Semi-implicit energy-preserving numerical schemes for stochastic wave equation via SAV approach

TL;DR

This paper introduces semi-implicit stochastic SAV schemes for nonlinear stochastic wave equations, ensuring energy preservation and convergence, advancing numerical methods for complex stochastic systems.

Contribution

It presents the first construction and analysis of energy-preserving semi-implicit schemes for nonlinear stochastic wave equations using the SAV approach.

Findings

Schemes preserve the modified energy evolution law.

Proved strong convergence rates of the schemes.

Established regularity estimates for the stochastic SAV.

Abstract

In this paper, we propose and analyze semi-implicit numerical schemes for the stochastic wave equation (SWE) with general nonlinearity and multiplicative noise. These numerical schemes, called stochastic scalar auxiliary variable (SAV) schemes, are constructed by transforming the considered SWE into a higher dimensional stochastic system with a stochastic SAV. We prove that they can be solved explicitly and preserve the modified energy evolution law and the regularity structure of the original system. These structure-preserving properties are the keys to overcoming the mutual effect of the noise and nonlinearity. By proving new regularity estimates of the introduced SAV, we establish the strong convergence rate of stochastic SAV schemes and the further fully-discrete schemes with the finite element method in spatial direction. To the best of our knowledge, this is the first result on the construction and strong convergence of semi-implicit energy-preserving schemes for nonlinear SWE.