Fully discrete approximation of the semilinear stochastic wave equation on the sphere

TL;DR

This paper introduces a fully discrete numerical scheme for the semilinear stochastic wave equation on the sphere, demonstrating strong and almost sure convergence with confirmed rates through numerical experiments.

Contribution

It develops a novel combination of stochastic trigonometric integrator and spectral Galerkin method for this specific stochastic PDE on the sphere, with proven convergence properties.

Findings

Strong convergence of the scheme is established.

Almost sure convergence is demonstrated.

Numerical experiments confirm theoretical convergence rates.

Abstract

The semilinear stochastic wave equation on the sphere driven by multiplicative Gaussian noise is discretized by a stochastic trigonometric integrator in time and a spectral Galerkin approximation in space based on the spherical harmonic functions. Strong and almost sure convergence of the explicit fully discrete numerical scheme are shown. Furthermore, these rates are confirmed by numerical experiments.