Largetime behavior of solutions of LaxFriedrichs finite difference equations for hyperbolic systems of conservation laws
Author:
ILiang Chern
Journal:
Math. Comp. 56 (1991), 107118
MSC:
Primary 65M12; Secondary 35L65, 39A12, 76L05
MathSciNet review:
1052088
Fulltext PDF Free Access
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Abstract: We study the largetime behavior of discrete solutions of the LaxFriedrichs finite difference equations for hyperbolic systems of conservation laws. The initial data considered here are small and tend to a constant state at . We show that the solutions tend to the discrete diffusion waves at the rate in , , with being an arbitrarily small constant. The discrete diffusion waves can be constructed from the selfsimilar solutions of the heat equation and the Burgers equation through an averaging process.
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 I.L. Chern and T.P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), 503517. MR 891950 (88g:35127)
 [2]
 , Erratum, convergence of diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 120 (1989), 525527. MR 981217 (90a:35138)
 [3]
 I.L. Chern, Multiplemode diffusion waves for viscous nonstrictly hyperbolic conservation laws, preprint, MCSP1340290, Math. Comp. Div., Argonno National Lab., 1990.
 [4]
 E. Hopf, The partial differential equation , Comm. Pure Appl. Math. 3 (1950), 201230. MR 0047234 (13:846c)
 [5]
 S. Kawashima, Systems of a hyperbolicparabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral thesis, Kyoto University, 1983.
 [6]
 , Largetime behavior of solutions to hyperbolicparabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 169194. MR 899951 (89d:35022)
 [7]
 T.P. Liu, Pointwise convergence to Nwaves for solutions of hyperbolic conservation laws, Bull. Inst. Math. Acad. Sinica 15 (1987), 117. MR 947772 (89h:35202)
 [8]
 A. Matzumura and T. Nishida, The initial value problem for the equations of motion of viscous and heatconductive gases, J. Math. Kyoto Univ. 20 (1980), 67104 MR 564670 (81g:35108)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199110520888
PII:
S 00255718(1991)10520888
Keywords:
LaxFriedrichs scheme,
hyperbolic systems of conservation laws,
discrete diffusion waves,
asymptotic behavior,
numerical viscosity
Article copyright:
© Copyright 1991 American Mathematical Society
