An Entropy Stable Central Solver for Euler Equations

TL;DR

This paper presents an improved entropy stable central solver for Euler equations that accurately captures shocks and contact discontinuities without requiring complex Riemann solvers, ensuring stability and precision.

Contribution

The authors enhance the MOVERS scheme to be entropy stable by incorporating an optimal numerical diffusion strategy and a limiter, eliminating the need for an entropy fix.

Findings

The new scheme is entropy stable and accurate.

It captures steady discontinuities exactly.

It avoids the need for an entropy fix.

Abstract

An exact discontinuity capturing central solver developed recently, named MOVERS (Method of Optimal Viscosity for Enhanced Resolution of Shocks, J Computat Phys 2009;228:770-798), is analyzed and improved further to make it entropy stable. MOVERS, which is designed to capture steady shocks and contact discontinuities exactly by enforcing the Rankine-Hugoniot jump condition directly in the discretization process, is a low diffusive algorithm in a simple central discretization framework, free of complicated Riemann solvers and flux splittings. However, this algorithm needs an entropy fix to avoid nonsmoothness in the expansion regions. The entropy conservation equation is used as a guideline to introduce an optimal numerical diffusion in the smooth regions and a limiter based switchover is introduced for numerical diffusion based on jump conditions at the large gradients. The resulting new scheme is entropy stable, accurate and captures steady discontinuities exactly while avoiding an entropy fix.