# Energy estimates for two-dimensional space-Riesz fractional wave equation

## TL;DR

This paper develops an energy method to analyze the stability and convergence of the two-dimensional space-Riesz fractional wave equation, providing theoretical proofs and numerical verification of its accuracy and stability.

## Contribution

It introduces a novel energy-based approach for error estimation of the 2D space-Riesz fractional wave equation with variable coefficients, including stability and convergence analysis.

## Key findings

- Unconditional stability of the scheme is proved.
- Convergence order is d7b5b2 in space and time.
- Numerical results confirm theoretical error estimates.

## Abstract

The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic viscoelasticity. In this paper, we first develop the energy method to estimate the one-dimensional space-Riesz fractional wave equation. For two-dimensional cases with the variable coefficients, the discretized matrices are proved to be commutative, which ensures to carry out of the priori error estimates. The unconditional stability and convergence with the global truncation error $\mathcal{O}(\tau^2+h^2)$ are theoretically proved and numerically verified. In particulary, the framework of the priori error estimates and convergence analysis are still valid for the compact finite difference scheme and the nonlocal wave equation.