An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes
type: Item
instance of: scholarly article
title: An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes
zbMATH Open document ID: 1158.65008
author: Robert Eymard
author: Alain Prignet
author: Sophie Mercier
published in: Journal of Computational and Applied Mathematics
publication date: 2008-11-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60K20
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60H35
zbMATH DE Number: 5368370
zbMATH Keywords: linear hyperbolic problems with measure solutions
zbMATH Keywords: weak bounded variation inequalities
zbMATH Keywords: Chapman-Kolmogorov equations
zbMATH Keywords: piecewise-deterministic Markov processes
zbMATH Keywords: growth-collapse Markov processes
zbMATH Keywords: convergence
zbMATH Keywords: finite volume scheme
zbMATH Keywords: numerical examples
reviewed by: Dominique Lépingle
MaRDI profile type: Publication
full work available at URL: j.cam.2007.10.053
OpenAlex ID: W2013392595
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DOI: 10.1016/J.CAM.2007.10.053
description: scientific article; zbMATH DE number 5368370