Symplectic Runge--Kutta Semidiscretization for Stochastic Schrödinger Equation

Based on a variational principle with a stochastic forcing, we indicate that the stochastic Schrödinger equation in the Stratonovich sense is an infinite-dimensional stochastic Hamiltonian system, whose phase flow preserves symplecticity. We propose a general class of stochastic symplectic Runge--Kutta methods in the temporal direction to the stochastic Schrödinger equation in the Stratonovich sense and show that the methods preserve the charge conservation law. We present a convergence theorem on the relationship between the mean-square convergence order of a semi-discrete method and its local accuracy order. Taking the stochastic midpoint scheme as an example of stochastic symplectic Runge--Kutta methods in the temporal direction, based on the theorem we show that the mean-square convergence order of the semidiscrete scheme is 1 under appropriate assumptions.