Analysis of a splitting scheme for a class of random nonlinear partial differential equations∗
1 Institut Mathématique de Toulouse, 118 route de Narbonne, 31062 Toulouse, cedex, France.
2 Universitéde Lorraine, CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, 54506, France.
Received: 4 May 2015
Revised: 11 March 2016
Accepted: 16 September 2016
In this paper, we consider a Lie splitting scheme for a nonlinear partial differential equation driven by a random time-dependent dispersion coefficient. Our main result is a uniform estimate of the error of the scheme when the time step goes to 0. Moreover, we prove that the scheme satisfies an asymptotic-preserving property. As an application, we study the order of convergence of the scheme when the dispersion coefficient approximates a (multi)fractional process.
Mathematics Subject Classification: 35Q55 / 65M15 / 60H15 / 60F17
Key words: Nonlinear partial differential equations / splitting / stochastic partial differential equations / asymptotic-Preserving schemes / fractional and multifractional processes
This work was partially supported by the French ANR grants MICROWAVE NT09460489 (http://microwave.math.cnrs.fr/) and BECASIM ANR-12-MONU-0007-02 (http://becasim.math.cnrs.fr/).
© EDP Sciences, SMAI, 2016