Analysis of a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems

This paper analyzes a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems with two time fractional derivatives of orders $\alpha$ and $\beta$ ($0 < \alpha < \beta < 1$). The stability of this method is established, the temporal accuracy of $O(\tau^{m+1-\beta/2})$ is derived, where $m$ denotes the degree of polynomials for the temporal discretization. It is shown that, even the solution has singularity near $t = {0+}$, this temporal accuracy can still be achieved by using the graded temporal grids. Numerical experiments are performed to verify the theoretical results

Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation

In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are studied. Schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Based on the idea of weighted and shifted Grunwald difference operator, we establish schemes with temporal and spatial accuracy order equal to two and four respectively.Comment: 20 pages, 1 figure

A discontinuous Petrov-Galerkin method for time-fractional diffusion equations

We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness and stability of approximate solutions, and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near~$t=0$ to compensate the singular (temporal) behaviour of the exact solution near $t=0$ caused by the weakly singular kernel, but the spatial mesh is quasiuniform. In the $L_\infty((0,T);L_2(\Omega))$-norm ($(0,T)$ is the time domain and $\Omega$ is the spatial domain), for sufficiently graded time meshes, a global convergence of order $k^{m+\alpha/2}+h^{r+1}$ is shown, where $0<\alpha<1$ is the fractional exponent, $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, and $m$ and $r$ are the degrees of approximate solutions in time and spatial variables, respectively. Numerical experiments indicate that our theoretical error bound is pessimistic. We observe that the error is of order ~$k^{m+1}+h^{r+1}$, that is, optimal in both variables.Comment: SIAM Journal on Numerical Analysis, 201

Finite difference/local discontinuous Galerkin method for solving the fractional diffusion-wave equation

In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and give a semidiscrete scheme in time with the truncation error $O((\Delta t)^2)$, where $\Delta t$ is the time step size. Further we develop a fully discrete scheme for the fractional diffusion-wave equation, and prove that the method is unconditionally stable and convergent with order $O(h^{k+1}+(\Delta t)^{2})$, where $k$ is the degree of piecewise polynomial. Extensive numerical examples are carried out to confirm the theoretical convergence rates.Comment: 18 pages, 2 figure

Subdiffusive discrete time random walks via Monte Carlo and subordination

A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Here we develop a Monte Carlo method for simulating discrete time random walks with Sibuya power law waiting times, providing another approximate solution of the fractional subdiffusion equation. The computation time scales as a power law in the number of time steps with a fractional exponent simply related to the order of the fractional derivative. We also provide an explicit form of a subordinator for discrete time random walks with Sibuya power law waiting times. This subordinator transforms from an operational time, in the expected number of random walk steps, to the physical time, in the number of time steps

Weighted average finite difference methods for fractional diffusion equations

Weighted averaged finite difference methods for solving fractional diffusion equations are discussed and different formulae of the discretization of the Riemann-Liouville derivative are considered. The stability analysis of the different numerical schemes is carried out by means of a procedure close to the well-known von Neumann method of ordinary diffusion equations. The stability bounds are easily found and checked in some representative examples.Comment: Communication presented at the FDA'04 Workshop (with some minor corrections and updates

Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data

This paper establishes the convergence of a time-steeping scheme for time fractional diffusion problems with nonsmooth data. We first analyze the regularity of the model problem with nonsmooth data, and then prove that the time-steeping scheme possesses optimal convergence rates in $L^2(0,T;L^2(\Omega))$-norm and $L^2(0,T;H_0^1(\Omega))$-norm with respect to the regularity of the solution. Finally, numerical results are provided to verify the theoretical results

Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

A Monotone Finite Volume Method for Time Fractional Fokker-Planck Equations

We develop a monotone finite volume method for the time fractional Fokker-Planck equations and theoretically prove its unconditional stability. We show that the convergence rate of this method is order 1 in space and if the space grid becomes sufficiently fine, the convergence rate can be improved to order 2. Numerical results are given to support our theoretical findings. One characteristic of our method is that it has monotone property such that it keeps the nonnegativity of some physical variables such as density, concentration, etc.Comment: 2 figures, accepted by SCIENCE CHINA Mathematic

Numerical schemes of the time tempered fractional Feynman-Kac equation

This paper focuses on providing the computation methods for the backward time tempered fractional Feynman-Kac equation, being one of the models recently proposed in [Wu, Deng, and Barkai, Phys. Rev. E, 84 (2016) 032151]. The discretization for the tempered fractional substantial derivative is derived, and the corresponding finite difference and finite element schemes are designed with well established stability and convergence. The performed numerical experiments show the effectiveness of the presented schemes.Comment: 21 pages, 2 figure