An overlapping domain decomposition splitting algorithm for stochastic nonlinear Schroedinger equation

TL;DR

This paper introduces a new overlapping domain decomposition splitting algorithm based on Crank-Nisolson for the stochastic nonlinear Schrödinger equation, offering improved computational efficiency while preserving key conservation laws.

Contribution

The paper develops a novel overlapping domain decomposition splitting algorithm tailored for stochastic nonlinear Schrödinger equations, demonstrating its efficiency and effectiveness compared to existing methods.

Findings

Significantly reduces computational cost.

Maintains conservation laws similar to original equations.

Outperforms existing stochastic numerical schemes.

Abstract

A novel overlapping domain decomposition splitting algorithm based on a Crank-Nisolson method is developed for the stochastic nonlinear Schroedinger equation driven by a multiplicative noise with non-periodic boundary conditions. The proposed algorithm can significantly reduce the computational cost while maintaining the similar conservation laws. Numerical experiments are dedicated to illustrating the capability of the algorithm for different spatial dimensions, as well as the various initial conditions. In particular, we compare the performance of the overlapping domain decomposition splitting algorithm with the stochastic multi-symplectic method in [S. Jiang, L. Wang and J. Hong, Commun. Comput. Phys., 2013] and the finite difference splitting scheme in [J. Cui, J. Hong, Z. Liu and W. Zhou, J. Differ. Equ., 2019]. We observe that our proposed algorithm has excellent computational efficiency and is highly competitive. It provides a useful tool for solving stochastic partial differential equations.