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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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Fast weak–KAM integrators for separable Hamiltonian systems
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by Anne Bouillard, Erwan Faou and Maxime Zavidovique PDF
Math. Comp. 85 (2016), 85-117 Request permission

Abstract:

We consider a numerical scheme for Hamilton–Jacobi equations based on a direct discretization of the Lax–Oleinik semi–group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak–KAM theorem which allows us to control its long time behavior. Taking advantage of a fast algorithm for computing (min,plus) convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.
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Additional Information
  • Anne Bouillard
  • Affiliation: Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France.
  • Email: anne.bouillard@ens.fr
  • Erwan Faou
  • Affiliation: INRIA & Ecole Normale Supérieure de Cachan Bretagne, Avenue Robert Schumann 35170 Bruz, France
  • MR Author ID: 656335
  • Email: Erwan.Faou@inria.fr
  • Maxime Zavidovique
  • Affiliation: IMJ-PRG, Université Pierre et Marie Curie, Case 247 4, place Jussieu, 75252 Paris Cedex 05, France
  • Email: Maxime.Zavidovique@upmc.fr
  • Received by editor(s): October 15, 2012
  • Received by editor(s) in revised form: December 5, 2013, and May 13, 2014
  • Published electronically: May 26, 2015
  • Additional Notes: The second author was supported by the ERC starting grant GEOPARDI
    The third author was supported by ANR-12-BLAN-WKBHJ
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 85-117
  • MSC (2010): Primary 35F21, 65M12, 06F05
  • DOI: https://doi.org/10.1090/mcom/2976
  • MathSciNet review: 3404444