A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

Thomas O Gallouët (1) , Quentin Mérigot (2, 3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Lagrangian numerical scheme; Hamiltonian; Optimal transport; Incompressible Euler equations

Domaines: Analyse numérique [math.NA]

Thomas O Gallouët, Quentin Mérigot. A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations. Foundations of Computational Mathematics, 2018, 18, pp.835-865. ⟨10.1007/s10208-017-9355-y⟩. ⟨hal-01425826v2⟩