An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes

Robert Eymard (1) , Sophie Mercier (1) , Alain Prignet (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We are interested here in the numerical approximation of a family of probability measures, solution of the Chapman-Kolmogorov equation associated to some non-diffusion Markov process with Uncountable state space. Such an equation contains a transport term and another term, which implies redistribution Of the probability mass on the whole space. All implicit finite Volume scheme is proposed, which is intermediate between an upstream weighting scheme and a modified Lax-Friedrichs one. Due to the seemingly unusual probability framework, a new weak bounded variation inequality had to be developed, in order to prove the convergence of the discretised transport term. Such an inequality may be used in other contexts, such as for the study of finite Volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in $L^1$. Also, due to the redistribution term, the tightness of the family of approximate probability measures had to be proven. Numerical examples are provided, showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context.

Mots clés: Chapman–Kolmogorov equations; Piecewise- deterministic Markov process; Growth–collapse Markov proces; Weak bounded variation inequalities; Linear hyperbolic problems with measure solutions

Domaines: Physique mathématique [math-ph]

Robert Eymard, Sophie Mercier, Alain Prignet. An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes. Journal of Computational and Applied Mathematics, 2008, 222 (2), pp.293-323. ⟨10.1016/j.cam.2007.10.053⟩. ⟨hal-00693134⟩