Semi-discrete Entropy Satisfying Approximate Riemann Solvers. The Case of the Suliciu Relaxation Approximation

In this work we establish conditions for an approximate simple Riemann solver to satisfy a semi-discrete entropy inequality. The semi-discrete approach is less restrictive than the fully-discrete case and allows to grant some other good properties for numerical schemes. First, conditions are established in an abstract framework for simple Riemann solvers to satisfy a semi-discrete entropy inequality and then the results are applied, as a particular case, to the Suliciu system. This will lead in particular to the definition of schemes for the isentropic gas dynamics and the full gas dynamics system that are stable and preserve the stationary shocks.

[1]  Eleuterio F. Toro,et al.  MUSTA fluxes for systems of conservation laws , 2006, J. Comput. Phys..

[2]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[3]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

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

[5]  I. Suliciu.  Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations , 1992 .

[6]  Antonio Marquina,et al.  Local Piecewise Hyperbolic Reconstruction of Numerical Fluxes for Nonlinear Scalar Conservation Laws , 1994, SIAM J. Sci. Comput..

[7]  P. LeFloch.  Hyperbolic Systems of Conservation Laws , 2002 .

[8]  B. Perthame,et al.  Some New Godunov and Relaxation Methods for Two-Phase Flow Problems , 2001 .

[9]  P. Lax.  Hyperbolic systems of conservation laws II , 1957 .

[10]  Roland Masson,et al.  A relaxation method for two-phase flow models with hydrodynamic closure law , 2005, Numerische Mathematik.

[11]  I. Suliciu.  On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure , 1990 .

[12]  T. Gallouët,et al.  Some approximate Godunov schemes to compute shallow-water equations with topography , 2003 .

[13]  Phillip Colella,et al.  Efficient Solution Algorithms for the Riemann Problem for Real Gases , 1985 .

[14]  François Bouchut,et al.  Entropy satisfying flux vector splittings and kinetic BGK models , 2003, Numerische Mathematik.

[15]  P. LeFloch,et al.  Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves , 2002 .