Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

TL;DR

This paper introduces three innovative multi-symplectic numerical methods for stochastic Hamiltonian PDEs, enhancing stability and accuracy by leveraging geometric properties, and demonstrates their effectiveness through numerical experiments on various stochastic equations.

Contribution

The paper develops three new multi-symplectic methods based on RBF collocation, splitting, and partitioned Runge-Kutta techniques for stochastic Hamiltonian PDEs, with practical implementations.

Findings

Methods effectively preserve multi-symplectic structure.

Numerical experiments confirm stability and accuracy.

Applicable to various stochastic PDEs like wave and Schrödinger equations.

Abstract

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique, and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.