Mathematical analysis and symmetric fractional-order reduction method for diffusion-wave equations

TL;DR

This paper introduces a symmetric fractional-order reduction method for fractional wave equations, developing efficient numerical algorithms on nonuniform meshes with validated accuracy through numerical experiments.

Contribution

It presents a novel symmetric fractional-order reduction method and optimal parameter choices for nonuniform meshes in fractional wave equations.

Findings

Algorithms demonstrate high accuracy in numerical experiments.

Optimal parameters improve stability and efficiency.

Method applicable to low regularity initial data.

Abstract

In this work, our aim is to introduce a symmetric fractional-order reduction (SFOR) method to develop numerical algorithms on nonuniform temporal meshes for fractional wave equations under lower regularity assumptions. The $L$-type methods--including $L1$ and $L2$-$1_\sigma$ schemes--are specifically designed for diffusion-wave equations, and we propose novel optimal parameter selections tailored to nonuniform meshes. Finally, several numerical experiments are conducted to validate the efficiency and accuracy of the algorithms.