July  2006, 6(4): 761-781. doi: 10.3934/dcdsb.2006.6.761

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

1. 

IRMAR and ENS de Cachan, antenne de Bretagne, Campus de Ker Lann, 35170 BRUZ Cedex, France

2. 

Université Paris XII - Centre de Mathématiques - CNRS UMR 8050, 61, avenue du Général de Gaulle, 94010 CRETEIL Cedex, France

Received  March 2005 Revised  October 2005 Published  April 2006

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.
Citation: Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
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