A variational time discretization for the compressible Euler equations

TL;DR

This paper presents a novel variational time discretization method for the compressible Euler equations, ensuring convergence to measure-valued solutions and capturing sticky particle behavior in one dimension.

Contribution

It introduces a new variational scheme based on minimizing acceleration over monotone transport maps, with proven convergence for compressible gas dynamics.

Findings

Convergence to measure-valued solutions for Euler equations.

Successful modeling of sticky particle solutions in one dimension.

Provides a new numerical approach for multi-dimensional gas dynamics.

Abstract

We introduce a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. For one space dimension, we obtain sticky particle solutions for the pressureless case.