Strong convergence of a Verlet integrator for the semi-linear stochastic wave equation

TL;DR

This paper proves optimal strong convergence rates for a fully discrete scheme combining discontinuous Galerkin finite elements in space and a Verlet integrator in time for the semi-linear stochastic wave equation, supported by numerical validation.

Contribution

It introduces a novel analysis of a combined spatial-temporal discretization scheme for stochastic wave equations, establishing stability and convergence under a CFL condition.

Findings

Optimal strong convergence rates proven theoretically.

Numerical experiments confirm theoretical predictions.

Expected energy bounds align with the exact solution.

Abstract

The full discretization of the semi-linear stochastic wave equation is considered. The discontinuous Galerkin finite element method is used in space and analyzed in a semigroup framework, and an explicit stochastic position Verlet scheme is used for the temporal approximation. We study the stability under a CFL condition and prove optimal strong convergence rates of the fully discrete scheme. Numerical experiments illustrate our theoretical results. Further, we analyze and bound the expected energy and numerically show excellent agreement with the energy of the exact solution.