Some recent finite volume schemes to compute Euler equations using real gas EOS

This paper deals with the resolution by finite volume methods of Euler equations in one space dimension, with real gas state laws (namely, perfect gas EOS, Tammann EOS and Van Der Waals EOS). All tests are of unsteady shock tube type, in order to examine a wide class of solutions, involving Sod shock tube, stationary shock wave, simple contact discontinuity, occurrence of vacuum by double rarefaction wave, propagation of a one-rarefaction wave over ‘vacuum’, … Most of the methods computed herein are approximate Godunov solvers: VFRoe, VFFC, VFRoe ncv (τ, u, p) and PVRS. The energy relaxation method with VFRoe ncv (τ, u, p) and Rusanov scheme have been investigated too. Qualitative results are presented or commented for all test cases and numerical rates of convergence on some test cases have been measured for first- and second-order (Runge–Kutta 2 with MUSCL reconstruction) approximations. Note that rates are measured on solutions involving discontinuities, in order to estimate the loss of accuracy due to these discontinuities. Copyright © 2002 John Wiley & Sons, Ltd.

[1]  Jean-Marc Hérard,et al.  An Approximate Roe-Type Riemann Solver for a Class of Realizable Second Order Closures , 2000 .

[2]  Marc Boucker,et al.  Modélisation numérique multidimensionnelle d'écoulements diphasiques liquide-gaz en régimes transitoire et permanent : méthodes et applications , 1998 .

[3]  Volker Boie.  Multiplication of distributions , 1998 .

[4]  Thierry Gallouët,et al.  On a rough Godunov scheme , 1996 .

[5]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[6]  Thierry Gallouët,et al.  On an Approximate Godunov Scheme , 1999 .

[7]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[8]  E. F. Toro,et al.  A linearized Riemann solver for the time-dependent Euler equations of gas dynamics , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[9]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[10]  Thierry Buffard.  Analyse de quelques methodes de volumes finis non structures pour la resolution des equations d'euler , 1993 .

[11]  Jean-Marc Hérard,et al.  A sequel to a rough Godunov scheme: application to real gases , 2000 .

[12]  Jean-Marc Hérard,et al.  A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems , 2002 .

[13]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[14]  Rémi Abgrall,et al.  A Simple Method for Compressible Multifluid Flows , 1999, SIAM J. Sci. Comput..

[15]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[16]  Arun In,et al.  Numerical Evaluation of an Energy Relaxation Method for Inviscid Real Fluids , 1999, SIAM J. Sci. Comput..

[17]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[18]  Lionel Sainsaulieu,et al.  Finite volume approximation of two phase-fluid flows based on an approximate Roe-type Riemann solver , 1995 .

[19]  A. Kumbaro,et al.  An Approximate Linearized Riemann Solver for a Two-Fluid Model , 1996 .

[20]  Jean-Marc Hérard,et al.  Solveur de type Godunov pour simuler les écoulements turbulents compressibles , 1997 .

[21]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[22]  B. Perthame,et al.  Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics , 1998 .

[23]  Eleuterio F. Toro,et al.  On Riemann solvers for compressible liquids , 1998 .

[24]  J. Hérard,et al.  Finite Volume Algorithm to Compute Dense Compressible Gas-Solid Flows , 1999 .

[25]  M. Uhlmann,et al.  An approximate solution of the Riemann problem for a realisable second-moment turbulent closure , 2002 .

[26]  Jean-Marie Masella.  Quelques méthodes numériques pour les écoulements diphasiques bi-fluide en conduites pétrolières , 1997 .

[27]  R. LeVeque.  Wave Propagation Algorithms for Multidimensional Hyperbolic Systems , 1997 .

[28]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.