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Existence and uniqueness of traveling waves and error estimates for Godunov schemes of conservation laws


Author: Haitao Fan
Journal: Math. Comp. 67 (1998), 87-109
MSC (1991): Primary 65M10, 35L65
DOI: https://doi.org/10.1090/S0025-5718-98-00921-1
MathSciNet review: 1451320
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Abstract: The existence and uniqueness of the Lipschitz continuous traveling wave of Godunov's scheme for scalar conservation laws are proved. The structure of the traveling waves is studied. The approximation error of Godunov's scheme on single shock solutions is shown to be $O(1)\Delta x$.


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  • [EO] B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), 321-351. MR 82c:65056
  • [EY] B. Engquist and Shih-Hsien Yu, Convergence of Lax-Wendroff scheme for piecewise smooth solutions with shocks, IMA preprint, (1994)
  • [Je] G. Jennings, Discrete shocks, Comm. Pure Appl. Math., 27 (1974) 25-37. MR 49:3358
  • [Ku] N. N. Kutnetsov, On stable methods for solving nonlinear first order partial differential equations in the class of discontinuous functions, Topics in Numerical Analysis III (Proc. Roy. Irish Acad. Conf.), J. J. H. Miller ed., Academic Press London, 1977, 183-197. MR 58:31874
  • [LX1] Jian-Guo Liu and Zhouping Xin, Nonlinear stability of discrete shocks for systems of conservation laws, Arch. Rat. Mech. Anal., 125 (1993) 217-256. MR 95c:35166
  • [LX2] Jian-Guo Liu and Zhouping Xin, $L^{1}$-stability of stationary discrete shocks, Math. Comp., 60 (1993) 233-244. MR 93d:35097
  • [Mi] D. Michelson, Discrete shocks for difference approximations to systems of conservation laws. Adv. Appl. Math., 5 (1984), 433-469. MR 86f:65159
  • [MR] A. Majda and J. Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math., 32 (1979) 445-482. MR 81i:35108
  • [OR] S. Osher and J. Ralston, $L^{1}$ stability of traveling waves with applications to convective porous media flow, Comm. Pure Appl. Math., 35 (1982) 737-749. MR 84j:35090
  • [Sm] Y. Smyrlis, Existence and stability of stationary profiles of the LW scheme, Comm. Pure Appl. Math., 43 (1990) 509-545. MR 91d:65143
  • [SX] A. Szepessy and Zhouping Xin, Nonlinear stability of viscous shock waves, Arch. Rat. Mech. Anal., 122 (1993) 53-103. MR 93m:35125
  • [Sz] A. Szepessy, On the stability of finite element methods for shock waves, Comm. Pure Appl. Math., 45 (1992) 923-946. MR 93f:65076
  • [TT] Tao Tang and Zhen-Huan Teng, The sharpness of Kuznetsov's $O(\sqrt {{\delta }x})$ $L^{1}$-error estimate for monotone difference schemes, Math. Comp. Vol 64 (1995), 581-589. MR 95f:65176
  • [TZ] Zhen-Huan Teng and Pingwen Zhang, to appear in SIAM J. Num. Anal.
  • [Yu] Shih-Hsien Yu, Stability of the traveling discrete shock profiles for Lax-Wendroff scheme, IMA preprint (1995).

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Additional Information

Haitao Fan
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057
Email: fan@gumath1.math.georgetown.edu

DOI: https://doi.org/10.1090/S0025-5718-98-00921-1
Received by editor(s): February 9, 1996
Received by editor(s) in revised form: August 19, 1996
Additional Notes: Research supported by NSF Fellowship under Grant DMS-9306064.
Article copyright: © Copyright 1998 American Mathematical Society

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