Structure-preserving splitting methods for stochastic logarithmic Schr\"odinger equation via regularized energy approximation

TL;DR

This paper develops and analyzes structure-preserving splitting methods for the stochastic logarithmic Schrödinger equation using a regularized energy approach, providing the first such numerical analysis for this class of equations.

Contribution

It introduces a regularized energy approximation and constructs the first numerical methods with proven convergence for stochastic Schrödinger equations with logarithmic nonlinearities.

Findings

Established a priori estimates, entropy, and energy conservation.

Proved strong convergence and energy convergence rates.

Demonstrated effectiveness of methods through numerical analysis.

Abstract

In this paper, we study two kinds of structure-preserving splitting methods, including the Lie--Trotter type splitting method and the finite difference type method, for the stochasticlogarithmic Schr\"odinger equation (SlogS equation) via a regularized energy approximation. We first introduce a regularized SlogS equation with a small parameter $0<\epsilon\ll1$ which approximates the SlogS equation and avoids the singularity near zero density. Then we present a priori estimates, the regularized entropy and energy, and the stochastic symplectic structure of the proposed numerical methods. Furthermore, we derive both the strong convergence rates and the convergence rates of the regularized entropy and energy. To the best of our knowledge, this is the first result concerning the construction and analysis of numerical methods for stochastic Schr\"odinger equations with logarithmic nonlinearities.