Effective numerical treatment of sub-diffusion equation with non-smooth solution

In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a smooth operator, we devise a high-order numerical scheme by combining the Nystrom method in temporal direction with the compact finite difference method and the spectral method in spatial direction. The distinct advantage of this approach in comparison with most current methods is its high convergence rate even though the solution of the anomalous sub-diffusion equation usually has lower regularity on the starting point. The effectiveness and efficiency of our proposed method are verified by several numerical experiments.Comment: 15 pages, 6 figure

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

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

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order $0<\alpha<1$. For each time $t \in [0,T]$, the HDG approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$\Omega$, the approximations to the exact solution $u$ in the $L_\infty\bigr(0,T;L_2(\Omega)\bigr)$-norm and to $\nabla u$ in the $L_\infty\bigr(0,T;{\bf L}_2(\Omega)\bigr)$-norm are proven to converge with the rate $h^{k+1}$ provided that $u$ is sufficiently regular, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate $h^{k+2}$ (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed

High order difference schemes for a time fractional differential equation with Neumann boundary conditions

Based on our recent results, in this paper, a compact finite difference scheme is derived for a time fractional differential equation subject to the Neumann boundary conditions. The proposed scheme is second order accurate in time and fourth order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. Stability and convergence of the schemes are analyzed using their matrix forms.Comment: 18 pages, 2 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

A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order $\mu\in (0,1)$ with variable coefficients. For the spatial discretization, we apply the standard piecewise linear continuous Galerkin method. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval~$(0,T)$ and a spatial domain~$\Omega$, our analysis suggest that the error in $L^2\bigr((0,T),L^2(\Omega)\bigr)$-norm is of order $O(k^{2-\frac{\mu}{2}}+h^2)$ (that is, short by order $\frac{\mu}{2}$ from being optimal in time) where $k$ denotes the maximum time step, and $h$ is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal $O(k^{2}+h^2)$ error bound in the stronger $L^\infty\bigr((0,T),L^2(\Omega)\bigr)$-norm. Variable time steps are used to compensate the singularity of the continuous solution near $t=0$

A second-order difference scheme for the time fractional substantial diffusion equation

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Gr\"{u}nwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully discrete Gr\"{u}nwald-Letnikov-formula-based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions. In addition, the problem with nonsmooth solution is also discussed, and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative. Numerical examples show the reliability and efficiency of the scheme.Comment: high-order finite difference method, fractional substantial derivative, weighted average operator, stability analysis, nonsmooth solutio

A Two-Grid Finite Element Approximation for A Nonlinear Time-Fractional Cable Equation

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Gr\"unwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved, Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with $O(\tau^{2-\alpha}+\tau^{2-\beta})$. Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.Comment: 23 pages, 5 figure

An exponential B-spline collocation method for fractional sub-diffusion equation

In this article, we propose an exponential B-spline collocation method to approximate the solution of the fractional sub-diffusion equation of Caputo type. The present method is generated by use of the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme in time and an efficient exponential B-spline based method in space. The unique solvability is rigorously discussed. Its stability is well illustrated via a procedure closely resembling the classic von Neumann approach. The resulting algebraic system is tri-diagonal that can rapidly be solved by the known algebraic solver with low cost and storage. A series of numerical examples are finally carried out and by contrast to the other algorithms available in the literature, numerical results confirm the validity and superiority of our method.Comment: 18 pages, 4 tables, 8 figure