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.

翻译：本文研究一维环面上带加性噪声的随机复Ginzburg-Landau方程的数值逼近问题。该方程的复值特性意味着许多针对随机偏微分方程发展的标准方法无法直接应用。在引入数值离散格式前，我们采用能量方法证明了随机偏微分方程解的存在唯一性，并获得了其矩估计。对于这样一个确定性情形下已被深入研究的方程，其随机情形的数值逼近问题此前未被探讨或许令人意外。本文方法基于空间上的谱离散与时间上的Lie-Trotter分裂法。在证明主要结果——任意大概率集合上的强收敛性之前，我们首先获得了数值格式的矩估计。由此进一步推导出概率收敛的结果。最后通过数值实验验证了所提方法的有效性。