More on stochastic and variational approach to the Lax-Friedrichs scheme
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- by Kohei Soga;
- Math. Comp. 85 (2016), 2161-2193
- DOI: https://doi.org/10.1090/mcom/3061
- Published electronically: February 10, 2016
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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.References
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Bibliographic Information
- 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
- Email: soga@math.keio.ac.jp
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 2161-2193
- MSC (2010): Primary 65M06, 35L65, 49L25, 60G50, 37J50
- DOI: https://doi.org/10.1090/mcom/3061
- MathSciNet review: 3511278