Section:
New Results
Generalized Solution of Euler
Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
Q. Mérigot and J.-M. Mirebeau introduced a numerical method for extracting minimal geodesics
along the group of volume preserving maps, equipped with the metric, which as observed
by Arnold solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete
optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Euler's equations, for which the dimension of the support of the flow is higher than the dimension of the domain, a striking and unavoidable consequence of this model. Our convergence results encompass this
generalized model, and our numerical experiments illustrate it for the first time in two space
dimensions (see Figure 14 ).
Figure
14. (First row) Beltrami flow in the unit square at
various timesteps, a classical solution to Euler's equation. The
color of the particles depend on their initial
position. (Second to fifth row) Generalized fluid flows that
are reconstructed by our algorithm, using boundary conditions
displayed in the first and last column. When
we recover the classical flow, while for the
solution is not classical any more and includes some mixing.
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