ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

On the discretization in time of parabolic stochastic partial differential equations

Printems, Jacques

Centre de Mathématiques de l'Université de Paris 12, EA 2343, Université de Paris 12, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. (printems@univ-paris12.fr)

Abstract

We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.

(Received July 24 2000)

(Revised January 8 2001)

(Revised September 17 2001)

(Online publication April 15 2002)

Key Words:

  • Stochastic partial differential equations;
  • semi-discretized scheme for stochastic partial differential equations;
  • Euler scheme.

Mathematics Subject Classification:

  • 60H15;
  • 60F25;
  • 60F99;
  • 65C20;
  • 60H35
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