Numerical resolution of Euler equations through semi-discrete optimal transport

Abstract

Geodesics along the group of volume preserving diffeomorphisms are solutions to Euler equations of inviscid incompressible fluids, as observed by Arnold. On the other hand, the projection onto volume preserving maps amounts to an optimal transport problem, as follows from the generalized polar decomposition of Brenier. We present, in the first section, the framework of semi-discrete optimal transport, initially developed for the study of generalized solutions to optimal transport and now regarded as an efficient approach to computational optimal transport. In a second and largely independent section, we present numerical approaches for Euler equations seen as a boundary value problem: knowing the initial and final positions of some fluid particles, reconstruct intermediate fluid states. Depending on the data, we either recover a classical solution to Euler equations, or a generalized flow for which the fluid particles motion is non-deterministic. See Minimal Geodesics along Volume Preserving Maps, through Semi-Discrete Optimal Transport, Merigot and Mirebeau 2015, for the proofs of the results presented in these proceedings.