Minimal geodesics along volume preserving maps, through semi-discrete optimal transport

International audienceWe introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L 2 metric, which as observed by Arnold [Arn66] solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier [Bre91], numerically implemented through semi-discrete optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Eu-ler's equations, for which the flow dimension is higher than the domain dimension, a striking and unavoidable consequence of this model [Shn94]. Our convergence results encompass this generalized model, and our numerical experiments illustrate it for the first time in two space dimensions