Wave Structure Similarity of the HLLC and Roe Riemann Solvers: Application to Low Mach Number Preconditioning

Abstract

The approximate Riemann solver of Roe and the solver of Harten--Lax--van Leer (HLL) and its variants, such as the HLLC solver, are widely used as building blocks of finite volume Godunov-type methods for the solution of the Euler equations of gas dynamics and related hyperbolic flow models. The HLLC solver has gained increasing popularity over the last two decades since it possesses some of the good properties of the Roe solver and in addition satisfies important entropy and positivity conditions with no need of special fixes. In this work, we rewrite the classical HLLC Riemann solver in a novel form that highlights the formal mathematical similarity of its wave structure with the one of the Roe solver. This similarity might be useful to extend to the HLLC method some numerical techniques devised specifically for the Roe method. As an example of application, we illustrate the design of a Turkel-type low Mach number preconditioning technique for the HLLC scheme by exploiting methodologies proposed in the literature for the Roe scheme.