Stochastic multi-symplectic Runge-Kutta methods for stochastic Hamiltonian PDEs

TL;DR

This paper develops stochastic Runge-Kutta methods that preserve multi-symplectic and energy conservation laws for stochastic Hamiltonian PDEs, demonstrated on stochastic Maxwell equations with multiplicative noise.

Contribution

It introduces sufficient conditions for multisymplecticity of stochastic Runge-Kutta methods applied to stochastic Hamiltonian PDEs, ensuring conservation laws are preserved.

Findings

Methods preserve discrete stochastic multi-symplectic conservation law almost surely.

Methods preserve discrete energy conservation law almost surely.

Application to stochastic Maxwell equations demonstrates effectiveness.

Abstract

In this paper, we consider stochastic Runge-Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for multisymplecticity of stochastic Runge-Kutta methods of stochastic Hamiltonian partial differential equations. Particularly, we apply these ideas to stochastic Maxwell equations with multiplicative noise, possessing the stochastic multi-symplectic conservation law and energy conservation law. Theoretical analysis shows that the methods can preserve both the discrete stochastic multi-symplectic conservation law and discrete energy conservation law almost surely.