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BV estimates of Lax-Friedrichs' scheme for a class of nonlinear hyperbolic conservation laws
Author(s):
Tong
Yang;
Huijiang
Zhao;
Changjiang
Zhu
Journal:
Proc. Amer. Math. Soc.
131
(2003),
1257-1266.
MSC (1991):
Primary 35L60, 35L65, 65N12
Posted:
October 1, 2002
MathSciNet review:
1948118
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Abstract:
We give uniform BV estimates and  -stability of Lax-Friedrichs' scheme for a class of  systems of strictly hyperbolic conservation laws whose integral curves of the eigenvector fields are straight lines, i.e., Temple class, under the assumption of small total variation. This implies that the approximate solutions generated via the Lax-Friedrichs' scheme converge to the solution given by the method of vanishing viscosity or the Godunov scheme, and then the Glimm scheme or the wave front tracking method.
References:
-
- 1.
- S. Bianchini and A. Bressan, A center manifold technique for tracing viscous waves, preprint 2001.
- 2.
- S. Bianchini and A. Bressan, BV solutions for a class of viscous hyperbolic systems, Indiana Univ. Math. J. 49 (4) (2000), 1673-1713. MR 2002c:35175
- 3.
- A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press, 2000. MR 2002d:35002
- 4.
- A. Bressan and H. K. Jenssen, On the convergence of Godunov scheme for nonlinear hyperbolic systems, Chin. Ann. Math. 21B (3) (2000), 269-284. CMP 2001:02
- 5.
- A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, Arch. Rational Mech. Anal. 140 (1997), 301-317. MR 98m:35125
- 6.
- A. Bressan, T.-P. Liu and T. Yang,
 stability estimates for  conservation laws, Arch. Rational Mech. Anal. 149 (1999), 1-22. MR 2000g:35139 - 7.
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27-70. MR 84k:35091
- 8.
- J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR 33:2976
- 9.
- P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 20:176
- 10.
- L.-W. Lin and T. Yang, Convergence of the Lax-Friedrichs' scheme for isentropic gas dynamics in Lagrangian coordinates, Commun. Partial Differential Equations 16 (8 & 9) (1991), 1441-1460. MR 92k:35181
- 11.
- T.-P. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (12) (1999), 1553-1586. MR 2000m:35126
- 12.
- O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2, 26 (1963), 95-172. MR 27:1721
- 13.
- D. Serre, Systems of Conservation Laws, I, II, Cambridge University Press, Cambridge, 2000. MR 2000g:35142, MR 2001c:35146
- 14.
- B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), 781-795. MR 84m:35080
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Additional Information:
Tong
Yang
Affiliation:
Department of Mathematics, City University of Hong Kong, Hong Kong
Email:
matyang@math.cityu.edu.hk
Huijiang
Zhao
Affiliation:
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan, People's Republic of China
Email:
hjzhao@wipm.ac.cn
Changjiang
Zhu
Affiliation:
Laboratory of Nonlinear Analysis and Department of Mathematics, Central China Normal University, Wuhan, People's Republic of China
Email:
cjzhu@ccnu.edu.cn
DOI:
10.1090/S0002-9939-02-06688-1
PII:
S 0002-9939(02)06688-1
Keywords:
Hyperbolic systems of conservation laws,
Lax-Friedrichs' scheme,
BV estimates
Received by editor(s):
December 5, 2001
Posted:
October 1, 2002
Additional Notes:
The first author was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (CityU 1032/98P)
The second author was supported in part by a grant from the National Natural Science Foundation of China (The Youth Foundation) and a grant from the Chinese Academy of Sciences entitled ``Yin Jin Guo Wai Jie Chu Ren Cai Ji Jing".
The third author was supported in part by a grant from the National Natural Science Foundation of China $\#$10171037 and two grants from the Ministry of Education of China entitled ``Liu Xue Hui Guo Ren Yuan Ji Jing" and ``Gao Deng Xue Xiao Zhong Dian Shi Yang Shi Ji Jing", respectively.
Communicated by:
Suncica Canic
Copyright of article:
Copyright
2002,
American Mathematical Society
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