Two approximations of solutions of HamiltonJacobi equations
Authors:
M. G. Crandall and P.L. Lions
Journal:
Math. Comp. 43 (1984), 119
MSC:
Primary 65M10; Secondary 35F20, 49C10
MathSciNet review:
744921
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Abstract: Equations of HamiltonJacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initialvalue problems almost never have globaltime 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.
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 M. G. Crandall, L. C. Evans & P. L. Lions, Some Properties of Viscosity Solutions of HamiltonJacobi Equations, Math. Res. Center TSR #2390, June 1982; Trans. Amer. Math. Soc. (To appear.) MR 732102 (86a:35031)
 [2]
 M. G. Crandall & P. L. Lions, "Viscosity solutions of HamiltonJacobi equations," Trans. Amer. Math. Soc., v. 277, 1983, pp. 142. MR 690039 (85g:35029)
 [3]
 M. G. Crandall & P. L. Lions, "Conditions d'unicité pour les solutions généraliseés d'équations de HamiltonJacobi de premier ordre," C. R. Acad. Sci. Paris, v. 292, 1981, pp. 183186. MR 610314 (82c:49020)
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 M. G. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [5]
 M. G. Crandall & L. Tartar, "Some relations between nonexpansive and order preserving mappings," Proc. Amer. Math. Soc., v. 79, 1979, pp. 7480. MR 553381 (81a:47054)
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 L. C. Evans, "Some maxmin methods for the HamiltonJacobi equation," Indiana Univ. Math. J., v. 33, 1984, pp. 3150. MR 726105 (85b:35009)
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 W. H. Fleming, "Nonlinear partial differential equationsProbabilistic and game theoretic methods," Problems in Nonlinear Analysis, CIME, Edizioni Cremonese, Roma, 1971. MR 0282048 (43:7762)
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 S. N. Kružkov, "The method of finite differences for a nonlinear equation of the first order with several independent variables," Ž. Vyčisl. Mat. i Mat. Fiz., v. 6, 1966, pp. 884894. (Russian) MR 0203205 (34:3058)
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 N. N. Kuznetsov, "On stable methods for solving nonlinear first order partial differential equations in the class of discontinuous functions," Topics in Numerical Analysis III (J. J. H. Miller, Ed.), Academic Press, New York, 1977, pp. 183197. MR 0657786 (58:31874)
 [10]
 P. L. Lions, Generalized Solutions of HamiltonJacobi Equations, Pitman Lecture Notes, London, 1982. MR 667669 (84a:49038)
 [11]
 P. E. Souganidis, "Approximation schemes for viscosity solutions of HamiltonJacobi equations," J. Differential Equations. (To appear.) MR 803085 (86k:35028)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407449218
PII:
S 00255718(1984)07449218
Keywords:
HamiltonJacobi equations,
difference approximations,
error estimates,
method of vanishing viscosity
Article copyright:
© Copyright 1984
American Mathematical Society
