Energy-preserving exponential integrable numerical method for stochastic cubic wave equation with additive noise

TL;DR

This paper introduces an energy-preserving exponential integrator for stochastic cubic wave equations with additive noise, combining spectral Galerkin, splitting, and averaged vector field methods to ensure energy and integrability preservation with proven convergence.

Contribution

It develops a novel structure-preserving numerical scheme that maintains energy and exponential integrability for stochastic wave equations, with proven strong convergence.

Findings

Numerical experiments confirm theoretical convergence rates.

The method preserves energy evolution law and exponential integrability.

Strong convergence rate is established and validated numerically.

Abstract

In this paper, we present an energy-preserving exponentially integrable numerical method for stochastic wave equation with cubic nonlinearity and additive noise. We first apply the spectral Galerkin method to discretizing the original equation and show that this spatial discretization possesses an energy evolution law and certain exponential integrability property. Then the exponential integrability property of the exact solution is deduced by proving the strong convergence of the semi-discretization. To propose a full discrete numerical method which could inherit both the energy evolution law and the exponential integrability, we use the splitting technique and averaged vector field method in the temporal direction. Combining these structure-preserving properties with regularity estimates of the exact and the numerical solutions, we obtain the strong convergence rate of the numerical method. Numerical experiments coincide with these theoretical results.