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

Charles-Edouard Bréhier (1, 2, 3) , David Cohen (4)

1 PSPM - Probabilités, statistique, physique mathématique
2 ICJ - Institut Camille Jordan
3 UPPA - Université de Pau et des Pays de l'Adour
4 Chalmers - Department of Mathematical Sciences

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.

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]

Charles-Edouard Bréhier, David Cohen. Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations. Applied Numerical Mathematics, 2023, 186, pp.57-83. ⟨10.1016/j.apnum.2023.01.002⟩. ⟨hal-02893328⟩