An entropy stable central solver for Euler equations

Abstract

An exact discontinuity capturing central solver developed recently, named method of optimal viscosity for enhanced resolution of shocks (MOVERS) (Jaisankar and Raghurama Rao in J Comput Phys 228(3):770–798, 2009), 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 non-smoothness 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.