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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Large-time behavior of solutions of Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws

Author: I-Liang Chern
Journal: Math. Comp. 56 (1991), 107-118
MSC: Primary 65M12; Secondary 35L65, 39A12, 76L05
MathSciNet review: 1052088
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Abstract: We study the large-time behavior of discrete solutions of the Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws. The initial data considered here are small and tend to a constant state at $ x = \pm \infty $. We show that the solutions tend to the discrete diffusion waves at the rate $ O({t^{ - 3/4 + 1/2p + \sigma }})$ in $ {l^p}$, $ 1 \leq p \leq \infty $, with $ \sigma > 0$ being an arbitrarily small constant. The discrete diffusion waves can be constructed from the self-similar solutions of the heat equation and the Burgers equation through an averaging process.

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  • [1] I-Liang Chern and Tai-Ping Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), no. 3, 503–517. MR 891950 (88g:35127)
  • [2] I-Liang Chern and Tai-Ping Liu, Erratum: “Convergence to diffusion waves of solutions for viscous conservation laws” [Comm.\ Math.\ Phys.\ {110} (1987), no. 3, 503–517; MR0891950 (88g:35127)], Comm. Math. Phys. 120 (1989), no. 3, 525–527. MR 981217 (90a:35138)
  • [3] I.-L. Chern, Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws, preprint, MCS-P134-0290, Math. Comp. Div., Argonno National Lab., 1990.
  • [4] Eberhard Hopf, The partial differential equation 𝑢_{𝑡}+𝑢𝑢ₓ=𝜇𝑢ₓₓ, Comm. Pure Appl. Math. 3 (1950), 201–230. MR 0047234 (13,846c)
  • [5] S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral thesis, Kyoto University, 1983.
  • [6] Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951 (89d:35022),
  • [7] Tai-Ping Liu, Pointwise convergence to 𝑁-waves for solutions of hyperbolic conservation laws, Bull. Inst. Math. Acad. Sinica 15 (1987), no. 1, 1–17. MR 947772 (89h:35202)
  • [8] Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), no. 1, 67–104. MR 564670 (81g:35108)

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Additional Information

PII: S 0025-5718(1991)1052088-8
Keywords: Lax-Friedrichs scheme, hyperbolic systems of conservation laws, discrete diffusion waves, asymptotic behavior, numerical viscosity
Article copyright: © Copyright 1991 American Mathematical Society