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Foundations of Computational Mathematics

, Volume 18, Issue 4, pp 835–865 | Cite as

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

  • Thomas O. Gallouët
  • Quentin Mérigot
Article

Abstract

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.

Keywords

Incompressible Euler equations Optimal transport Lagrangian numerical scheme Hamiltonian 

Mathematics Subject Classification

35Q31 65M12 65M50 65Z05 

Notes

Acknowledgements

We would like to thank Yann Brenier who pointed out to us the Reference [8] on which this article elaborates, and for several interesting discussions at various stages of this work. We also thank Pierre Bousquet who indicated the Reference [16] to us.

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© SFoCM 2017

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité de LiègeLiègeBelgique
  2. 2.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

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