Maximum principle on the entropy and second-order kinetic schemes

We consider kinetic schemes for the multidimensional inviscid gas dynamics equations (compressible Euler equations). We prove that the discrete maximum principle holds for the specific entropy. This fixes the choice of the equilibrium functions necessary for kinetic schemes. We use this property to perform a second-order oscillation-free scheme, where only one slope limitation (for three conserved quantities in 1D) is necessary. Numerical results exhibit stability and strong convergence of the scheme. INTRODUCTION We consider the gas dynamics equations in one or two space dimensions, atp + div(pu) = 0, (1) & atpu ? + div(puju) + a, p = 0, j = 1, 2, 9,E +div[(E +p)u]= 0, where x = (xl, x2), u = (ul, u2), and the total energy E piu12/2+pT/(y-1) is related to the pressure by the relation p = pT, 1 Oh}, and, in 11D, Godunov and Lax-Friedrichs schemes preserve this property at the discretized level because they solve exactly the system (1). A reason why (4) should hold is that S satisfies the (meaningless) equation ats + u * Vxs < 0. Received by the editor March 10, 1992 and, in revised form, January 7, 1993. 1991 Mathematics Subject Classification. Primary 35L65, 76N10, 65M06, 76PO5.

[1]  B. Perthame.  Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions , 1992 .

[2]  B. Perthame,et al.  Boltzmann type schemes for gas dynamics and the entropy property , 1990 .

[3]  S. M. Deshpande,et al.  A second-order accurate kinetic-theory-based method for inviscid compressible flows , 1986 .

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

[5]  Thomas W. Roberts,et al.  The behavior of flux difference splitting schemes near slowly moving shock waves , 1990 .

[6]  Eitan Tadmor,et al.  The numerical viscosity of entropy stable schemes for systems of conservation laws. I , 1987 .

[7]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[8]  B. Perthame,et al.  Kinetic formulation of the isentropic gas dynamics andp-systems , 1994 .

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

[10]  S. Osher.  Riemann Solvers, the Entropy Condition, and Difference , 1984 .

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

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

[13]  Eitan Tadmor,et al.  A minimum entropy principle in the gas dynamics equations , 1986 .

[14]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .