×

zbMATH — the first resource for mathematics

Equilibrium schemes for scalar conservation laws with stiff sources. (English) Zbl 1017.65070
The authors consider a simple model case of stiff source terms in hyperbolic conservation laws, and introduce new numerical schemes, so-called equilibrium schemes, to solve numerically this problem. Based on a new tool for the identification of solutions to kinetic equations corresponding to entropy solutions of the problem under consideration, the authors give a new convergence proof. They also give some numerical tests to demonstrate the computational efficiency of these schemes. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alfredo Bermúdez, Alain Dervieux, Jean-Antoine Desideri, and M. Elena Vázquez, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. Methods Appl. Mech. Engrg. 155 (1998), no. 1-2, 49 – 72. · Zbl 0961.76047 · doi:10.1016/S0045-7825(97)85625-3 · doi.org
[2] F. Bouchut and B. Perthame, Kružkov’s estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2847 – 2870 (English, with English and French summaries). · Zbl 0955.65069
[3] Yann Brenier, Résolution d’équations d’évolution quasilinéaires en dimension \? d’espace à l’aide d’équations linéaires en dimension \?+1, J. Differential Equations 50 (1983), no. 3, 375 – 390 (French). · Zbl 0549.35055 · doi:10.1016/0022-0396(83)90067-0 · doi.org
[4] Gui Qiang Chen, C. David Levermore, and Tai-Ping Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), no. 6, 787 – 830. · Zbl 0806.35112 · doi:10.1002/cpa.3160470602 · doi.org
[5] Frédéric Coquel and Benoît Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), no. 6, 2223 – 2249. · Zbl 0960.76051 · doi:10.1137/S0036142997318528 · doi.org
[6] Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223 – 270. · Zbl 0616.35055 · doi:10.1007/BF00752112 · doi.org
[7] Björn Engquist and Stanley Osher, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp. 34 (1980), no. 149, 45 – 75. · Zbl 0438.76051
[8] R. Eymard, T. Gallouët, and R. Herbin, Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Ann. Math. Ser. B 16 (1995), no. 1, 1 – 14. A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 1, 119. · Zbl 0830.35077
[9] Laurent Gosse and Alain-Yves Leroux, Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 5, 543 – 546 (French, with English and French summaries). · Zbl 0858.65091
[10] Gosse L., Localization effects and measure source terms in numerical schemes for balance laws, Preprint. · Zbl 0997.65108
[11] J. M. Greenberg, A. Y. Leroux, R. Baraille, and A. Noussair, Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal. 34 (1997), no. 5, 1980 – 2007. · Zbl 0888.65100 · doi:10.1137/S0036142995286751 · doi.org
[12] Yoshikazu Giga and Tetsuro Miyakawa, A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J. 50 (1983), no. 2, 505 – 515. · Zbl 0519.35053 · doi:10.1215/S0012-7094-83-05022-6 · doi.org
[13] S. N. Kružkov, Generalized solutions of the Cauchy problem in the large for first order nonlinear equations, Dokl. Akad. Nauk. SSSR 187 (1969), 29 – 32 (Russian).
[14] N. N. Kuznecov, Finite-difference methods for the solution of a first-order multidimensional quasilinear equation in a class of discontinuous functions, Problems in mathematical physics and numerical mathematics (Russian), ”Nauka”, Moscow, 1977, pp. 181 – 194, 326 (Russian).
[15] J. O. Langseth, A. Tveito, and R. Winther, On the convergence of operator splitting applied to conservation laws with source terms, SIAM J. Numer. Anal. 33 (1996), no. 3, 843 – 863. · Zbl 0866.65059 · doi:10.1137/0733042 · doi.org
[16] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603 – 634. · Zbl 0268.35014
[17] Randall J. LeVeque, Numerical methods for conservation laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. · Zbl 0847.65053
[18] P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169 – 191. · Zbl 0820.35094
[19] Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795 – 823. , https://doi.org/10.1002/(SICI)1097-0312(199608)49:83.0.CO;2-3 · Zbl 0872.35064
[20] B. Perthame, Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure, J. Math. Pures Appl. (9) 77 (1998), no. 10, 1055 – 1064 (English, with English and French summaries). · Zbl 0919.35088 · doi:10.1016/S0021-7824(99)80003-8 · doi.org
[21] Benoit Perthame and Athanasios E. Tzavaras, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal. 155 (2000), no. 1, 1 – 48. · Zbl 0980.35092 · doi:10.1007/s002050000109 · doi.org
[22] Russo G., personal communication.
[23] Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91 – 106. · Zbl 0533.65061
[24] A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 749 – 782 (English, with French summary). · Zbl 0751.65061
[25] A. Vasseur, Time regularity for the system of isentropic gas dynamics with \?=3, Comm. Partial Differential Equations 24 (1999), no. 11-12, 1987 – 1997. · Zbl 0940.35169 · doi:10.1080/03605309908821491 · doi.org
[26] María Elena Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys. 148 (1999), no. 2, 497 – 526. · Zbl 0931.76055 · doi:10.1006/jcph.1998.6127 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.