Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Order of convergence of second order schemes based on the minmod limiter


Authors: Bojan Popov and Ognian Trifonov
Journal: Math. Comp. 75 (2006), 1735-1753
MSC (2000): Primary 65M15; Secondary 65M12
DOI: https://doi.org/10.1090/S0025-5718-06-01875-8
Published electronically: May 23, 2006
MathSciNet review: 2240633
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu-Tadmor scheme. It is well known that the $ L_p$-error of monotone finite difference methods for the linear advection equation is of order $ 1/2$ for initial data in $ W^1(L_p)$, $ 1\leq p\leq \infty$. For second or higher order nonoscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the $ L_2$-error for a class of second order schemes based on the minmod limiter is of order at least $ 5/8$ in contrast to the $ 1/2$ order for any formally first order scheme.


References [Enhancements On Off] (What's this?)

  • 1. Y. Brenier and S. Osher,
    The one-sided Lipschitz condition for convex scalar conservation laws,
    SIAM J. Numer. Anal., 25: 8-23, 1988. MR 0923922 (89a:65134)
  • 2. P. Brenner, V. Thomée and L. B. Wahlbin,
    Besov spaces and applications to difference methods for initial value problems, (A. Dold and B. Eckmann, eds.)
    Lecture Notes in Math., vol. 434, Springer-Verlag, Berlin and New York, 1975. MR 0461121 (57:1106)
  • 3. R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. MR 1261635 (95f:41001)
  • 4. J. Goodman and R. LeVeque,
    A geometric approach to high resolution TVD schemes,
    SIAM J. Numer. Anal., 25: 268-284, 1988.MR 0933724 (89c:65097)
  • 5. A. Harten and S. Osher,
    Uniformly high order accurate non-oscillatory schemes, I,
    SIAM J. Numer. Anal. 24: 279-309, 1987. MR 0881365 (90a:65198)
  • 6. A. Harten, B. Enquist, S. Osher and S.R. Chakravarthy,
    Uniformly high order accurate essentially non-oscillatory schemes, III,
    J. Comp. Phys., 71: 231-303, 1987. MR 0897244 (90a:65199)
  • 7. G.-S. Jiang, D. Levi, C.-T. Lin, S. Osher and E. Tadmor,
    High-resolution non-oscillatory central schemes with nonstaggered grids for hyperbolic conservation laws,
    SIAM J. Numer. Anal., 35: 2147-2169, 1998. MR 1655841 (99j:65145)
  • 8. G.-S. Jiang and E. Tadmor,
    Nonoscillatory central schemes for hyperbolic conservation laws,
    SIAM J. Sci. Comput., 19: 1892-1917, 1998. MR 1638064 (99f:65128)
  • 9. Yu. V. Kryakin,
    On the theorem of H. Whitney in spaces $ L^p$, $ 1\leq p\leq\infty$,
    Mathematica Balkanika, New series, Vol. 4, Fasc. 3: 258-270, 1990.MR 1169221 (93i:41002)
  • 10. P. Lax and B. Wendroff,
    Systems of conservation laws,
    Comm. Pure Appl. Math., 13: 217-237, 1960. MR 0120774 (22:11523)
  • 11. S. Konyagin, B. Popov and O. Trifonov,
    On Convergence of Minmod-Type Schemes,
    SIAM J. Numer. Anal., 42: 1978-1997, 2005. MR 2139233 (2006b:65129)
  • 12. S.N. Kruzhkov,
    First order quasi-linear equations in several independent variables,
    Math. USSR Sbornik, 10: 217-243, 1970.
  • 13. A. Kurganov and E. Tadmor,
    New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations,
    J. Comp. Phys., 160: 241-282, 2000. MR 1756766 (2001d:65135)
  • 14. H. Nessyahu and E. Tadmor,
    Non-oscillatory central differencing for hyperbolic conservation laws,
    J. Comp. Phys., 87: 408-463, 1990. MR 1047564 (91i:65157)
  • 15. F. Sabac,
    The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws,
    SIAM J. Numer. Anal., 34: 2306-2318, 1997. MR 1480382 (98j:65064)
  • 16. C.-W. Shu,
    Numerical experiments on the accuracy of ENO and modified ENO schemes,
    J. Comp. Phys., 5: 127-149, 1990.
  • 17. P. Sweby,
    High resolution schemes using flux limiters for hyperbolic conservation laws,
    SIAM J. Numer. Anal., 21: 995-1011, 1984. MR 0760628 (85m:65085)
  • 18. T. Tang and Z.-H. Teng,
    The sharpness of Kuznetsov's $ O(\sqrt{\Delta x})$ $ L^1$-errror estimate for monotone difference scheme,
    Math. Comp., 64: 581-589, 1995. MR 1270625 (95f:65176)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M15, 65M12

Retrieve articles in all journals with MSC (2000): 65M15, 65M12


Additional Information

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

Ognian Trifonov
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: trifonov@math.sc.edu

DOI: https://doi.org/10.1090/S0025-5718-06-01875-8
Keywords: Conservation laws, error estimates, second order schemes, minmod limiter
Received by editor(s): April 22, 2004
Received by editor(s) in revised form: July 6, 2005
Published electronically: May 23, 2006
Additional Notes: The first author was supported in part by NSF DMS Grant #0510650.
The second author was supported in part by NSF DMS Grant #9970455.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society