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 norm is conserved. The proof of convergence uses a compactness argument in the framework of L 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.