Entropy Conservative Schemes and the Receding Flow Problem
This work clarifies the limitations of entropy conservation in numerical schemes for fluid dynamics, important for researchers developing robust finite-volume methods.
The authors analyze entropy conservative (EC) schemes for the Euler equations, focusing on the receding flow problem. They show that even with fully discrete entropy conservation, abnormal temperature spikes can still occur, contradicting previous assumptions that entropy conservation prevents such issues.
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.