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This note presents a numerical method based on optimal transport to construct minimal geodesics along the group of volume preserving maps, equipped with the L metric. As observed by Arnold, such geodesics solve the Euler equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete optimal transport and it is robust enough to extract non-classical, multi-valued solutions of Euler’s equations predicted by Brenier and Schnirelman [Mérigot and Mirebeau, SIAM J. Num. Anal., 54(6), 2016]. In a second part, we explain how this approach also leads to a numerical scheme able to approximate regular solutions to the Cauchy problem for Euler’s equations [Gallouët and Mérigot, J. Found Comput Math, 2017].

Discretization of Euler’s equations using optimal transport: Cauchy and boundary value problems