Maximum Principle on the Entropy and Second-Order Kinetic Schemes

Abstract

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.