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

Abstract : We 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.
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Submitted on : Thursday, December 15, 2016 - 2:30:09 PM
Last modification on : Friday, January 21, 2022 - 3:09:44 AM

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Quentin Mérigot, Jean-Marie Mirebeau. Minimal geodesics along volume preserving maps, through semi-discrete optimal transport. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2016, 54 (6), pp.3465-3492. ⟨10.1137/15M1017235⟩. ⟨hal-01152168v2⟩

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