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Fast weak-KAM integrators for separable Hamiltonian systems

Authors: Anne Bouillard, Erwan Faou and Maxime Zavidovique
Journal: Math. Comp. 85 (2016), 85-117
MSC (2010): Primary 35F21, 65M12, 06F05
Published electronically: May 26, 2015
MathSciNet review: 3404444
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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.

Erwan Faou
Affiliation: INRIA & Ecole Normale Supérieure de Cachan Bretagne, Avenue Robert Schumann 35170 Bruz, France

Maxime Zavidovique
Affiliation: IMJ-PRG, Université Pierre et Marie Curie, Case 247 4, place Jussieu, 75252 Paris Cedex 05, France

Keywords: Hamilton--Jacobi equations, weak--KAM theorem, Geometric integration, (min, plus) convolution
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
Article copyright: © Copyright 2015 American Mathematical Society