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We present a space–time fractional Allen–Cahn phase-field model that describes the transport of the fluid mixture of two immiscible fluid phases. The space and time fractional order parameters control the sharpness and the decay behavior of the interface via a seamless transition of the parameters. Although they are shown to provide more accurate description of anomalous diffusion processes and sharper interfaces than traditional integer-order phase-field models do, fractional models yield numerical methods with dense stiffness matrices. Consequently, the resulting numerical schemes have significantly increased computational work and memory requirement. We develop a lossless fast numerical method for the accurate and efficient numerical simulation of the space–time fractional phase-field model. Numerical experiments shows the utility of the fractional phase-field model and the corresponding fast numerical method.

Many biological fluids, like mucus and cytoplasm, have prominent viscoelastic properties. As a consequence, immersed particles exhibit subdiffusive behavior, which is to say, the variance of the particle displacement grows sublinearly with time. In this work, we propose a viscoelastic generalization of the LandauLifschitz NavierStokes fluid model and investigate the properties of particles that are passively advected by such a medium. We exploit certain exact formulations that arise from the Gaussian nature of the fluid model and introduce analysis of memory in the fluid statistics, marking an important step toward capturing fluctuating hydrodynamics among subdiffusive particles. The proposed method is spectral, meshless and is based on the numerical evaluation of the covariance matrix associated with individual fluid modes. With this method, we probe a central hypothesis of passive microrheology, a field premised on the idea that the statistics of particle trajectories can reveal fundamental information about their surrounding fluid environment.

We develop a fast finite volume method for variable-coefficient, conservative space-fractional diffusion equations in convex domains via a volume-penalization approach. The method has an optimal storage and an almost linear computational complexity. The method retains second-order accuracy without requiring a Richardson extrapolation. Numerical results are presented to show the utility of the method.

Diffusion processes in heterogeneous media, and biological systems in particular, are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory, or a special combination of the two. Accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well. This problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time-dependent coefficients. We propose an iterative scheme for solving fractional differential equations with time-dependent coefficients that is based on a parametric expansion in the fractional index. We demonstrate how this method can be used to predict the long-time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources.

Dispersive transport of charge carriers in disordered nanostructured semiconductors is described in terms of integral diffusion equations nonlocal in time. Transient photocurrent kinetics is analyzed for different situations. Relation to the fractional differential approach is demonstrated. Using this relation provides specifications in interpretation of the time-of-flight data. Joint influence of morphology and energy distribution of localized states is described in frames of the trap-limited advection-diffusion on a comb structure modeling a percolation cluster.

Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.

The continuous time random walk, introduced in the physics literature by Montroll and Weiss, has been widely used to model anomalous diffusion in external force fields. One of the features of this model is that the governing equations for the evolution of the probability density function, in the diffusion limit, can generally be simplified using fractional calculus. This has in turn led to intensive research efforts over the past decade to develop robust numerical methods for the governing equations, represented as fractional partial differential equations.Here we introduce a discrete time random walk that can also be used to model anomalous diffusion in an external force field. The governing evolution equations for the probability density function share the continuous time random walk diffusion limit. Thus the discrete time random walk provides a novel numerical method for solving anomalous diffusion equations in the diffusion limit, including the fractional Fokker-Planck equation. This method has the clear advantage that the discretisation of the diffusion limit equation, which is necessary for numerical analysis, is itself a well defined physical process. Some examples using the discrete time random walk to provide numerical solutions of the probability density function for anomalous subdiffusion, including forcing, are provided.

Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O ( N 3 ) of computational work per time step and O ( N 2 ) of memory to store where N is the number of spatial grid points in the discretization.In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O ( N ) of memory and O ( N log ? N ) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.

The observation of an increasing number of anomalous diffusion phenomena motivates the study to reveal the actual reason for such stochastic processes. When it is difficult to get analytical solutions or necessary to track the trajectory of particles, lattice Monte Carlo (LMC) simulation has been shown to be particularly useful. To develop such an LMC simulation algorithm for the Galilei variant anomalous diffusion, we derive explicit solutions for the conditional and unconditional first passage time (FPT) distributions with double absorbing barriers. According to the theory of random walks on lattices and the FPT distributions, we propose an LMC simulation algorithm and prove that such LMC simulation can reproduce both the mean and the mean square displacement exactly in the long-time limit. However, the error introduced in the second moment of the displacement diverges according to a power law as the simulation time progresses. We give an explicit criterion for choosing a small enough lattice step to limit the error within the specified tolerance. We further validate the LMC simulation algorithm and confirm the theoretical error analysis through numerical simulations. The numerical results agree with our theoretical predictions very well.

Fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior. Recently, spectral Galerkin methods were developed for space-fractional diffusion equations aiming at achieving exponential convergence. An optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity. An optimal order error estimate in the L 2 norm was proved via the well known Nitsche lifting technique under the assumption that the true solution to the corresponding boundary-value problem of the fractional diffusion equation has the required regularity for each right-hand side.In this paper we show that the true solution to the Dirichlet boundary-value problem of a conservative fractional diffusion equation of order 2 - β with 0 < β < 1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional Sobolev space H 3 / 2 - β in general, but is still in the Besov space B ∞ 3 / 2 - β ( L 2 ) . Hence, the provable convergence rate of a spectral Galerkin method in the L 2 norm is at most of the order O ( N - ( 3 / 2 - β ) ) , where N is the degree of the polynomial space in the numerical method. Numerical experiments show that the spectral Galerkin method exhibits a subquadratic convergence in the L 2 norm for any 0 < β < 1 .We develop a high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of one-sided variable-coefficient conservative fractional diffusion equations. The method has a proved high-order convergence rate of arbitrary order (i) without requiring the smoothness of the true solution u to the given boundary-value problem, but only assuming that the diffusivity coefficient and the right-hand source term have the desired regularity; (ii) for a variable diffusivity coefficient; and (iii) for an inhomogeneous Dirichlet boundary condition. Numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right-hand source term have the desired regularity.

This paper is devoted to the construction and analysis of finite difference methods for solving a class of time-fractional subdiffusion equations. Based on the certain superconvergence at some particular points of the fractional derivative by the traditional first-order Grunwald-Letnikov formula, some effective finite difference schemes are derived. The obtained schemes can achieve the global second-order numerical accuracy in time, which is independent of the values of anomalous diffusion exponent α ( 0 < α < 1 ) in the governing equation. The spatial second-order scheme and the spatial fourth-order compact scheme, respectively, are established for the one-dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes. The superconvergent points of the traditional first-order Grunwald-Letnikov formula to approximate the Riemann-Liouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time-fractional sub-diffusion equations. The obtained schemes can achieve the global second-order accuracy in time, which is independent of the anomalous diffusion exponent α ( 0 < α < 1 ) in the governing equation.The spatial second-order scheme and the spatial fourth-order compact scheme are presented, respectively. The strict stability and convergence analysis on the resulting difference schemes are given by the discrete energy method.The extension to the two-dimensional case is also discussed. The compact difference scheme with the convergence of second-order in time and fourth-order in space is derived along with the strict stability and convergence analysis.Several numerical experiments are implemented to illustrate the computational efficiency, numerical accuracy and robustness of the proposed schemes.

An error propagation equation has been obtained for the numerical solution of an initial-boundary value problem governed by linear diffusion equation. The error propagation equation is analyzed to identify different types of errors as those due to sub- or super-diffusion and dispersive diffusion errors. Quantification of errors has also been used for numerical analysis of some central difference methods to solve this equation.

A general state-based peridynamic formulation is presented for convective single-phase flow of a liquid of small and constant compressibility in heterogeneous porous media. In addition to local fluid transport, possible anomalous diffusion due to non-local fluid transport is considered and simulated. The governing integral equations of the peridynamic formulation are computationally easier to solve in domains with discontinuities than the traditional conservation models containing spatial derivatives. A bond-based peridynamic formulation is also developed and demonstrated to be a special case of the state-based formulation. The non-local model does not assume continuity in the field variables, satisfies mass conservation over an arbitrary bounded body and approaches the corresponding local model as the non-local region goes to zero. The exact solution of the local model closely matches the non-local model for a classical two-dimensional flow problem with fluid sources and sinks and for both Neumann and Dirichlet boundary conditions. The model is shown to capture arbitrary flow discontinuities/heterogeneities without any fundamental changes to the model and with small incremental computational costs.

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved via Gaussian elimination, which requires computational work of O ( N 3 ) per time step and O ( N 2 ) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-,?y-,?z-directions commute. We base on the ADI formulation to develop a fast iterative ADI finite difference method, which has a computational work count of O ( N log N ) per iteration at each time step and a memory requirement of O ( N ) . We also develop a fast multistep ADI finite difference method, which has a computational work count of O ( N log 2 N ) per time step and a memory requirement of O ( N log N ) . Numerical experiments of a three-dimensional space-fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three-dimensional implicit finite difference method, but have significantly improved computational cost and memory requirement. These numerical experiments show the utility of the fast method.

In this paper, we are concerned with numerical methods for the solution of initial-boundary value problems of anomalous diffusion equations of order α ? ( 1 , 2 ) . The classical Crank-Nicholson method is used to discretize the fractional diffusion equation and then the spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Two preconditioned iterative methods, namely, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient for normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. Numerical experiments are given to illustrate the efficiency of the methods.

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the non-local property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved with Gaussian elimination, which requires computational work of O(N^3) per time step and O(N^2) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. In this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time-dependent space-fractional diffusion equations in three space dimensions, by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices. The fast method has a computational work count of O(NlogN) per iteration and a memory requirement of O(N), while retaining the same accuracy as the underlying finite difference method solved with Gaussian elimination. Numerical experiments of a three-dimensional space-fractional diffusion equation show the utility of the fast method.

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of O(N^2) and computational cost of O(N^3) for a problem of size N. We develop a superfast-preconditioned conjugate gradient squared method for the efficient solution of steady-state space-fractional diffusion equations. The method reduces the computational work from O(N^2) to O(NlogN) per iteration and reduces the memory requirement from O(N^2) to O(N). Furthermore, the method significantly reduces the number of iterations to be mesh size independent. Preliminary numerical experiments for a one-dimensional steady-state diffusion equation with 2^1^3 nodes show that the fast method reduces the overall CPU time from 3h and 27min for the Gaussian elimination to 0.39s for the fast method while retaining the accuracy of Gaussian elimination. In contrast, the regular conjugate gradient squared method diverges after 2days of simulations and more than 20,000 iterations.

One of the ongoing issues with fractional diffusion models is the design of an efficient high-order numerical discretization. This is one of the reasons why fractional diffusion models are not yet more widely used to describe complex systems. In this paper, we derive a radial basis functions (RBF) discretization of the one-dimensional space-fractional diffusion equation. In order to remove the ill-conditioning that often impairs the convergence rate of standard RBF methods, we use the RBF-QR method [1,33]. By using this algorithm, we can analytically remove the ill-conditioning that appears when the number of nodes increases or when basis functions are made increasingly flat. The resulting RBF-QR-based method exhibits an exponential rate of convergence for infinitely smooth solutions that is comparable to the one achieved with pseudo-spectral methods. We illustrate the flexibility of the algorithm by comparing the standard RBF and RBF-QR methods for two numerical examples. Our results suggest that the global character of the RBFs makes them well-suited to fractional diffusion equations. They naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from a second-order to a fractional-order diffusion model. As such, they should be considered as one of the methods of choice to discretize fractional diffusion models of complex systems.

Highlights? Error analysis and numerical experiments demonstrate the efficiency of our method for time fractional diffusion equations. ? Our method avoids costly convolution integral calculation and dilemmatic selection of time step size. ? A truly boundary-only meshless method is applied to Laplace-transformed inhomogeneous problem. ? Our method effectively simulates 3D long time-history fractional diffusion systems. This paper develops a novel boundary meshless approach, Laplace transformed boundary particle method (LTBPM), for numerical modeling of time fractional diffusion equations. It implements Laplace transform technique to obtain the corresponding time-independent inhomogeneous equation in Laplace space and then employs a truly boundary-only meshless boundary particle method (BPM) to solve this Laplace-transformed problem. Unlike the other boundary discretization methods, the BPM does not require any inner nodes, since the recursive composite multiple reciprocity technique (RC-MRM) is used to convert the inhomogeneous problem into the higher-order homogeneous problem. Finally, the Stehfest numerical inverse Laplace transform (NILT) is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding BPM solutions. In comparison with finite difference discretization, the LTBPM introduces Laplace transform and Stehfest NILT algorithm to deal with time fractional derivative term, which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability. Consequently, it can effectively simulate long time-history fractional diffusion systems. Error analysis and numerical experiments demonstrate that the present LTBPM is highly accurate and computationally efficient for 2D and 3D time fractional diffusion equations.

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O(N^3) per time step and memory of O(N^2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O(N log^2N) per time step and memory of O(N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

We analyze the different degrees of accuracy of two Monte Carlo methods for the simulation of one-dimensional diffusion processes with homogeneous or spatial dependent diffusion coefficient that we assume correctly described by a differential equation. The methods analyzed correspond to fixed and Gaussian steplengths. For a homogeneous diffusion coefficient it is known that the Gaussian steplength generates exact results at fixed time steps @Dt. For spatial dependent diffusion coefficients the symmetric character of the Gaussian distribution introduces an error that increases with time. As an example, we consider a diffusion coefficient with constant gradient and show that the error is not present for fixed steplength with appropriate asymmetric jump probabilities.

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O(N^2) and computational cost of O(N^3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O(N) and computational cost of O(Nlog^2N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor-corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor-corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.

We present a new code aiming at giving a global and coherent approach for transport and turbulence issues in the edge plasma of Tokamaks. The TOKAM-3D code solves 3D fluid drift equations in full-torus geometry including both closed field lines and SOL physics. No scale separation is assumed so that interactions between large scale flows and turbulence are coherently treated. Moreover, the code can be run in transport regimes ranging from purely anomalous diffusion to fully established turbulence. Specific numerical schemes have been developed which can solve the model equations whether the presence of a limiter in the plasma is taken into account or not. Example cases giving an overview of the field of application of the code as well as verification results are also presented.

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^m^i^n^{^1^+^2^@a^,^2^})+O(h^2), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for @a=1.0 with the ones of directly discretizing classical Fokker-Planck equation, some numerical results for time fractional Fokker-Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for @a=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.

We present a new approach for treating molecular diffusion in filtered density function (FDF) methods for modeling turbulent reacting flows. The diffusion term accounts for molecular transport in physical space and molecular mixing in composition space. Conventionally, the FDF is represented by an ensemble of particles and transport is modeled by a random walk in physical space. There are two significant shortcomings in this transport model: (1) the resulting composition variance equation contains a spurious production term and (2) because the random walk is governed by a single diffusion coefficient, the formulation cannot account for differential diffusion, which can have a first-order effect in reacting flows. In our new approach transport is simply modeled by a mean drift in the particle composition equation. The resulting variance equation contains no spurious production term and differential diffusion is treated easily. Hence, the new formulation reduces to a direct numerical simulation (DNS) in the limit of vanishing filter width, a desirable property of any large-eddy simulation (LES) approach. We use the IEM model for mixing. It is shown that there is a lower bound on the specified mixing rate such that the boundedness of the compositions is ensured. We present a numerical method for solving the particle equations which is second-order accurate in space and time, obeys detailed conservation, enforces the realizability and boundedness constraints and is unconditionally stable.