Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics

Abstract

We consider the Euler equations for a compressible inviscid fluid with a general pressure law $p(\rho,\varepsilon)$, where $\rho$ represents the density of the fluid and $\varepsilon$ its specific internal energy. We show that it is possible to introduce a relaxation of the nonlinear pressure law introducing an energy decomposition under the form $\varepsilon= \varepsilon _1 + \varepsilon _2.$ The internal energy $\varepsilon _1$ is associated with a (simpler) pressure law $p_1(\rho,\varepsilon_1)$; the energy $\varepsilon _2$ is advected by the flow. These two energies are also subject to a relaxation process and in the limit of an infinite relaxation rate, we recover the initial pressure law p. We show that, under some conditions of subcharacteristic type, for any convex entropy associated with the pressure p, we can find a global convex and uniform entropy for the relaxation system. From our construction, we also deduce the extension to general pressure laws of classical approximate Riemann solvers for polytropic gases, which only use a single call to the pressure law (per mesh point and time step). For the Godunov scheme, we show that this extension satisfies stability, entropy, and accuracy conditions.