Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation

In this article, we prove the convergence of a semi-discrete scheme applied to the stochastic Korteweg--de Vries equation driven by an additive and localized noise. It is the Crank--Nicholson scheme for the deterministic part and is implicit. This scheme was used in previous numerical experiments on the influence of a noise on soliton propagation [8, 9]. Its main advantage is that it is conservative in the sense that in the absence of noise, the $L^2$ norm is conserved. The proof of convergence uses a compactness argument in the framework of $L^2$ weighted spaces and relies mainly on the path-wise uniqueness in such spaces for the continuous equation. The main difficulty relies in obtaining a priori estimates on the discrete solution. Indeed, contrary to the continuous case, Ito formula is not available for the discrete equation.