High Order Finite Difference Methods on Non-uniform Meshes for Space Fractional Operators

TL;DR

This paper develops and analyzes high-order finite difference schemes on non-uniform meshes for space fractional operators, addressing a gap in existing methods that are limited to uniform meshes, and demonstrates their effectiveness through theoretical and numerical validation.

Contribution

It introduces a novel strategy for constructing high-order finite difference schemes on non-uniform meshes for fractional operators, expanding the applicability of these methods.

Findings

The schemes achieve expected convergence orders.

Error estimates and stability are rigorously established.

Numerical experiments confirm theoretical results.

Abstract

In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform meshes. The nonlocal property of space fractional operator makes it difficult to design the finite difference scheme on non-uniform meshes. This paper provides a basic strategy to derive the first and high order discretization schemes on non-uniform meshes for fractional operators. And the obtained first and second schemes on non-uniform meshes are used to solve space fractional diffusion equations. The error estimates and stability analysis are detailedly performed; and extensive numerical experiments confirm the theoretical analysis or verify the convergence orders.