Error estimates for scalar conservation laws by a kinetic approach. (English) Zbl 1119.35339
Summary: We use the kinetic approach of B. Perthame and E. Tadmor [Commun. Math. Phys. 136, No. 3, 501–517 (1991; Zbl 0729.76070)] to calculate the error estimates for general scalar conservation laws governing problems in gas dynamics or fluid mechanics in general. The Kružkov and Kuznetsov techniques are generalized to this method, and an error bound of order \(\sqrt {\varepsilon} \) (where \(\varepsilon \) is the mean free path) is obtained.
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Citations:
Zbl 0729.76070References:
| [1] | Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases, Physical Review, 94, 3, 511-525 (1954) · Zbl 0055.23609 · doi:10.1103/PhysRev.94.511 |
| [2] | Bouchut, F.; Golse, F.; Pulvirenti, M., Kinetic Equations and Asymptotic Theory. Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics (Paris), 4, x+162 (2000), Paris: Gauthier-Villars, Paris · Zbl 0979.82048 |
| [3] | Bouchut, F.; Perthame, B., Kružkov’s estimates for scalar conservation laws revisited, Transactions of the American Mathematical Society, 350, 7, 2847-2870 (1998) · Zbl 0955.65069 · doi:10.1090/S0002-9947-98-02204-1 |
| [4] | Brenier, Y., Résolution d’équations d’évolution quasilinéaires en dimension d’espace à l’aide d’équations linéaires en dimension <mml:math alttext=“\(N + 1\)” id=“C2”>N+1, Journal of Differential Equations, 50, 3, 375-390 (1983) · Zbl 0549.35055 · doi:10.1016/0022-0396(83)90067-0 |
| [5] | Brenier, Y., Averaged multivalued solutions for scalar conservation laws, SIAM Journal on Numerical Analysis, 21, 6, 1013-1037 (1984) · Zbl 0565.65054 · doi:10.1137/0721063 |
| [6] | Cercignani, C., Mathematical Methods in Kinetic Theory (1984), New York: Plenum Press, New York · Zbl 0191.25103 |
| [7] | Chen, G.-Q.; Karlsen, K. H., <mml:math alttext=“\(L^1 \)” id=“C3”>L1-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations, Transactions of the American Mathematical Society, 358, 3, 937-963 (2006) · Zbl 1096.35072 · doi:10.1090/S0002-9947-04-03689-X |
| [8] | Chen, G.-Q.; Perthame, B., Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 20, 4, 645-668 (2003) · Zbl 1031.35077 |
| [9] | Giga, Y.; Miyakawa, T., A kinetic construction of global solutions of first order quasilinear equations, Duke Mathematical Journal, 50, 2, 505-515 (1983) · Zbl 0519.35053 · doi:10.1215/S0012-7094-83-05022-6 |
| [10] | Kružkov, S. N., First order quasilinear equations in several independent variables, Mathematics of the USSR, Sbornik, 10, 217-243 (1970) · Zbl 0215.16203 |
| [11] | Kuznetsov, N. N., Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Computational Mathematics and Mathematical Physics, 16, 6, 105-119 (1976) · Zbl 0381.35015 · doi:10.1016/0041-5553(76)90046-X |
| [12] | Kuznetsov, N. N.; Voloshin, S. A., On monotone difference approximations for a first-order quasi-linear equation, Soviet Mathematics. Doklady, 17, 1203-1206 (1976) · Zbl 0361.65082 |
| [13] | Lions, P.-L.; Perthame, B.; Tadmor, E., A kinetic formulation of multidimensional scalar conservation laws and related equations, Journal of the American Mathematical Society, 7, 1, 169-191 (1994) · Zbl 0820.35094 · doi:10.2307/2152725 |
| [14] | Miyakawa, T., A kinetic approximation of entropy solutions of first order quasilinear equations, Recent Topics in Nonlinear PDE (Hiroshima, 1983). Recent Topics in Nonlinear PDE (Hiroshima, 1983), North-Holland Math. Stud., 98, 93-105 (1984), Amsterdam: North-Holland, Amsterdam · Zbl 0567.35014 |
| [15] | Omrane, A., Estimations d’erreur pour les lois de conservation scalaires par approches cinétiques, et étude du cas des conditions aux limites, M.S. thesis (1998) |
| [16] | Perthame, B.; Tadmor, E., A kinetic equation with kinetic entropy functions for scalar conservation laws, Communications in Mathematical Physics, 136, 3, 501-517 (1991) · Zbl 0729.76070 · doi:10.1007/BF02099071 |
| [17] | Smoller, J., Shock Waves and Reaction-Diffusion Equations. Shock Waves and Reaction-Diffusion Equations, Fundamental Principles of Mathematical Science, 258, xxi+581 (1983), New York: Springer, New York · Zbl 0508.35002 |
| [18] | Vila, J.-P., Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes, Mathematical Modelling and Numerical Analysis, 28, 3, 267-295 (1994) · Zbl 0823.65087 |
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