A New Regularization of the One-Dimensional Euler and Homentropic Euler Equations

TL;DR

This paper introduces a novel regularization technique for one-dimensional Euler and homentropic Euler equations using low-pass filtering of convective velocities, aiming to improve solution smoothness while preserving key physical properties.

Contribution

The paper proposes a new regularization method based on averaging nonlinear fluxes with filtered velocities, maintaining conservation and wave solutions, and demonstrating convergence to original solutions.

Findings

Regularized equations are smooth and capture shock tube behavior.

Solutions converge to weak solutions of original equations as regularization diminishes.

Numerical simulations validate the effectiveness of the regularization.

Abstract

This paper examines an averaging technique in which the nonlinear flux term is expanded and the convective velocities are passed through a low-pass filter. It is the intent that this modification to the nonlinear flux terms will result in an inviscid regularization of the homentropic Euler equations and Euler equations. The physical motivation for this technique is presented and a general method is derived, which is then applied to the homentropic Euler equations and Euler equations. These modified equations are then examined, discovering that they share the conservative properties and traveling wave solutions with the original equations. As the averaging is diminished it is proven that the solutions converge to weak solutions of the original equations. Finally, numerical simulations are conducted finding that the regularized equations appear smooth and capture the relevant behavior in shock tube simulations.