A numerical conception of entropy for quasi-linear equations

Author:
A. Y. le Roux

Journal:
Math. Comp. **31** (1977), 848-872

MSC:
Primary 65M10; Secondary 35F25

DOI:
https://doi.org/10.1090/S0025-5718-1977-0478651-3

MathSciNet review:
0478651

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Abstract: A family of difference schemes solving the Cauchy problem for quasi-linear equations is studied. This family contains well-known schemes such as the decentered, Lax, Godounov or Lax-Wendroff schemes. Two conditions are given, the first assures the convergence to a weak solution and the second, more restrictive, implies the convergence to the solution in Kružkov's sense, which satisfies an entropy condition that guarantees uniqueness. Some counterexamples are proposed to show the necessity of such conditions.

**[1]**Edward Conway and Joel Smoller,*Clobal solutions of the Cauchy problem for quasi-linear first-order equations in several space variables*, Comm. Pure Appl. Math.**19**(1966), 95–105. MR**0192161**, https://doi.org/10.1002/cpa.3160190107**[2]**A. DOUGLIS, "Lectures on discontinuous solutions of first order non linear partial differential equations,"*North British Symposium on Partial Differential Equations*, 1972.**[3]**Eberhard Hopf,*On the right weak solution of the Cauchy problem for a quasilinear equation of first order*, J. Math. Mech.**19**(1969/1970), 483–487. MR**0251357****[4]**S. N. Kružkov,*First order quasilinear equations with several independent variables.*, Mat. Sb. (N.S.)**81 (123)**(1970), 228–255 (Russian). MR**0267257****[5]**Peter D. Lax,*Weak solutions of nonlinear hyperbolic equations and their numerical computation*, Comm. Pure Appl. Math.**7**(1954), 159–193. MR**0066040**, https://doi.org/10.1002/cpa.3160070112**[6]**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774**, https://doi.org/10.1002/cpa.3160130205**[7]**A. Y. LE ROUX,*Résolution numérique du problème de Cauchy pour une equation hyperbolique quasilineaire à une ou plusieurs variables d'espace*, Thèse 3e cycle, Rennes, 1974.**[8]**S. OHARU & T. TAKAHASHI, "A convergence theorem of nonlinear semigroups, and its applications to first order quasi-linear equations." (To appear.)**[9]**O. A. Oleĭnik,*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737****[10]**O. A. OLEĬNIK, "Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation,"*Amer. Math. Soc. Transl.*(2), v. 33, 1963, pp. 285-290. MR**22**#8187.**[11]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[12]**L. F. Shampine and R. J. Thompson,*Difference methods for nonlinear first-order hyperbolic systems of equations*, Math. Comp.**24**(1970), 45–56. MR**0263269**, https://doi.org/10.1090/S0025-5718-1970-0263269-1

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0478651-3

Article copyright:
© Copyright 1977
American Mathematical Society