Second order finite difference approximations for the two-dimensional time-space Caputo-Riesz fractional diffusion equation

TL;DR

This paper develops and analyzes second-order finite difference schemes for the two-dimensional time-space Caputo-Riesz fractional diffusion equation, proving stability and convergence properties both theoretically and numerically.

Contribution

It introduces new finite difference schemes for the fractional diffusion equation and provides rigorous stability and convergence analysis, including variable coefficients and two-dimensional cases.

Findings

Implicit scheme is unconditionally stable.

Explicit scheme is conditionally stable with a specific stability condition.

The schemes are second-order convergent in space and (2−γ)-th order in time.

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 $\frac{\tau^{\gamma}}{(\Delta x)^{\alpha}}+\frac{\tau^{\gamma}}{(\Delta y)^{\beta}} <C$) and 2nd order convergent in space direction, and $(2-\gamma)$-th order convergent in time direction, where $\gamma \in(0,1]$.