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
Energy Science and Technology Software Center (ESTSC)
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.
Energy Science and Technology Software Center (ESTSC)
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.