Title: Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrödinger Equations,
Abstract: We prove the optimal strong convergence rate of a fully discrete scheme,
based on a splitting approach, for a stochastic nonlinear Schrödinger (NLS)
equation. The main novelty of our method lies on the uniform a priori estimate
and exponential integrability of a sequence of splitting processes which are
used to approximate the solution of the stochastic NLS equation. We show that
the splitting processes converge to the solution with strong order $1/2$. Then
we use the Crank--Nicolson scheme to temporally discretize the splitting
process and get the temporal splitting scheme which also possesses strong order
$1/2$. To obtain a full discretization, we apply this splitting Crank--Nicolson
scheme to the spatially discrete equation which is achieved through the
spectral Galerkin approximation. Furthermore, we establish the convergence of
this fully discrete scheme with optimal strong convergence rate
$\mathcal{O}(N^{-2}+\tau^\frac12)$, where $N$ denotes the dimension of the
approximate space and $\tau$ denotes the time step size. To the best of our
knowledge, this is the first result about strong convergence rates of
temporally numerical approximations and fully discrete schemes for stochastic
NLS equations, or even for stochastic partial differential equations (SPDEs)
with non-monotone coefficients. Numerical experiments verify our theoretical
result.