Entropy Conservative Schemes and the Receding Flow Problem

TL;DR

This paper investigates entropy conservative schemes for the Euler equations, highlighting the impact of time-integration on solution behavior and demonstrating that entropy conservation alone does not ensure well-behaved solutions in receding flow problems.

Contribution

It develops an entropy conservative time-integration scheme and analyzes the effects of fully discrete entropy conservation on solution stability.

Findings

Spurious entropy rise causes temperature spikes in finite-volume schemes.

Fully discrete entropy conservation does not guarantee stable solutions.

Time-integration impacts the effectiveness of entropy conservative schemes.

Abstract

This work delves into the family of entropy conservative (EC) schemes introduced by Tadmor. The discussion is centered around the Euler equations of fluid mechanics and the receding flow problem extensively studied by Liou. This work is motivated by Liou's recent findings that an abnormal spike in temperature observed with finite-volume schemes is linked to a spurious entropy rise, and that it can be prevented in principle by conserving entropy. While a semi-discrete analysis suggests EC schemes are a good fit, a fully discrete analysis based on Tadmor's framework shows the non-negligible impact of time-integration on the solution behavior. An EC time-integration scheme is developed to show that enforcing conservation of entropy at the fully discrete level does not necessarily guarantee a well-behaved solution.