Stochastic finite differences for elliptic diffusion equations in stratified domains

Sylvain Maire (1) , Giang Nguyen (2)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 LSIS - Laboratoire des Sciences de l'Information et des Systèmes

We describe Monte Carlo algorithms to solve elliptic partial differen- tial equations with piecewise constant diffusion coefficients and general boundary conditions including Robin and transmission conditions as well as a damping term. The treatment of the boundary conditions is done via stochastic finite differences techniques which possess an higher order than the usual methods. The simulation of Brownian paths inside the domain relies on variations around the walk on spheres method with or without killing. We check numerically the efficiency of our algorithms on various examples of diffusion equations illustrating each of the new techniques introduced here.

Mots clés: Stratified me- dia; Monte Carlo method; Elliptic diffusion equations; Walk on spheres; Stochastic finite differences; Walk on spheres.

Domaines: Equations aux dérivées partielles [math.AP]

Sylvain Maire, Giang Nguyen. Stochastic finite differences for elliptic diffusion equations in stratified domains. Mathematics and Computers in Simulation, 2016, 121, ⟨10.1016/j.matcom.2015.09.008⟩. ⟨hal-00809203v2⟩