A Finite Difference Method on Quasi-uniform Mesh for Time-Fractional Advection-Diffusion Equations with Source Term

TL;DR

This paper introduces a stable finite difference method on quasi-uniform meshes for solving time-fractional advection-diffusion equations, improving accuracy and demonstrating stability, convergence, and effectiveness through theoretical analysis and numerical experiments.

Contribution

It proposes a novel finite difference scheme on quasi-uniform grids that enhances accuracy for time-fractional PDEs involving Caputo derivatives.

Findings

The method is unconditionally stable.

Error estimates show improved accuracy on quasi-uniform grids.

Numerical experiments confirm the theoretical results.

Abstract

The present paper deals with the numerical solution of time-fractional advection-diffusion equations involving the Caputo derivative with source term by means of an unconditionally stable implicit finite difference method on quasi-uniform grids. We use a special quasi-uniform mesh in order to improve the numerical accuracy of the classical discrete fractional formula for the Caputo derivative. The stability and the convergence of the method are discussed. The error estimates established for a quasi-uniform grid and a uniform one are reported to support the theoretical results. Numerical experiments are carried out to demonstrate the effectiveness of the method.