Relaxation approximation of the Euler equations

Abstract

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.

A relaxation scheme for hybrid modelling of gas-particle flows

“…A proof of this result can be found in [9] for instance, with a slightly different form of the relaxation system. The proof is obtained by construction and we recall it below.…”

“…The choice of Ψ associated with (A2) ensures the uniqueness of the jump conditions and results in the relaxation system that corresponds exactly to the system introduced for the simulation of barotropic Euler equations, when ρR L = π(ρ). When restricting to (A2) and though equation (9.3) is slightly different from the formulations presented for instance in [4] and [9] for barotropic Euler equations (see Appendix A too), relaxation systems are equivalent for smooth solutions and the underlying idea is the same.…”

“…Some energy estimates for relaxation systems corresponding to (A1) and (A2) are given in Appendix B, in a one-dimensional framework (and in [18,16] in a 2D framework). These should not be confused with entropy estimates for barotropic Euler equations for instance, such as those provided in [4], [35] and [9] among others.…”

“…This approach, which will be referred to as A1, relies on the combined use of relaxation-type techniques [1,4,5,9,12,13,28,35,36,38] and upwinding schemes [14,19,22,23,29,30] and was first introduced in [25] in order to predict single-phase turbulent flows [32]. In the latter approach, an extended hyperbolic system was introduced in a natural way, following what is done for previous associated Eulerian models (see [2,3,6,21]).…”

Large Time-Step Numerical Scheme for the Seven-Equation Model of Compressible Two-Phase Flows

“…This system is treated using a pressure relaxation approach that consists in introducing a linearized pressure π k (see for instance [5] and especially the references therein), such that (π k ) n j = (p k ) n j , and in solving the partial differential system…”