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Fast Weak-Kam Integrators for separable Hamiltonian systems

Anne Bouillard 1, 2 Erwan Faou 3, 4 Maxime Zavidovique 5
2 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
3 IPSO - Invariant Preserving SOlvers
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
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 {\em geometric integrator} satisfying a discrete weak-KAM theorem which allows 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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Anne Bouillard, Erwan Faou, Maxime Zavidovique. Fast Weak-Kam Integrators for separable Hamiltonian systems. Mathematics of Computation, American Mathematical Society, 2016, 85 (297), pp.85-117. ⟨hal-00743462⟩

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