Convergence of a misanthrope process to the entropy solution of 1D problems

We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of...

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Bibliographic Details
Published in:Stochastic processes and their applications 2012-11, Vol.122 (11), p.3648-3679
Main Authors: Eymard, R., Roussignol, M., Tordeux, A.
Format: Article
Language:English
Subjects:
Entropy Young measure solution
Mathematics
Misanthrope stochastic process
Non linear scalar hyperbolic equation
Numerical Analysis
Traffic flow simulation
Weak BV inequality
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Summary:We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2012.07.002