Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations

TL;DR

This paper introduces new variational particle schemes for the porous medium and isentropic Euler equations, leveraging optimal transport theory to achieve stable, high-order numerical methods that effectively capture nonlinear flow features.

Contribution

It develops novel discretization schemes based on variational principles in optimal transport, enabling stable and accurate simulations of complex nonlinear PDEs.

Findings

Successfully captures shocks and rarefaction waves in Euler equations

Demonstrates stability and accuracy of the proposed schemes

Extends to higher order methods using BDF and Runge-Kutta techniques

Abstract

Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for the isentropic Euler equations. We also show how to design higher order methods for these problems in the optimal transport setting using backward differentiation formula (BDF) multi-step methods or diagonally implicit Runge-Kutta methods.