Two approximations of solutions of Hamilton-Jacobi equations

Authors:
M. G. Crandall and P.-L. Lions

Journal:
Math. Comp. **43** (1984), 1-19

MSC:
Primary 65M10; Secondary 35F20, 49C10

MathSciNet review:
744921

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Abstract | References | Similar Articles | Additional Information

Abstract: Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initial-value problems almost never have global-time classical solutions, and one must deal with suitable generalized solutions. The correct class of generalized solutions has only recently been established by the authors. This article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates. Analogous results are proved by similar means for the method of vanishing viscosity.

**[1]**M. G. Crandall, L. C. Evans, and P.-L. Lions,*Some properties of viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 487–502. MR**732102**, 10.1090/S0002-9947-1984-0732102-X**[2]**Michael G. Crandall and Pierre-Louis Lions,*Viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc.**277**(1983), no. 1, 1–42. MR**690039**, 10.1090/S0002-9947-1983-0690039-8**[3]**Michael G. Crandall and Pierre-Louis Lions,*Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 3, 183–186 (French, with English summary). MR**610314****[4]**Michael G. Crandall and Andrew Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, 10.1090/S0025-5718-1980-0551288-3**[5]**Michael G. Crandall and Luc Tartar,*Some relations between nonexpansive and order preserving mappings*, Proc. Amer. Math. Soc.**78**(1980), no. 3, 385–390. MR**553381**, 10.1090/S0002-9939-1980-0553381-X**[6]**Lawrence C. Evans,*Some min-max methods for the Hamilton-Jacobi equation*, Indiana Univ. Math. J.**33**(1984), no. 1, 31–50. MR**726105**, 10.1512/iumj.1984.33.33002**[7]**W. H. Fleming,*Nonlinear partial differential equations—Probabilistic and game theoretic methods*, Problems in Non-Linear Analysis (C.I.M.E., IV Ciclo, Varenna, 1970) Edizioni Cremonese, Rome, 1971, pp. 95–128. MR**0282048****[8]**S. N. Kružkov,*The method of finite differences for a nonlinear equation of the first order with several independent variables*, Z. Vyčisl. Mat. i Mat. Fiz.**6**(1966), 884–894 (Russian). MR**0203205****[9]**N. N. Kuznetsov,*On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions*, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 183–197. MR**0657786****[10]**Pierre-Louis Lions,*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**667669****[11]**Panagiotis E. Souganidis,*Approximation schemes for viscosity solutions of Hamilton-Jacobi equations*, J. Differential Equations**59**(1985), no. 1, 1–43. MR**803085**, 10.1016/0022-0396(85)90136-6

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1984-0744921-8

Keywords:
Hamilton-Jacobi equations,
difference approximations,
error estimates,
method of vanishing viscosity

Article copyright:
© Copyright 1984
American Mathematical Society