Symplectic Runge-Kutta Semi-discretization for Stochastic Schrödinger Equation

2014/10/23 by Chen, Chuchu, Hong, Jialin
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1410.6228

Abstract

Based on a variational principle with a stochastic forcing, we indicate that the stochastic Schrödinger equation in 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 temporal direction to the stochastic Schrödinger equation in 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 stochastic midpoint scheme as an example of stochastic symplectic Runge-Kutta methods in temporal direction, based on the theorem we show that the mean-square convergence order of the semi-discrete scheme is 1 under appropriate assumptions.

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