Long-time behavior of exact and numerical solutions of stochastic evolution equations on the sphere

TL;DR

This paper studies the long-time behavior of solutions and numerical methods for stochastic evolution equations on the sphere, highlighting the effectiveness of stochastic exponential integrators.

Contribution

It demonstrates that certain numerical schemes fail to capture long-term behavior, while stochastic exponential integrators succeed, supported by theoretical analysis and numerical experiments.

Findings

Euler-Maruyama schemes fail to reproduce correct long-time behavior.

Stochastic exponential integrator preserves physical quantities over time.

Numerical experiments confirm theoretical predictions.

Abstract

We investigate the long-time behavior of exact solutions and numerical approximations of linear stochastic evolution equations defined on the sphere. We focus on three classical models arising in mathematical physics: the stochastic wave equation, the stochastic Schr\"odinger equation, and the stochastic Maxwell's equations. For these SPDEs, we analyze several widely used time integrators with respect to trace formulas describing the evolution of physically relevant quantities such as energy, mass, and momentum dependent on the forcing term. In particular, we prove that the forward and backward Euler-Maruyama schemes fail to reproduce the correct long-time behavior of the exact solutions. In addition, we prove that the stochastic exponential integrator preserves the correct long-time behavior of the physical quantities of interest. Finally, several numerical experiments are provided to illustrate our theoretical findings.