Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise

TL;DR

This paper develops energy-preserving numerical schemes for nonlinear stochastic wave equations with multiplicative noise, ensuring the discrete averaged energy law is maintained, validated through theoretical proofs and numerical experiments.

Contribution

It introduces novel fully-discrete schemes combining finite difference and finite element methods that preserve the energy evolution law in stochastic wave equations.

Findings

Schemes preserve discrete averaged energy law.

Exact energy law inheritance for additive noise cases.

Numerical results confirm theoretical energy preservation.

Abstract

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Pad\'e approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the averaged energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings.