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 $\mathcal{O}(N^{-2}+{\tau }^{\frac{1}{2}})$, 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.

39 pages

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• Well-posedness and Optimal Regularity of Stochastic Evolution Equations with Multiplicative Noises
• Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE
• $L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs
• Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation
• Wellposedness and regularity estimate for stochastic Cahn--Hilliard equation with unbounded noise diffusion
• Energy-preserving exponential integrable numerical method for stochastic cubic wave equation with additive noise
• Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation
• On Global Existence and Blow-up for Damped Stochastic Nonlinear Schrödinger Equation
• Analysis of A Splitting Scheme for Damped Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise
• Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-sided Lipschitz coefficient
• Stochastic logarithmic Schrodinger equations: energy regularized approach
• Absolute continuity and numerical approximation of stochastic Cahn--Hilliard equation with unbounded noise diffusion
• Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
• A convergent stochastic scalar auxiliary variable method
• Structure-preserving splitting methods for stochastic logarithmic Schrödinger equation via regularized energy approximation
• A new efficient operator splitting method for stochastic Maxwell equations
• High-accuracy time discretization of stochastic fractional diffusion equation
• Numerical integrators for continuous disordered nonlinear Schrödinger equation