A Lagrangian scheme for the incompressible Euler equation using optimal transport
Provides a convergent numerical scheme for the incompressible Euler equation based on optimal transport, but the results are limited to 2D and simple test cases.
The paper develops a numerical scheme for the incompressible Euler equation by combining Arnold's geodesic interpretation with Brenier's extrinsic approximation, using semi-discrete optimal transport solvers. It proves convergence to regular solutions and demonstrates the method on 2D test cases.
We approximate the regular solutions of the incompressible Euler equation by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold's interpretation of the solution of Euler's equation for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields numerical scheme able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of these scheme towards regular solutions of the incompressible Euler equation, and to provide numerical experiments on a few simple testcases in 2D.