A skew-symmetric energy and entropy stable formulation of the compressible Euler equations

TL;DR

This paper introduces a skew-symmetric formulation for the compressible Euler equations that ensures energy and entropy stability both analytically and in discrete numerical schemes, improving the robustness of simulations.

Contribution

It develops a novel skew-symmetric form for the Euler equations that guarantees energy and entropy bounds in both continuous and discrete settings.

Findings

Skew-symmetric form leads to energy and entropy bounds.

Discrete schemes based on summation-by-parts preserve stability.

Method applicable to primitive variable formulations.

Abstract

We show that a specific skew-symmetric form of nonlinear hyperbolic problems leads to energy and entropy bounds. Next, we exemplify by considering the compressible Euler equations in primitive variables, transform them to skew-symmetric form and show how to obtain energy and entropy estimates. Finally we show that the skew-symmetric formulation lead to energy and entropy stable discrete approximations if the scheme is formulated on summation-by-parts form.