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

Abstract

In this paper, we consider stochastic Runge–Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for stochastic multi-symplecticity of stochastic Runge–Kutta methods. To present more clearly, we apply these ideas to three dimensional stochastic Maxwell equations driven by multiplicative noise, which play an important role in stochastic electromagnetism and statistical radiophysics areas. Theoretical analysis shows that the methods inherit the energy conservation law of the original system, and preserve the discrete stochastic multi-symplectic conservation law almost surely.