Upwinding of the source term at interfaces for Euler equations with high friction

We consider Euler equations with a friction term that describe an isentropic gas flow in a porous domain. More precisely, we consider the transition between low and high friction regions. In the high friction region the system is reduced to a parabolic equation, the porous media equation. In this paper we present a hyperbolic approach based on a finite volume technique to compute numerical solutions for the system in both regimes. The Upwind Source at Interfaces (USI) scheme that we propose satisfies the following properties. Firstly it preserves the nonnegativity of gas density. Secondly, and this is the motivation, the scheme is asymptotically consistent with the limit model (porous media equation) when the friction coefficient goes to infinity. We show analytically and through numerical results that the above properties are satisfied. We shall also compare results given with the use of USI, hyperbolic-parabolic coupling and classical centered sources schemes.

[1]  Laurent Gosse,et al.  A priori error estimate for a well-balanced scheme designed for inhomogeneous scalar conservation laws† , 1998 .

[2]  An asymptotic preserving scheme for Hydrodynamics Radiative Transfert Models Numerical Scheme for radiative transfert , 2005 .

[3]  François Golse,et al.  The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method , 1999 .

[4]  Doron Levy,et al.  CENTRAL-UPWIND SCHEMES FOR THE SAINT-VENANT SYSTEM , 2002 .

[5]  B. Perthame,et al.  A kinetic formulation of multidimensional scalar conservation laws and related equations , 1994 .

[6]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[7]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

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

[9]  Pierangelo Marcati,et al.  The One-Dimensional Darcy's Law as the Limit of a Compressible Euler Flow , 1990 .

[10]  Laurent Gosse,et al.  Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation , 2004, Numerische Mathematik.

[11]  Benoît Perthame,et al.  Kinetic formulation of conservation laws , 2002 .

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

[13]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[14]  Shi Jin,et al.  Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations , 2000, SIAM J. Numer. Anal..

[15]  Chiara Simeoni,et al.  Convergence of the Upwind Interface Source Method for Hyperbolic Conservation Laws , 2003 .

[16]  Shi Jin,et al.  A steady-state capturing method for hyperbolic systems with geometrical source terms , 2001 .

[17]  Chiara Simeoni,et al.  First and second order error estimates for the Upwind Source at Interface method , 2004, Math. Comput..

[18]  M. Vázquez-Cendón.  Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

[19]  Philip L. Roe,et al.  Upwind differencing schemes for hyperbolic conservation laws with source terms , 1987 .

[20]  P. Souganidis,et al.  Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates , 1998 .

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

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

[23]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[24]  Stéphane Cordier,et al.  Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models , 2004 .

[25]  Chiara Simeoni,et al.  Upwinding sources at interfaces in conservation laws , 2004, Appl. Math. Lett..

[26]  B. Perthame,et al.  A kinetic scheme for the Saint-Venant system¶with a source term , 2001 .