Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equations

Abstract

In this paper, we show that solutions of stochastic nonlinear Schrödinger (NLS) equations can be approximated by solutions of coupled splitting systems. Based on these systems, we propose a new kind of fully discrete splitting schemes which possess algebraic strong convergence rates for stochastic NLS equations. Key ingredients of our approach are using the exponential integrability and stability of the corresponding splitting systems and numerical approximations. In particular, under very mild conditions, we derive the optimal strong convergence rate O(N−2+τ12) of the spectral splitting Crank–Nicolson scheme, where N and τ denote the dimension of the approximate space and the time step size, respectively.