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More on stochastic and variational approach to the Lax-Friedrichs scheme

Author: Kohei Soga
Journal: Math. Comp. 85 (2016), 2161-2193
MSC (2010): Primary 65M06, 35L65, 49L25, 60G50, 37J50
Published electronically: February 10, 2016
MathSciNet review: 3511278
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Abstract: A stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws and Hamilton-Jacobi equations generated by space-time dependent flux functions of the Tonelli type was clarified by Soga (2015). The results for the Lax-Friedrichs scheme are extended here to show its time-global stability, the large-time behavior, and error estimates. Also provided is a weak KAM-like theorem for discrete equations that is useful in the numerical analysis and simulation of the weak KAM theory. As one application, a finite difference approximation to effective Hamiltonians and KAM tori is rigorously treated. The proofs essentially rely on the calculus of variations in the Lax-Friedrichs scheme and on the theory of viscosity solutions of Hamilton-Jacobi equations.

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Kohei Soga
Affiliation: Unité de mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France
Address at time of publication: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
MR Author ID: 909684

Keywords: Lax-Friedrichs scheme, scalar conservation law, Hamilton-Jacobi equation, calculus of variations, random walk, weak KAM theory
Received by editor(s): September 17, 2013
Received by editor(s) in revised form: April 21, 2014, and October 7, 2014
Published electronically: February 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society