Euler-Maruyama approximations of the stochastic heat equation on the sphere

TL;DR

This paper develops spectral and Euler-Maruyama numerical schemes for approximating the stochastic heat equation on the sphere, establishing optimal convergence rates and validating them through simulations.

Contribution

It introduces a spectral method combined with Euler-Maruyama schemes for the stochastic heat equation on the sphere, providing rigorous convergence analysis and numerical validation.

Findings

Optimal strong convergence rates derived for the schemes

Convergence of expectation and second moment shown

Numerical simulations confirm theoretical results

Abstract

The stochastic heat equation on the sphere driven by additive isotropic Wiener noise is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. Optimal strong convergence rates for a given regularity of the initial condition and driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown, where the approximation of the second moment converges with twice the strong rate. Numerical simulations confirm the theoretical results.