Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrödinger Equations
Provides the first strong convergence rate analysis for numerical approximations of stochastic NLS equations, addressing a gap for SPDEs with non-monotone coefficients.
The authors prove optimal strong convergence rate (1/2 in time, spectral in space) for a splitting-based fully discrete scheme applied to stochastic nonlinear Schrödinger equations, achieving rate O(N^{-2} + τ^{1/2}). This is the first such result for SPDEs with non-monotone coefficients.
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 $\mathcal{O}(N^{-2}+τ^\frac12)$, where $N$ denotes the dimension of the approximate space and $τ$ 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.