On the choice of wavespeeds for the HLLC Riemann solver

This paper considers a class of approximate Riemann solver devised by Harten, Lax, and van Leer (denoted HLL) for the Euler equations of inviscid gas dynamics. In their 1983 paper, Harten, Lax, and van Leer showed how, with a priori knowledge of the signal velocities, a single-state approximate Riemann solver could be constructed so as to automatically satisfy the entropy condition and yield exact resolution of isolated shock waves. Harten, Lax, and van Leer further showed that a two-state approximation could be devised, such that both shock and contact waves would be resolved exactly. However, the full implementation of this two-state approximation was never given. We show that with an appropriate choice of acoustic and contact wave velocities, the two-state so-called HLLC construction of Toro, Spruce, and Speares will yield this exact resolution of isolated shock and contact waves. We further demonstrate that the resulting scheme is positively conservative. This property, which cannot be guaranteed by any linearized approximate Riemann solver, forces the numerical method to preserve initially positive pressures and densities. Numerical examples are given to demonstrate that the solutions generated are comparable to those produced with an exact Riemann solver, only with a stronger enforcement of the entropy condition across expansion waves.