Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 761 - 781, Volume 6, Issue 4, July 2006      doi:10.3934/dcdsb.2006.6.761

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Arnaud Debussche - IRMAR and ENS de Cachan, antenne de Bretagne, Campus de Ker Lann, 35170 BRUZ Cedex, France (email)
Jacques Printems - Université Paris XII - Centre de Mathématiques - CNRS UMR 8050, 61, avenue du Général de Gaulle, 94010 CRETEIL Cedex, France (email)

Abstract: 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.

Keywords:  Korteweg-de Vries, stochastic PDE, numerical scheme.
Mathematics Subject Classification:  35Q53, 60H15, 60H35.

Received: March 2005;      Revised: October 2005;      Available Online: April 2006.