A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations
Gallouët, Thomas; Mérigot, Quentin (2018), A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations, Foundations of Computational Mathematics, 18, 4, p. 835–865. 10.1007/s10208-017-9355-y
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01425826
Journal nameFoundations of Computational Mathematics
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Abstract (EN)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.
Subjects / KeywordsHamiltonian; Lagrangian numerical scheme; Optimal transport; Incompressible Euler equations
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