Weighted average finite difference methods for fractional diffusion equations

TL;DR

This paper explores weighted average finite difference methods for fractional diffusion equations, analyzing stability and discretization schemes to improve numerical solutions of these complex equations.

Contribution

It introduces and compares various discretization formulas for the Riemann-Liouville derivative and provides stability analysis for the resulting numerical schemes.

Findings

Stability bounds are derived and verified.

Different discretization schemes are evaluated.

The methods are applicable to fractional diffusion equations.

Abstract

Weighted averaged finite difference methods for solving fractional diffusion equations are discussed and different formulae of the discretization of the Riemann-Liouville derivative are considered. The stability analysis of the different numerical schemes is carried out by means of a procedure close to the well-known von Neumann method of ordinary diffusion equations. The stability bounds are easily found and checked in some representative examples.