Superlinearly convergent algorithms for the two-dimensional space–time Caputo–Riesz fractional diffusion equation

Abstract

In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify that the implicit finite difference scheme is unconditionally stable (the explicit scheme is conditionally stable with the stability condition(∆y) β < C) and 2nd order convergent in space direction, and (2 − γ)-th order convergent in time direction, where γ ∈ (0, 1].

A weighted finite difference method for the fractional diffusion equation based on the Riemann–Liouville derivative

“…Here, we consider a discretization of the Riemann-Liouville fractional derivative in an unbounded domain and prove its second order consistency. We would like to point out that during the time this work was under revision, some authors have been using the discretization of the fractional derivative introduced here in different problems [3,4,8,13,33]. At the same time, second order and higher order approximations for the fractional derivative, based in different ideas, have been appearing in literature [5,6,39].…”

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

“…The situation becomes more complicated for partial differential equations with both temporal and spatial fractional derivatives. In the present study, we consider efficient numerical method for the following two-dimensional time-space fractional diffusion equations in a rectangular domain: Study of this kind of equations can be found in [3][4][5][6] and the references therein. Previously we applied the second order time discretization developed in [7] to some time-space fractional differential equations in [6].…”

“…We remark that compact difference schemes were successfully applied to improve the spatial accuracy of fractional diffusion equations in recent years (see [9][10][11][12][13] and the references therein). As a whole, we established a scheme which converges with O(τ 2 + h 4 1 + h 4 2 ), where τ is the temporal step size and h 1 , h 2 are the spatial step sizes respectively. We also consider technique of Richardson extrapolation [14][15][16] which improves the accuracy of our proposed method.…”

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

“…Generally, the Grünwald-Letnikov derivative is used to approximate to the Riesz fractional derivative, most of which are finite difference methods [21][22][23][24]26,27,29,37,38]. In addition, the matrix transform method [25,35], Galerkin finite element method [28], predictor-corrector method [36], variational iteration method [39] and alternating direction method [30] are also proposed to applied to the fractional diffusion equations with Riesz space fractional derivative.…”