A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations

TL;DR

This paper introduces a new class of second order difference schemes using weighted and shifted Grünwald operators for numerically solving space fractional diffusion equations, demonstrating their stability, convergence, and efficiency.

Contribution

The paper develops and analyzes a novel second order approximation method for space fractional diffusion equations, with proven stability and convergence in multiple dimensions.

Findings

Numerical schemes are stable and convergent for constant coefficient problems.

The methods achieve the expected second order accuracy.

Numerical results confirm effectiveness for variable coefficient problems.

Abstract

A class of second order approximations, called the weighted and shifted Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions. The stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established. Several numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence order, and the numerical results for variable coefficients problem are also presented.