Strong Convergence of a Splitting Method for the Stochastic Complex Ginzburg-Landau Equation

TL;DR

This paper introduces a spectral splitting numerical method for the stochastic complex Ginzburg-Landau equation, proving strong convergence and demonstrating effectiveness through numerical experiments.

Contribution

It is the first to analyze the numerical approximation of the stochastic complex Ginzburg-Landau equation using a spectral splitting approach with proven strong convergence.

Findings

Proven strong convergence of the numerical method.

Established moment bounds for the discretization.

Numerical experiments confirm the method's effectiveness.

Abstract

We consider the numerical approximation of the stochastic complex Ginzburg-Landau equation with additive noise on the one dimensional torus. The complex nature of the equation means that many of the standard approaches developed for stochastic partial differential equations can not be directly applied. We use an energy approach to prove an existence and uniqueness result as well to obtain moment bounds on the stochastic PDE before introducing our numerical discretization. For such a well studied deterministic equation it is perhaps surprising that its numerical approximation in the stochastic setting has not been considered before. Our method is based on a spectral discretization in space and a Lie-Trotter splitting method in time. We obtain moment bounds for the numerical method before proving our main result: strong convergence on a set of arbitrarily large probability. From this we obtain a result on convergence in probability. We conclude with some numerical experiments that illustrate the effectiveness of our method.