Numerical resolution of Euler equations through semi-discrete optimal transport
Journées équations aux dérivées partielles (2015), article no. 7, 16 p.

Geodesics along the group of volume preserving diffeomorphisms are solutions to Euler equations of inviscid incompressible fluids, as observed by Arnold [4]. 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 [14].

We present, in the first section, the framework of semi-discrete optimal transport, initially developed for the study of generalized solutions to optimal transport [1] 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 [16, 7, 33]: 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 [15] for which the fluid particles motion is non-deterministic, as predicted by [39].

DOI: 10.5802/jedp.636
Mirebeau, Jean-Marie 1

1 CNRS et Université Paris-Sud Université Paris-Saclay Départment de Mathématiques Bâtiment 425 91405 Orsay Cedex France
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Mirebeau, Jean-Marie. Numerical resolution of Euler equations  through semi-discrete optimal transport. Journées équations aux dérivées partielles (2015), article  no. 7, 16 p. doi : 10.5802/jedp.636. http://www.numdam.org/articles/10.5802/jedp.636/

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