Finite difference schemes for multi-term time-fractional mixed diffusion-wave equations

The multi-term time-fractional mixed diffusion-wave equations (TFMDWEs) are considered and the numerical method with its error analysis is presented in this paper. First, a $L2$ approximation is proved with first order accuracy to the Caputo fractional derivative of order $\beta \in (1,2).$ Then the approximation is applied to solve a one-dimensional TFMDWE and an implicit, compact difference scheme is constructed. Next, a rigorous error analysis of the proposed scheme is carried out by employing the energy method, and it is proved to be convergent with first order accuracy in time and fourth order in space, respectively. In addition, some results for the distributed order and two-dimensional extensions are also reported in this work. Subsequently, a practical fast solver with linearithmic complexity is provided with partial diagonalization technique. Finally, several numerical examples are given to demonstrate the accuracy and efficiency of proposed schemes.Comment: $L2$ approximation compact difference scheme distributed order fast solver convergenc

A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional Allen-Cahn equations

In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically verified.Comment: 24 pages, 7 figures, 10 table

An efficient second-order convergent scheme for one-side space fractional diffusion equations with variable coefficients

In this paper, a second order finite difference scheme is investigated for time-dependent one-side space fractional diffusion equations with variable coefficients. The existing schemes for the equation with variable coefficients have temporal convergence rate no better than second order and spatial convergence rate no better than first order, theoretically. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Gr\"unwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme

Efficient computation of the Grunwald-Letnikov fractional diffusion derivative using adaptive time step memory

Computing numerical solutions to fractional differential equations can be computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In general, numerical approaches that depend on truncating part of the system history while efficient, can suffer from high degrees of error and inaccuracy. Here we present an adaptive time step memory method for smooth functions applied to the Grunwald-Letnikov fractional diffusion derivative. This method is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points backward in time, a temporally weighted history is computed that includes contributions from the entire past of the system, maintaining accuracy, but with fewer points actually calculated, greatly improving computational efficiency.Comment: 25 pages; in press in The Journal of Computational Physic

A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term

An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the $L_2$-norm is $\mathcal{O}(\tau^2 + h^2)$ with time step $\tau$ and mesh size $h$. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.Comment: 27 pages, 5 figures, 5 table

An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.Comment: 22 pages, 26 figure

Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.Comment: 20 pages, 4 figures, 8 table

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

Numerical methods for nonlocal and fractional models

Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.Comment: Revised/Improved version. 126 pages, 18 figures, review pape

Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation

The functional distributions of particle trajectories have wide applications, including the occupation time in half-space, the first passage time, and the maximal displacement, etc. The models discussed in this paper are for characterizing the distribution of the functionals of the paths of anomalous diffusion described by time-space fractional diffusion equation. This paper focuses on providing effective computation methods for the models. Two kinds of time stepping schemes are proposed for the fractional substantial derivative. The multiresolution Galerkin method with wavelet B-spline is used for space approximation. Compared with the finite element or spectral polynomial bases, the wavelet B-spline bases have the advantage of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The unconditional stability and convergence of the provided schemes are theoretically proved and numerically verified. Finally, we also discuss the efficient implementations and some extensions of the schemes, such as the wavelet preconditioning and the non-uniform time discretization.Comment: 31 pages, 1 figur