Maximum principle on the entropy and second-order kinetic schemes
HTML articles powered by AMS MathViewer
- by Brahim Khobalatte and Benoît Perthame PDF
- Math. Comp. 62 (1994), 119-131 Request permission
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.References
-
S. Deshpande, A second order accurate, kinetic theory based, method for inviscid compressible flows, NASA Langley Technical Paper no. 2613, 1986.
- Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231–303. MR 897244, DOI 10.1016/0021-9991(87)90031-3
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
- P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790
- Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI 10.1137/0721016
- B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal. 27 (1990), no. 6, 1405–1421 (English, with French summary). MR 1080328, DOI 10.1137/0727081
- B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal. 29 (1992), no. 1, 1–19 (English, with French summary). MR 1149081, DOI 10.1137/0729001 B. Perthame and Y. Qiu, A new variant of Van Leer’s method for multidimensional systems of conservation laws, INRIA report no. 1562.
- Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32–78. MR 1010162, DOI 10.1016/0021-9991(89)90222-2
- Eitan Tadmor, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math. 2 (1986), no. 3-5, 211–219. MR 863987, DOI 10.1016/0168-9274(86)90029-2
- Eitan Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comp. 49 (1987), no. 179, 91–103. MR 890255, DOI 10.1090/S0025-5718-1987-0890255-3 B. Van Leer, Towards the ultimate conservative difference scheme. V, A second order sequel of Godunov’s method, J. Comput. Phys. 32 (1979), 101-136.
- Thomas W. Roberts, The behavior of flux difference splitting schemes near slowly moving shock waves, J. Comput. Phys. 90 (1990), no. 1, 141–160. MR 1070474, DOI 10.1016/0021-9991(90)90200-K
- B. Einfeldt, C.-D. Munz, P. L. Roe, and B. Sjögreen, On Godunov-type methods near low densities, J. Comput. Phys. 92 (1991), no. 2, 273–295. MR 1094256, DOI 10.1016/0021-9991(91)90211-3
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp. 62 (1994), 119-131
- MSC: Primary 65M06; Secondary 35L65, 76M25, 76N15
- DOI: https://doi.org/10.1090/S0025-5718-1994-1208223-4
- MathSciNet review: 1208223