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Weakly nonoscillatory schemes for scalar conservation laws


Authors: Kirill Kopotun, Marian Neamtu and Bojan Popov
Journal: Math. Comp. 72 (2003), 1747-1767
MSC (2000): Primary 65M15; Secondary 35L65, 35B05, 35B30
DOI: https://doi.org/10.1090/S0025-5718-03-01524-2
Published electronically: April 29, 2003
MathSciNet review: 1986803
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Abstract: A new class of Godunov-type numerical methods (called here weakly nonoscillatory or WNO) for solving nonlinear scalar conservation laws in one space dimension is introduced. This new class generalizes the classical nonoscillatory schemes. In particular, it contains modified versions of Min-Mod and UNO. Under certain conditions, convergence and error estimates for WNO methods are proved.


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  • 1. F. Bouchut, Ch. Bourdarias and B. Perthame, A MUSCL method satisfying all entropy inequalities, Math. Comp. 65 (1996), 1439-1461. MR 97a:65080
  • 2. F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2847-2870. MR 98m:65156
  • 3. Y. Brenier and S. Osher, The one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal. 25 (1988), 8-23. MR 89a:65134
  • 4. P. Colella and P. Woodward, The piecewise parabolic method for gas-dynamical simulations, J. Comput. Phys. 54 (1984), 174-201.
  • 5. F. Coquel and P. LeFloch, An entropy satisfying MUSCL scheme for systems of conservation laws, Numer. Math. 74 (1996), 1-33. MR 97g:65179
  • 6. R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. MR 95f:41001
  • 7. A. Harten and S. Osher, Uniformly high order accurate nonoscillatory schemes, I, SIAM J. Numer. Anal. 24 (1987), no. 2, 279-309. MR 90a:65198
  • 8. A. Harten, B. Enquist, S. Osher and S.R. Chakravarthy, Uniformly high order accurate essentially nonoscillatory schemes, III, J. Comput. Phys. 71 (1987), no. 2, 231-303. MR 90a:65199
  • 9. G.-S. Jiang, C.-T. Lin, S. Osher and E. Tadmor High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal. 35 (1998), no. 6, 2147-2169. MR 99j:65145
  • 10. G.-S. Jiang and E. Tadmor Nonoscillatory central schemes for hyperbolic conservation laws, SIAM J. Sci. Comput. 19 (1998), no. 6, 1892-1917. MR 99f:65128
  • 11. S.N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik 10 (1970), no. 2, 217-243. MR 42:2159
  • 12. N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first order quasi-linear equations, USSR Comput. Math. and Math. Phys. 16 (1976), no. 6, 105-119. MR 58:3510
  • 13. P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217-237. MR 22:11523
  • 14. X. Liu and S. Osher, Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes, I, SIAM J. Numer. Anal. 33 (1996), 760-779. MR 97h:65110
  • 15. X. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. Math. 79 (1997), 397-425. MR 99h:65149
  • 16. B. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), no. 6, 1074-1081. MR 88a:65104
  • 17. H. Nessyahu and E. Tadmor, The convergence rate of nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 1505-1519. MR 93j:65139
  • 18. S. Osher and E. Tadmor, On the convergence rate of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), 19-51. MR 89m:65086
  • 19. R. Rockafellar and R. Wets, Variational analysis, Springer-Verlag, Berlin, 1998. MR 98m:49001
  • 20. C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput. 5 (1990), no. 2, 127-149.
  • 21. E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891-906. MR 92d:35190
  • 22. -, Approximate solutions of Nonlinear Conservation Laws and Related Equations, Proc. Sympos. Appl. Math., 54 (1998), 325-368. MR 99c:35149
  • 23. W. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, Springer-Verlag, 1989. MR 91e:46046

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Additional Information

Kirill Kopotun
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2 Canada
Email: kopotunk@cc.umanitoba.ca

Marian Neamtu
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: neamtu@math.vanderbilt.edu

Bojan Popov
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77845
Email: popov@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-03-01524-2
Keywords: Scalar conservation laws, Godunov-type schemes, error estimates, weak nonoscillation, relaxed entropic projection
Received by editor(s): March 1, 2001
Received by editor(s) in revised form: January 28, 2002
Published electronically: April 29, 2003
Additional Notes: The first author was supported by NSERC of Canada and by NSF of USA under grant DMS-9705638
The second author was supported by NSF under grant DMS-9803501
The third author was supported by the ONR Grant No. N00014-91-J-1076
Article copyright: © Copyright 2003 American Mathematical Society

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