Discretization of non linear Langevin SDEs

Mireille Bossy 1
1 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We consider a family of Langevin equations arising in the Lagrangian approach for fluid mechanics models. Here we are interested by the numerical discretization of a Langevin system of SDEs (position and velocity) which are non-linear in the McKean sense, moreover the position must be confined in a bounded domain and must take into account a given velocity at the boundary. Such numerical simulations are motivated by some meteorological applications: we use such Lagrangian model inside a computational cell of "classical meteorological solver" in order to refine locally the wind computation. We will present our numerical scheme for the Langevin system and we will detail two of the main numerical difficulties. First, SDEs have non-Lipschitz diffusion coefficients, typically of the form xa, 1/2≤a≤ 1. Considering weak convergence, we give a rate of convergence result for a symmetrized scheme in a generic but 1D situation, without confinement. Second, we propose a scheme for the confined process, but like for reflected processes, the rate of weak convergence depends on the a priori regularity of solutions of the associated Kolmogorov PDE.
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Submitted on : Monday, July 4, 2011 - 10:48:02 AM
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Mireille Bossy. Discretization of non linear Langevin SDEs. Minisymposium at the 5-th European Congress of Mathematics, Jul 2008, Amsterdam, Netherlands. 〈hal-00605716〉

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