A trigonometric method for the linear stochastic wave equation

TL;DR

This paper introduces a fully discrete numerical scheme combining finite element spatial discretization with a stochastic trigonometric time integrator for the linear stochastic wave equation, offering error bounds independent of spatial discretization and preserving a trace formula.

Contribution

The paper proposes a novel explicit stochastic trigonometric scheme that improves error bounds and preserves key properties for the linear stochastic wave equation.

Findings

Error bounds are independent of spatial discretization.

The scheme preserves a trace formula similar to the exact solution.

Numerical experiments confirm the theoretical advantages.

Abstract

A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretisation and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretisation and thus do not have a step size restriction as in the often used St\"ormer-Verlet-leap-frog scheme. Moreover it enjoys a trace formula as does the exact solution of our problem. These favourable properties are demonstrated with numerical experiments.