Stochastic Global Momentum-Preserving Schemes for Two-Dimensional Stochastic Partial Differential Equations

Abstract

In this paper, the global momentum conservation laws and the global momentum evolution laws are presented for the two-dimensional stochastic nonlinear Schrödinger equation with multiplicative noise and the two-dimensional stochastic Klein-Gordon equation with additive noise, respectively. In order to preserve the global momenta or their changing trends in numerical simulation, the schemes are constructed by using a stochastic multi-symplectic formulation. It is shown that under periodic boundary conditions, the schemes have discrete global momentum conservation laws or the discrete global momentum evolution laws. Numerical experiments confirm global momentum-preserving properties of the schemes and their mean square convergence in the time direction.