Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
This provides a novel computational tool for fluid dynamics, enabling the study of non-classical solutions that are theoretically predicted but previously inaccessible numerically.
The paper introduces a numerical method to compute minimal geodesics in the group of volume-preserving maps under the L2 metric, solving Euler's equations for inviscid incompressible fluids. The method extracts multi-valued solutions in 2D for the first time, with convergence results supporting the generalized model.
We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L2 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 flow dimension is higher than the domain dimension, 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.