ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

[1]  A. I. Vol'pert.  THE SPACES BV AND QUASILINEAR EQUATIONS , 1967 .

[2]  Blake Temple,et al.  Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system , 1995 .

[3]  Stefan Diehl,et al.  On scalar conservation laws with point source and discontinuous flux function , 1995 .

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

[5]  N. Risebro,et al.  ON A NONLINEAR DEGENERATE PARABOLIC TRANSPORT-DIFFUSION EQUATION WITH A DISCONTINUOUS COEFFICIENT , 2002 .

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

[7]  Christian Klingenberg,et al.  Stability of a Resonant System of Conservation Laws Modeling Polymer Flow with Gravitation , 2001 .

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

[9]  Christian Klingenberg,et al.  Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior , 1995 .

[10]  E. Giusti.  Minimal surfaces and functions of bounded variation , 1977 .

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

[12]  T. Gallouët,et al.  Numerical modeling of two-phase flows using the two-fluid two-pressure approach , 2004 .

[13]  Thierry BuÄard.  A sequel to a rough Godunov scheme: application to real gases , 2000 .

[14]  B. Temple.  Global solution of the cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws , 1982 .

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

[16]  Nils Henrik Risebro,et al.  STABILITY OF CONSERVATION LAWS WITH DISCONTINUOUS COEFFICIENTS , 1999 .

[17]  John D. Towers.  A Difference Scheme for Conservation Laws with a Discontinuous Flux: The Nonconvex Case , 2001, SIAM J. Numer. Anal..

[18]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[19]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

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

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

[22]  E. Isaacson,et al.  Analysis of a singular hyperbolic system of conservation laws , 1986 .

[23]  Blake Temple,et al.  The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems 1 , 1988 .

[24]  Alexis Vasseur,et al.  Strong Traces for Solutions of Multidimensional Scalar Conservation Laws , 2001 .

[25]  Jean-Marc Hérard,et al.  Some recent finite volume schemes to compute Euler equations using real gas EOS , 2002 .

[26]  A. Leroux,et al.  Riemann Solvers for some Hyperbolic Problems with a Source Term , 1999 .

[27]  Edwige Godlewski,et al.  The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case , 2004, Numerische Mathematik.

[28]  L. Lin,et al.  A comparison of convergence rates for Godunov's method and Glimm's method in resonant nonlinear systems of conservation laws , 1995 .

[29]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[30]  F. James,et al.  One-dimensional transport equations with discontinuous coefficients , 1998 .

[31]  John D. Towers.  Convergence of a Difference Scheme for Conservation Laws with a Discontinuous Flux , 2000, SIAM J. Numer. Anal..

[32]  Blake Temple,et al.  The Classification of Solutions of Quadratic Riemann Problems. I. , 1985 .

[33]  E. Isaacson,et al.  The Riemann problem near a hyperbolic singularity II , 1988 .