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

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 cannot be directly applied. We use an energy approach to prove an existence and uniqueness result as well as to obtain moment bounds on the stochastic partial differential equation (SPDE) 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 also obtain the convergence in probability. The numerical experiments illustrate the effectiveness of our method.