An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements

Abstract

The numerical solution for a class of sub-diffusion equations involving a parameter in the range −1 < α < 0 is studied. For the time discretization, we use an implicit finite-difference Crank-Nicolson method and show that the error is of order k 2+α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h 2 max(1, log k −1), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.

Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation

“…Theorem 5 asserts an error bound of order k 2+α + h 2 (k) using a suitable mesh grading in time (but still a quasiuniform spatial mesh). Recent work [9] by the first author proved the same convergence rate for a generalized Crank-Nicolson scheme.…”

“…(3) and (4). We apply the piecewise-linear, DG method (13) with time steps satisfying (9) and (10), and with S n =Ḣ 1 so there is no spatial discretization. If γ * := (2 + α)/σ , then for 1 ≤ n ≤ N,…”

“…In a series of papers [5,6,[8][9][10] we have considered numerical methods for the initial value problem…”

Numerical asymptotic stability for the integro-differential equations with the multi-term kernels

“…This issue is relevant from a computational point of view since, as has been observed in a number of contexts (fractional wave equation, evolution equation with positive type memory term, sub-diffusion equation, etc. [20][21][22][23] ), the convergence property in the classical sense of numerical analysis (a property that concerns finite-time horizons) is not sufficient to ensure the asymptotic behavior of the PDE solutions to be captured correctly. The fact that the numerical approximation schemes preserve the decay properties of continuous solutions can be considered as a manifestation of the property of PDE solutions.…”

A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

“…In the literature, many works have been devoted to this study [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Based on the Grünwald-Letnikov approximation, Yuste and Acedo [5] constructed an explicit difference scheme for fractional diffusion equations, and investigated the stability using the von Neumann method.…”