An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes
Growth-collapse Markov processes describe the time evolution of a quantity with successive phases of deterministic growth and random instantaneous jumps. The paper is devoted to the study of convergence of an implicit finite volume scheme towards the solution of the Chapman-Kolmogorov equations associated to a growth-collapse process. This equation possesses a transport term and an integral term. A new weak bounded variation inequality is developed in order to prove the convergence of the discretised transport term. Such an inequality may be used in other contexts, such as the study of finite volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in \(L^1\). The method is easily extended to the more general setting of piecewise-deterministic Markov processes. Numerical examples are provided, showing the efficiency of the implicit scheme and its ability to be useful in an industrial reliability context.
linear hyperbolic problems with measure solutions
piecewise-deterministic Markov processes
scientific article; zbMATH DE number 5368370