Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps

Antoine Lejay 1, 2 Géraldine Pichot 3
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
3 SAGE - Simulations and Algorithms on Grids for Environment
Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
Abstract : In this article, we propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian Motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.
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Antoine Lejay, Géraldine Pichot. Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps. Journal of Computational Physics, Elsevier, 2012, 231 (21), pp.7299-7314. ⟨10.1016/j.jcp.2012.07.011⟩. ⟨hal-00649170v3⟩

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