Unconditional energy dissipation and error estimates of the SAV Fourier spectral method for nonlinear fractional generalized wave equation

TL;DR

This paper introduces a second-order SAV Fourier spectral method for nonlinear fractional wave equations, proving unconditional energy dissipation and optimal error estimates without restrictive assumptions, validated by numerical experiments.

Contribution

It develops a novel second-order SAV Fourier spectral scheme with unconditional energy dissipation and optimal error bounds for nonlinear fractional wave equations, overcoming previous restrictions.

Findings

Unconditional energy dissipation of the scheme is established.

Optimal error estimates are achieved without global Lipschitz restrictions.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we consider a second-order scalar auxiliary variable (SAV) Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all previous works for fractional undamped or damped wave equations. Finally, some numerical experiments are presented to confirm our theoretical analysis.