High-order fractional-compact finite difference method for Riesz spatial telegraph equation

TL;DR

This paper develops high-order compact finite difference schemes for Riesz derivatives and applies a 4th-order scheme to the Riesz spatial telegraph equation, demonstrating stability, convergence, and numerical accuracy.

Contribution

The paper introduces even order compact schemes for Riesz derivatives and applies them to the telegraph equation, with proven stability and convergence.

Findings

Convergence orders in space and time are both 4th order.

Numerical experiments confirm the schemes' accuracy and stability.

High-order schemes effectively solve the Riesz spatial telegraph equation.

Abstract

In this paper, we establish even order compact numerical schemes (4th-order, 6th-order, 8th-order, 10th-order) for Riesz derivatives by using the symmetrical fractional centred difference operator. Then we apply the derived 4th-order algorithm to the Riesz spatial telegraph equation. We carefully study the stability and convergence by matrix method, and show that convergence orders in temporal and spatial directions are both 4th order. Numerical experiments are displayed which support the compact difference schemes for Riesz derivatives and the Riesz spatial telegraph equation.