Decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws
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- by Hailiang Liu and Jinghua Wang PDF
- Math. Comp. 66 (1997), 69-84 Request permission
Abstract:
This paper is to study the decay rate for perturbations of stationary discrete shocks for the Lax-Friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method. If the summation of the initial perturbation over $(-\infty , j)$ is small and decays at the algebraic rate as $|j|\rightarrow \infty$, then the solution approaches the stationary discrete shock profiles at the corresponding rate as $n\rightarrow \infty$. This rate seems to be almost optimal compared with the rate in the continuous counterpart. Proofs are given by applying the weighted energy integration method to the integrated scheme of the original one. The selection of the time-dependent discrete weight function plays a crucial role in this procedure.References
- I-Liang Chern, Large-time behavior of solutions of Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws, Math. Comp. 56 (1991), no. 193, 107–118. MR 1052088, DOI 10.1090/S0025-5718-1991-1052088-8
- Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI 10.1090/S0025-5718-1980-0551288-3
- Björn Engquist and Stanley Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), no. 154, 321–351. MR 606500, DOI 10.1090/S0025-5718-1981-0606500-X
- Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI 10.1007/BF00276840
- Eduard Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527–536. MR 929127
- A. M. Il′in and O. A. Oleĭnik, Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. (N.S.) 51 (93) (1960), 191–216 (Russian). MR 0120469
- Gray Jennings, Discrete shocks, Comm. Pure Appl. Math. 27 (1974), 25–37. MR 338594, DOI 10.1002/cpa.3160270103
- Christopher K. R. T. Jones, Robert Gardner, and Todd Kapitula, Stability of travelling waves for nonconvex scalar viscous conservation laws, Comm. Pure Appl. Math. 46 (1993), no. 4, 505–526. MR 1211740, DOI 10.1002/cpa.3160460404
- Shuichi Kawashima and Akitaka Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97–127. MR 814544
- Shuichi Kawashima and Akitaka Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math. 47 (1994), no. 12, 1547–1569. MR 1303220, DOI 10.1002/cpa.3160471202
- H.L. Liu, Asymptotic stability of shock profiles for non-convex convection-diffusion equation, (preprint)(1994).
- Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, DOI 10.1090/memo/0328
- H.L. Liu and J.H. Wang, Nonlinear stability of stationary discrete shocks for non-convex scalar conservation laws, Math. Comp. 65(1996), 1137–1153.
- Jian-Guo Liu and Zhou Ping Xin, Nonlinear stability of discrete shocks for systems of conservation laws, Arch. Rational Mech. Anal. 125 (1993), no. 3, 217–256. MR 1245071, DOI 10.1007/BF00383220
- Jian-Guo Liu and Zhou Ping Xin, $L^1$-stability of stationary discrete shocks, Math. Comp. 60 (1993), no. 201, 233–244. MR 1159170, DOI 10.1090/S0025-5718-1993-1159170-7
- Andrew Majda and James Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math. 32 (1979), no. 4, 445–482. MR 528630, DOI 10.1002/cpa.3160320402
- Akitaka Matsumura and Kenji Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 2 (1985), no. 1, 17–25. MR 839317, DOI 10.1007/BF03167036
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- Stanley Osher and James Ralston, $L^{1}$ stability of travelling waves with applications to convective porous media flow, Comm. Pure Appl. Math. 35 (1982), no. 6, 737–749. MR 673828, DOI 10.1002/cpa.3160350602
- D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
- Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI 10.1090/S0025-5718-1983-0679435-6
- D. Serre, $L^1$ stabilty of viscous shock profile for scalar non-convex conservation law, private communication(1994).
- Yiorgos Sokratis Smyrlis, Existence and stability of stationary profiles of the LW scheme, Comm. Pure Appl. Math. 43 (1990), no. 4, 509–545. MR 1047334, DOI 10.1002/cpa.3160430405
- Anders Szepessy and Zhou Ping Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993), no. 1, 53–103. MR 1207241, DOI 10.1007/BF01816555
- Eitan Tadmor, The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme, Math. Comp. 43 (1984), no. 168, 353–368. MR 758188, DOI 10.1090/S0025-5718-1984-0758188-8
Additional Information
- Hailiang Liu
- Affiliation: Department of Mathematics, Henan Normal University, Xinxiang, 453002, P. R. China
- Email: guozm@sun.ihep.ac.cn
- Jinghua Wang
- Affiliation: Institute of Systems Science, Academia Sinica, Beijing 100080, P. R. China
- Email: jwang@iss06.iss.ac.cn
- Received by editor(s): May 19, 1995
- Received by editor(s) in revised form: January 31, 1996
- Additional Notes: The first author was supported in part by the National Natural Science Foundation of China and by the Institute of Mathematics, Academia Sinica.
The second author was supported in part by the National Natural Science Foundation of China and by The Texs Coordinating Board for Higher Education, Advanced Research Program. - © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 69-84
- MSC (1991): Primary 39A11; Secondary 35L65
- DOI: https://doi.org/10.1090/S0025-5718-97-00804-1
- MathSciNet review: 1377663