) simulations when two types of sources (ideal pencil beam and Gaussian beam) and multiple optical properties were tested. Results show that with standard optical properties of human tissue, photon densities obtained using RTE are, on average, around 5% of those predicted by MC simulations in the entire/deeper region. These results suggest that this implementation of the finite element method-RTE is an accurate forward model for optical tomography in human tissues. PMID:26953662
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.
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.
Convergence of a random walk method for the Burgers equation
Roberts, S.
1985-10-01
In this paper we consider a random walk algorithm for the solution of Burgers' equation. The algorithm uses the method of fractional steps. The non-linear advection term of the equation is solved by advecting ''fluid'' particles in a velocity field induced by the particles. The diffusion term of the equation is approximated by adding an appropriate random perturbation to the positions of the particles. Though the algorithm is inefficient as a method for solving Burgers' equation, it does model a similar method, the random vortex method, which has been used extensively to solve the incompressible Navier-Stokes equations. The purpose of this paper is to demonstrate the strong convergence of our random walk method and so provide a model for the proof of convergence for more complex random walk algorithms; for instance, the random vortex method without boundaries.
Computer difference scheme for a singularly perturbed convection-diffusion equation
NASA Astrophysics Data System (ADS)
Shishkin, G. I.
2014-08-01
The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ɛ (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is developed in the case when the data of the grid problem include perturbations and additional perturbations are introduced in the course of the computations on a computer. In the absence of perturbations, the standard difference scheme converges at an (δ st ) rate, where δ st = (ɛ + N -1)-1 N -1 and N + 1 is the number of grid nodes; the scheme is not ɛ-uniformly well conditioned or stable to perturbations of the data. Even if the convergence of the standard scheme is theoretically proved, the actual accuracy of the computed solution in the presence of perturbations degrades with decreasing ɛ down to its complete loss for small ɛ (namely, for ɛ = (δ-2max i, j |δ a {/i j }| + δ-1 max i, j |δ b {/i j }|), where δ = δ st and δ a {/i j }, δ b {/i j } are the perturbations in the coefficients multiplying the second and first derivatives). For the boundary value problem, we construct a computer difference scheme, i.e., a computing system that consists of a standard scheme on a uniform mesh in the presence of controlled perturbations in the grid problem data and a hypothetical computer with controlled computer perturbations. The conditions on admissible perturbations in the grid problem data and on admissible computer perturbations are obtained under which the computer difference scheme converges in the maximum norm for ɛ ∈ (0, 1] at the same rate as the standard scheme in the absence of perturbations.
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
NASA Technical Reports Server (NTRS)
Kennedy, Christopher A.; Carpenter, Mark H.
2002-01-01
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first 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 (ARK(sub 2)) 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 of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) 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 ARK(sub 2) methods.
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.
Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
NASA Astrophysics Data System (ADS)
Schehr, Grégory; Majumdar, Satya N.
2008-07-01
We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n - θ( d) where θ( d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n -2( θ(2)+ θ( d)). For Weyl polynomials and Binomial polynomials, this probability decays respectively like exp{(-2θ_{infty}}sqrt{n}) and exp{(-πθ _{infty}sqrt{n})} where θ ∞ is such that θ(d)=2^{-3/2}θ_{infty}sqrt{d} in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [ a, b] has a scaling form given by exp{(-N_{ab}tilde{\\varphi}(k/N_{ab}))} where N ab is the mean number of real roots in [ a, b] and tilde{\\varphi}(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent -2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, 2007).
Giraldo, F.X.; Neta, B.
1995-04-21
An Eulerian and semi-Lagrangian finite element methods for the solution of the two dimensional advection equation were developed. Bilinear rectangular elements were used. Linear stability analysis of the method is given.
Consistency problem with tracer advection in the Atmospheric Model GAMIL
NASA Astrophysics Data System (ADS)
Zhang, Kai; Wan, Hui; Wang, Bin; Zhang, Meigen
2008-03-01
The radon transport test, which is a widely used test case for atmospheric transport models, is carried out to evaluate the tracer advection schemes in the Grid-Point Atmospheric Model of IAP-LASG (GAMIL). Two of the three available schemes in the model are found to be associated with significant biases in the polar regions and in the upper part of the atmosphere, which implies potentially large errors in the simulation of ozone-like tracers. Theoretical analyses show that inconsistency exists between the advection schemes and the discrete continuity equation in the dynamical core of GAMIL and consequently leads to spurious sources and sinks in the tracer transport equation. The impact of this type of inconsistency is demonstrated by idealized tests and identified as the cause of the aforementioned biases. Other potential effects of this inconsistency are also discussed. Results of this study provide some hints for choosing suitable advection schemes in the GAMIL model. At least for the polar-region-concentrated atmospheric components and the closely correlated chemical species, the Flux-Form Semi-Lagrangian advection scheme produces more reasonable simulations of the large-scale transport processes without significantly increasing the computational expense.
NASA Astrophysics Data System (ADS)
Cabrera Fernandez, Delia; Salinas, Harry M.; Somfai, Gabor; Puliafito, Carmen A.
2006-03-01
Optical coherence tomography (OCT) is a rapidly emerging medical imaging technology. In ophthalmology, OCT is a powerful tool because it enables visualization of the cross sectional structure of the retina and anterior eye with higher resolutions than any other non-invasive imaging modality. Furthermore, OCT image information can be quantitatively analyzed, enabling objective assessment of features such as macular edema and diabetes retinopathy. We present specific improvements in the quantitative analysis of the OCT system, by combining the diffusion equation with the free Shrödinger equation. In such formulation, important features of the image can be extracted by extending the analysis from the real axis to the complex domain. Experimental results indicate that our proposed novel approach has good performance in speckle noise removal, enhancement and segmentation of the various cellular layers of the retina using the OCT system.
Phase Segregation of Passive Advective Particles in an Active Medium.
Das, Amit; Polley, Anirban; Rao, Madan
2016-02-12
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. PMID:26919022
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 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.
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
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/vμt‧ 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.
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 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.
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. PMID:27610294
NASA Astrophysics Data System (ADS)
Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.
2015-07-01
In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.
COMPARISON OF NUMERICAL SCHEMES FOR SOLVING A SPHERICAL PARTICLE DIFFUSION EQUATION
A new robust iterative numerical scheme was developed for a nonlinear diffusive model that described sorption dynamics in spherical particle suspensions. he numerical scheme had been applied to finite difference and finite element models that showed rapid convergence and stabilit...
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. PMID:21575582
Piserchia, Andrea; Barone, Vincenzo
2016-08-01
A generalization to arbitrary large amplitude motions of a recent approach to the evaluation of diffusion tensors [ J. Comput. Chem. , 2009 , 30 , 2 - 13 ] is presented and implemented in a widely available package for electronic structure computations. A fully black-box tool is obtained, which, starting from the generation of geometric structures along different kinds of paths, proceeds toward the evaluation of an effective diffusion tensor and to the solution of one-dimensional Smoluchowski equations by a robust numerical approach rooted in the discrete variable representation (DVR). Application to a number of case studies shows that the results issuing from our approach are identical to those delivered by previous software (in particular DiTe) for rigid scans along a dihedral angle, but can be improved by employing relaxed scans (i.e., constrained geometry optimizations) or even more general large amplitude paths. The theoretical and numerical background is robust and general enough to allow quite straightforward extensions in several directions (e.g., inclusion of reactive paths, solution of Fokker-Planck or stochastic Liouville equations, multidimensional problems, free-energy rather than electronic-energy driven processes). PMID:27403666
Correlation between the self-diffusion coefficient of lithium and the equation of state
NASA Astrophysics Data System (ADS)
Eftaxias, K.; Grammatikakis, J.; Varotsos, P.
1985-10-01
Anderson and Swenson [Phys. Rev. B 31, 668 (1985)] have recently presented new isothermal elastic data for lithium for temperatures up to 350 K. It is shown that these data are closely connected to the temperature variation of the self-diffusion coefficient D. Although the latter varies by six orders of magnitude (in the temperature region 195-350 K), however, the elastic data can successfully reproduce the self-diffusion curve without the use of any adjustable parameter.
NASA Astrophysics Data System (ADS)
Lin, Guoxing
2015-10-01
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 K2(t) D (rad2/s) where K ( t ) = ∫0 t γ g ( t ' ) d t ' is a wavenumber and D is the physical diffusion coefficient of the spin carrier in the real space. The term K(ttot) z1 needs to be added to the APS when K(ttot) is not zero. Most of the time, K(ttot) equals zero. However, in iMQC experiments, the condition K(ttot) 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(ttot), 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 Dapp is used to analyze the signal attenuation, it equals nD for zero K(ttot) which agrees with current theoretical and experimental reports, while for non-zero K(ttot), 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 Dapp equals (2n - 1) D is important because the higher value of Dapp means that non-zero K(ttot) iMQC can potentially provide more contrast and measure slower diffusion rates than zero K(ttot) iMQC. The EPSDE method provides a new way to analyze iMQC diffusion experiments.
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. PMID:20365134
NASA Astrophysics Data System (ADS)
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.
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.
NASA Technical Reports Server (NTRS)
Kaplan, M.; Gooden, A.; Turner, R.; Wong, Y.
1977-01-01
The reported investigation is concerned with mesoscale squall-line simulation utilizing numerical techniques. Particular attention is given to fourth-order results and their implications for the dynamical evolution of mesoscale squall-line systems which contain severe local storms. An adiabatic inviscid set of prognostic equations is utilized in a z coordinate system for the fundamental experiment. The fourth-order advection is employed for all time-dependent equations. The Euler-backward time marching scheme is utilized with a 60 second time step. The horizontal mesh length is 42 km. Horizontal diffusion is accomplished by utilizing a smoother-desmoother. Lateral boundary conditions are designed to reduce the development of strong gradients in dependent variables near the boundaries by bringing interior values from the grid to the boundaries.
Diffusion, Peer Pressure, and Tailed Distributions
NASA Astrophysics Data System (ADS)
Cecconi, Fabio; Marsili, Matteo; Banavar, Jayanth R.; Maritan, Amos
2002-08-01
We present a general, physically motivated nonlinear and nonlocal advection equation in which the diffusion of interacting random walkers competes with a local drift arising from a kind of peer pressure. We show, using a mapping to an integrable dynamical system, that on varying a parameter the steady-state behavior undergoes a transition from the standard diffusive behavior to a localized stationary state characterized by a tailed distribution. Finally, we show that recent empirical laws on economic growth can be explained as a collective phenomenon due to peer pressure interaction.
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.
Sharp spatiotemporal patterns in the diffusive time-periodic logistic equation
NASA Astrophysics Data System (ADS)
Du, Yihong; Peng, Rui
To reveal the complex influence of heterogeneous environment on population systems, we examine the asymptotic profile (as ɛ→0) of the positive solution to the perturbed periodic logistic equation {
Lu Benzhuo; Andrew McCammon, J.; Zhou, Y.C.
2010-09-20
In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.
Lu, Benzhuo; Holst, Michael J.; McCammon, J. Andrew; Zhou, Y. C.
2010-01-01
In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems. PMID:21709855
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.
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
Correlation operators based on the iterative solution of an implicitly formulated diffusion equation
NASA Astrophysics Data System (ADS)
Weaver, Anthony; Tshimanga, Jean; Piacentini, Andrea
2015-04-01
Correlation operators are used in variational data assimilation (VDA) for defining background error covariance models and in hybrid ensemble-VDA for localizing, via a Schur product, low-rank sample estimates of background error covariance matrices. This presentation describes new approaches for defining correlation operators based on diffusion operators. The starting point is a two dimensional (2D) implicitly formulated diffusion operator on the sphere, which has been shown in previous works to support symmetric and positive definite smoothing kernels that are closely related to those from the Matern correlation family. Different iterative and preconditioning methods are proposed for solving the 2D implicit diffusion problem, and are compared with respect to their efficiency, accuracy, memory cost, and parallel properties on high-performance computers. The algorithms described in this presentation are evaluated in a global ocean VDA system.
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)
Bai, Cheng-Zu; Zhang, Ren; Hong, Mei; Qian, Long-xia; Wang, Zhengxin
2015-07-01
In this paper, to naturally fill the gap in incomplete data, a new algorithm is proposed for estimating the risk of natural disasters based on the information diffusion theory and the equation of the vibrating string. Two experiments are performed with small samples to investigate its effectiveness. Furthermore, to demonstrate the practicality of the new algorithm, it is applied to study the relationship between epicentral intensity and earthquake magnitude, with strong-motion earthquake observations measured in Yunnan Province in China. The regression model, the back-propagation neural network and the conventional information diffusion model are also involved for comparison. All results show that the new algorithm, which can unravel fuzzy information in incomplete data, is better than the main existing methods for risk estimation.
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.
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.
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)
Li, Fang; Liang, Xing; Shen, Wenxian
2016-08-01
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading-vanishing dichotomy occurs for such free boundary problems (see [16]). In this second part of the series, we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs. We first prove the existence of a unique time almost periodic semi-wave solution associated to such a free boundary problem. Using the semi-wave solution, we then prove that the free boundary problem has a unique spreading speed.
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.
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
Comparison of thermal advection measurements by clear-air radar and radiosonde techniques
Crochet, M.; Rougier, G.; Bazile, G. Meteorologie Nationale, Trappes )
1990-10-01
Vertical profiles of the horizontal wind have been measured every 4 min by a clear-air radar (stratospheric-troposphere radar), and vertical profiles of temperature have been obtained every 2 hours by three radiosonde soundings in the same zone in Brittany during the Mesoscale Frontal Dynamics Project FRONTS 87 campaign. Radar thermal advection is deduced from the thermal wind equation using the measured real horizontal wind instead of the geostrophic wind. Radiosonde thermal advection is determined directly from the sounding station data sets of temperature gradients and also approximately from the thermodynamic equation by the temperature tendency. These approximations, applied during a frontal passage, show the same general features and magnitude of the thermal advection, giving a preliminary but encouraging conclusion for a possible real-time utilization of clear-air radars to monitor thermal advection and to identify its characteristic features. 6 refs.
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
Vertical Structure of Advection-dominated Accretion Flows
NASA Astrophysics Data System (ADS)
Zahra Zeraatgari, Fateme; Abbassi, Shahram
2015-08-01
We solve the set of hydrodynamic equations for optically thin advection-dominated accretion flows by assuming a radially self-similar spherical coordinate system (r,θ ,φ ). The disk is considered to be in steady state and axisymmetric. We define the boundary conditions at the pole and the equator of the disk and, to avoid singularity at the rotation axis, the disk is taken to be symmetric with respect to this axis. Moreover, only the {τ }rφ component of the viscous stress tensor is assumed, and we have set {v}θ =0. The main purpose of this study is to investigate the variation of dynamical quantities of the flow in the vertical direction by finding an analytical solution. As a consequence, we found that the advection parameter, {f}{adv}, varies along the θ direction and reaches its maximum near the rotation axis. Our results also show that, in terms of the no-outflow solution, thermal equilibrium still exists and consequently advection cooling can balance viscous heating.
Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models
NASA Astrophysics Data System (ADS)
Ji, Yanyan Claire; Fenton, Flavio H.
2016-08-01
We describe the implementation of the explicit Euler, Crank-Nicolson, and implicit alternating direction methods for solving partial differential equations and apply these methods to obtain numerical solutions of three excitable-media models used to study neurons and cardiomyocyte dynamics. We discuss the implementation, accuracy, speed, and stability of these numerical methods.
Generalization of the diffusion equation by using the maximum entropy principle
NASA Astrophysics Data System (ADS)
Jumarie, Guy
1985-06-01
By using the so-called maximum entropy principle in information theory, one derives a generalization of the Fokker-Planck-Kolmogorov equation which applies when the n first transition moments of the process are proportional to Δt, while the other ones can be neglected.
Analysis of some identification problems for the reaction-diffusion-convection equation
NASA Astrophysics Data System (ADS)
Alekseev, G. V.; Mashkov, D. V.; Yashenko, E. N.
2016-04-01
Identification problems for a linear stationary reaction-diffusion-convection model, considered in the bounded domain under Dirichlet boundary condition, are studied. Using an optimization method these problems are reduced to respective control problems. The reaction coefficient and the volume density of substance source play the role of controls in this control problem. The solvability of the direct and control problems is proved, the optimality system, which describes the necessary optimality conditions, is derived and the numerical algorithm is developed.
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.
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.
Bleibel, Johannes; Domínguez, Alvaro; Oettel, Martin
2016-06-22
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 [Formula: see text]. 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. PMID:27115236
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.
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...
NASA Astrophysics Data System (ADS)
Ruggeri, Michele; Abert, Claas; Hrkac, Gino; Suess, Dieter; Praetorius, Dirk
2016-04-01
We consider the coupling of the Landau-Lifshitz-Gilbert equation with a quasilinear diffusion equation to describe the interplay of magnetization and spin accumulation in magnetic-nonmagnetic multilayer structures. For this problem, we propose and analyze a convergent finite element integrator, where, in contrast to prior work, we consider the stationary limit for the spin diffusion. Numerical experiments underline that the new approach is more effective, since it leads to the same experimental results as for the model with time-dependent spin diffusion, but allows for larger time-steps of the numerical integrator.