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Weak solutions of given problems are sometimes not necessarily unique. Relevant solutions are then picked out of the set of weak solutions by so-called entropy conditions. Connections between the original and the numerical entropy condition were often discussed in the particular case of scalar conservation laws, and also a general theory was presented in the literature for general scalar problems. The entropy conditions were realized by certain inequalities not generalizable to systems of equations in a trivial way. It is a concern of this article to extend the theory in such a way that inequalities can be replaced by general relations, and this not only in an abstract way but also realized by examples.
[1] A.  Harten, J. M. Heyman and P. D. Lax: On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976), 297–322. DOI 10.1002/cpa.3160290305 | MR 0413526
[2] B. Engquist, St. Osher: Stable and entropy satisfying approximations for transonic flow calculations. Math. Comp. 34 (1980), 45–75. DOI 10.1090/S0025-5718-1980-0551290-1 | MR 0551290
[3] R.  Ansorge, J.  Lei: The convergence of discretization methods if applied to weakly formulated problems. Theory and applications. Z. Angew. Math. Mech. 71 (1991), 207–221. DOI 10.1002/zamm.19910710702 | MR 1121485
[4] R.  Ansorge: Convergence of discretizations of nonlinear problems. Z. Angew. Math. Mech. 73 (1993), 239–253. DOI 10.1002/zamm.19930731002 | MR 1248574 | Zbl 0801.65054
[5] P.  Lax, B.  Wendroff: Systems of conservation laws. Comm. Pure Appl. Math. 13 (1960), 217–237. DOI 10.1002/cpa.3160130205 | MR 0120774
[6] M. G.  Crandall, A.  Majda: Monotone difference approximations and entropy conditions for shocks. Math. Comp. 34 (1980), 1–21. MR 0551288
[7] S. N.  Kruzhkov: Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order. Soviet Math. Dokl. 10 (1969), 785–788. Zbl 0202.37701
[8] J.  Glimm: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 95–105. DOI 10.1002/cpa.3160180408 | MR 0194770 | Zbl 0141.28902
[9] D. Kröner: Numerical Schemes for Conservation Laws. Wiley-Teubner, Chichester, Stuttgart, 1997. MR 1437144
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