An improved algorithm based on finite difference schemes for fractional boundary value problems with non-smooth solution

TL;DR

This paper introduces an extrapolation-based algorithm that enhances the accuracy of finite difference schemes, specifically WSGD and FCD, for fractional boundary value problems with non-smooth solutions, achieving near second-order accuracy.

Contribution

It proposes an extrapolation technique to improve finite difference schemes' accuracy for fractional boundary problems with non-smooth solutions, validated by theoretical analysis and numerical experiments.

Findings

Improved accuracy and convergence rate of WSGD and FCD schemes.

Theoretical error estimates in maximum norm.

Numerical examples confirm second-order accuracy for non-smooth solutions.

Abstract

In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with non-smooth solution. Two popular finite difference schemes, the weighted shifted Gr\"{u}nwald difference (WSGD) scheme and the fractional centered difference (FCD) scheme, are revisited and the error estimate of the schemes is provided in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, it is demonstrated that, with the use of the proposed algorithm, the improved WSGD and FCD schemes can recover the second-order accuracy for non-smooth solution. Several numerical examples are given to validate our theoretical prediction. It is shown that both accuracy and convergence rate of numerical solutions can be significantly improved by using the proposed algorithm.