A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation

TL;DR

This paper introduces a second-order finite difference method for solving two-dimensional two-sided space fractional convection diffusion equations, demonstrating stability and convergence through theoretical proof and numerical verification.

Contribution

The paper presents a practical alternating directions implicit method with proven stability and second-order accuracy for complex fractional PDEs in two dimensions.

Findings

Unconditionally von Neumann stable scheme

Second-order convergence in space and time

Numerical verification confirms theoretical results

Abstract

Space fractional convection diffusion equation describes physical phenomena where particles or energy (or other physical quantities) are transferred inside a physical system due to two processes: convection and superdiffusion. In this paper, we discuss the practical alternating directions implicit method to solve the two-dimensional two-sided space fractional convection diffusion equation on a finite domain. We theoretically prove and numerically verify that the presented finite difference scheme is unconditionally von Neumann stable and second order convergent in both space and time directions.