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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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More on stochastic and variational approach to the Lax-Friedrichs scheme
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by Kohei Soga PDF
Math. Comp. 85 (2016), 2161-2193 Request permission

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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Additional 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