Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

Abstract

We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schrödinger equations driven by additive Itô noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

Explicit Numerical Methods for High Dimensional Stochastic Nonlinear Schrödinger Equation: Divergence, Regularity and Convergence

“…To deal with the strong nonlinearity in SNLSEs, many authors use the stopping time techniques and truncated SNLSEs to consider the convergence rates of numerical methods in probability or in pathwise sense (see, e.g., [19,28,11]) which is weaker than the strong one. Some progress has been achieved by studying exponential integrability of exact and numerical solutions (see, e.g., [13,12,14,5]). For 1D stochastic cubic Schrödinger equation, the authors in [13,12,14] derive the optimal strong and weak convergence rates of a kind of temporal splitting Crank-Nicolson schemes and their full discretizations.…”

Structure-preserving splitting methods for stochastic logarithmic Schrödinger equation via regularized energy approximation

“…Despite various and fruitful numerical results of stochastic Schrödinger equations with smooth nonlinearities (see e.g. [1,5,8,9,10,11,13,15] and references therein), the numerical analysis of stochastic Schrödinger equations with non-locally Lipschitz nonlinearities, especially for the SlogS equations, is far from being well understood and confronts several challenges. One is that the direct numerical discretization often produces the numerical vacuum which are difficult to deal with when computing the logarithmic nonlinearity.…”