On nonlocal monotone difference schemes for scalar conservation laws
Author:
Bradley J. Lucier
Journal:
Math. Comp. 47 (1986), 1936
MSC:
Primary 65M10; Secondary 35L65
MathSciNet review:
842121
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Abstract: We provide error analyses for explicit, implicit, and semiimplicit monotone finitedifference schemes on uniform meshes with nonlocal numerical fluxes. We are motivated by finitedifference discretizations of certain longwave (Sobolev) regularizations of the conservation laws that explicitly add a dispersive term as well as a nonlinear dissipative term. We also develop certain relationships between dispersion and stability in finitedifference schemes. Specifically, we find that discretization and explicit dispersion have identical effects on the amount of artificial dissipation necessary for stability.
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 [1]
 T. B. Benjamin, J. L. Bona & J. J. Mahony, "Model equations for long waves in nonlinear dispersive systems," Philos. Trans. Roy. Soc. London Ser. A, v. 272, 1972, pp. 4778. MR 0427868 (55:898)
 [2]
 J. L. Bona, W. G. Pritchard & L. R. Scott, "An evaluation of a model equation for water waves." Philos. Trans. Roy. Soc. London Ser. A, v. 302, 1981, pp. 457510. MR 633485 (83a:35088)
 [3]
 M. G. Crandall & T. M. Liggett, "Generation of semigroups of nonlinear transformations on general Banach spaces," Amer. J. Math., v. 93, 1971, pp. 265298. MR 0287357 (44:4563)
 [4]
 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. 78, 1980, pp. 385390. MR 553381 (81a:47054)
 [6]
 K. Deimling, Ordinary Differential Equations in Banach Spaces, SpringerVerlag, New York, 1977. MR 0463601 (57:3546)
 [7]
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 J. Douglas, Jr., R. P. Kendall & M. F. Wheeler, "Long wave regularization of onedimensional, twophase, immiscible flow in porous media," Finite Element Methods for Convection Dominated Flows, AMDv. 34, ASME, New York, 1979, pp. 201211. MR 571679 (81c:76001)
 [9]
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 [10]
 B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. Comp., v. 34, 1980, pp. 4575. MR 551290 (81b:65082)
 [11]
 E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Australian National University, 1977. MR 0638362 (58:30685)
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 A. Harten, J. M. Hyman & P. D. Lax, "On finite difference approximations and entropy conditions for shocks," Comm. Pure Appl. Math., v. 29, 1976, pp. 297322. MR 0413526 (54:1640)
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 G. W. Hedstrom, "Models of difference schemes for by partial differential equations," Math. Comp., v. 29, 1975, pp. 969977. MR 0388797 (52:9631)
 [14]
 A. Jameson & T. J. Baker, Solution of the Euler Equations for Complex Configurations, AIAA paper 831929, 1983.
 [15]
 S. N. Kruzhkov, "First order quasilinear equations with several independent variables," Math. USSR Sb., v. 10, 1970, pp. 217243.
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 N. N. Kuznetsov, "Accuracy of some approximate methods for computing the weak solutions of a firstorder quasilinear equation," USSR Comput. Math. and Math. Phys., v. 16, no. 6, 1976, pp. 105119.
 [17]
 N. N. Kuznetsov & S. A. Voloshin, "On the stability of a class of implicit finitedifference schemes," Dokl. Akad. Nauk SSSR, v. 242, no. 3, 1978, pp. 525528. MR 507136 (80b:65122)
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 P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 0120774 (22:11523)
 [19]
 B. J. Lucier, Dispersive Approximations for Hyperbolic Conservation Laws, ANL8174, Argonne National Laboratory, 1981.
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 B. J. Lucier, "On Sobolev regularizations of hyperbolic conservation laws," Comm. Partial Differential Equations, v. 10, no. 1, 1985, pp. 128. MR 773210 (86h:35082)
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 R. D. Richtmyer & K. W. Morton, Difference Methods for InitialValue Problems, 2nd ed., Wiley, New York, 1967. MR 0220455 (36:3515)
 [22]
 R. Sanders, "On convergence of monotone finite difference schemes with variable spatial differencing," Math. Comp., v. 40, 1983, pp. 91106. MR 679435 (84a:65075)
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 L. N. Trefethen, "Group velocity in finite difference schemes," SIAM Rev., v. 24, 1982, pp. 113136. MR 652463 (83b:65141)
 [24]
 S. A. Voloshin, "On a class of monotonic finite difference approximations of a firstorder quasilinear equation," Dokl. Akad. Nauk SSSR, v. 242, no. 1, 1978, pp. 1416. MR 506456 (80i:65099)
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 S. A. Voloshin, "On a class of implicit finitedifference schemes," USSR Comput. Math. and Math. Phys., v. 23, no. 2, 1983, pp. 5963. MR 698222 (85f:65091)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608421216
PII:
S 00255718(1986)08421216
Article copyright:
© Copyright 1986
American Mathematical Society
