Compact Finite Difference Approximations for Space Fractional Diffusion Equations

TL;DR

This paper develops compact finite difference schemes for space fractional diffusion equations using WSGD operators, achieving high-order accuracy and providing stability and error analysis.

Contribution

It introduces new compact discretization schemes for fractional operators with proven stability and convergence orders, advancing numerical methods for fractional PDEs.

Findings

Schemes achieve third-order spatial accuracy

Schemes achieve second-order temporal accuracy

Numerical results confirm theoretical convergence rates

Abstract

Based on the weighted and shifted Gr\"{u}nwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the one and two dimensional space fractional diffusion equations. The detailed numerical stability and error analysis are theoretically performed. We theoretically prove and numerically verify that the provided numerical schemes have the convergent orders 3 in space and 2 in time.