Numerical resolution of scalar convex equations : explicit stability, entropy and convergence conditions

In this paper we extend to scalar convex equations a previous work done for linear advection numerical resolution (cf. [3], [11]). We do a study of finite volume schemes, giving some explicit conditions on the fluxes for such a scheme to be convergent. These conditions are of three types: consistency, stability, entropy. Then we propose one particular scheme, taking the most downwind possible fluxes under the convergence conditions, applying the idea of [3] to derive anti-dissipative schemes. The proposed scheme is, as expected, low dissipative. We show some numerical results. AMS Subject Classification. 65M06, 65N06, 82C40 . .

[1]  Bruno Després,et al.  Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics , 2002, J. Sci. Comput..

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

[3]  Jian-Guo Liu,et al.  Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions , 1999, Math. Comput..

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

[5]  P. Souganidis,et al.  Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations , 1995 .

[6]  Clint Dawson,et al.  High Resolution Schemes for Conservation Laws with Locally Varying Time Steps , 2000, SIAM J. Sci. Comput..

[7]  Philippe G. LeFloch,et al.  Convergence of finite difference schemes for conservation laws in several space dimensions , 1991 .

[8]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[9]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[10]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[11]  Frédéric Lagoutière,et al.  Modelisation mathematique et resolution numerique de problemes de fluides compressibles a plusieurs constituants , 2000 .

[12]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[13]  Frederic COQUELtt.  CONVERGENCE OF FINITE DIFFERENCE SCHEMES FOR CONSERVATION LAWS IN SEVERAL SPACE DIMENSIONS: A GENERAL THEORY* , 1993 .

[14]  B. Després,et al.  Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire , 1999 .

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

[16]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .