On Godunov-Type Methods for Gas Dynamics

In this paper we describe a new approximate Riemann solver for compressible gas flow. In contrast to previous Riemann solvers, where a numerical approximation for the pressure and the velocity at the contact discontinuity is computed, we derive a numerical approximation for the largest and smallest signal velocity in the Riemann problem. Having obtained the numerical signal velocities, we use theoretical results by Harten, Lax and van Leer to obtain the full approximation. A stability condition for the numerical signal velocities is derived. We also demonstrate a relation between the signal velocities and the dissipation contained in the corresponding Godunov-type method. The computation of signal velocities for a general (convex) equation of state is discussed. Numerical results for the one- and two-dimensional compressible gas dynamics equations are also given.