Order of convergence of second order schemes based on the minmod limiter
HTML articles powered by AMS MathViewer
- by Bojan Popov and Ognian Trifonov PDF
- Math. Comp. 75 (2006), 1735-1753 Request permission
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
- Yann Brenier and Stanley Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal. 25 (1988), no. 1, 8–23. MR 923922, DOI 10.1137/0725002
- Philip Brenner, Vidar Thomée, and Lars B. Wahlbin, Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, Vol. 434, Springer-Verlag, Berlin-New York, 1975. MR 0461121, DOI 10.1007/BFb0068125
- Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635, DOI 10.1007/978-3-662-02888-9
- Jonathan B. Goodman and Randall J. LeVeque, A geometric approach to high resolution TVD schemes, SIAM J. Numer. Anal. 25 (1988), no. 2, 268–284. MR 933724, DOI 10.1137/0725019
- Ami Harten and Stanley Osher, Uniformly high-order accurate nonoscillatory schemes. I, SIAM J. Numer. Anal. 24 (1987), no. 2, 279–309. MR 881365, DOI 10.1137/0724022
- Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231–303. MR 897244, DOI 10.1016/0021-9991(87)90031-3
- G.-S. Jiang, D. Levy, 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, 2147–2168. MR 1655841, DOI 10.1137/S0036142997317560
- Guang-Shan Jiang and Eitan Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 19 (1998), no. 6, 1892–1917. MR 1638064, DOI 10.1137/S106482759631041X
- Yu. V. Kryakin, On the theorem of H. Whitney in spaces $L^p,\;1\leq p\leq \infty$, Math. Balkanica (N.S.) 4 (1990), no. 3, 258–271 (1991). MR 1169221
- Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
- Sergei Konyagin, Bojan Popov, and Ognian Trifonov, On convergence of minmod-type schemes, SIAM J. Numer. Anal. 42 (2005), no. 5, 1978–1997. MR 2139233, DOI 10.1137/S0036142903423861
- S.N. Kruzhkov, First order quasi-linear equations in several independent variables, Math. USSR Sbornik, 10: 217–243, 1970.
- Alexander Kurganov and Eitan Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000), no. 1, 241–282. MR 1756766, DOI 10.1006/jcph.2000.6459
- Haim Nessyahu and Eitan Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 2, 408–463. MR 1047564, DOI 10.1016/0021-9991(90)90260-8
- Florin Şabac, The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws, SIAM J. Numer. Anal. 34 (1997), no. 6, 2306–2318. MR 1480382, DOI 10.1137/S003614299529347X
- C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Comp. Phys., 5: 127–149, 1990.
- P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), no. 5, 995–1011. MR 760628, DOI 10.1137/0721062
- T. Tang and Zhen Huan Teng, The sharpness of Kuznetsov’s $O(\sqrt {\Delta x})\ L^1$-error estimate for monotone difference schemes, Math. Comp. 64 (1995), no. 210, 581–589. MR 1270625, DOI 10.1090/S0025-5718-1995-1270625-9
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
- 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. - © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 75 (2006), 1735-1753
- MSC (2000): Primary 65M15; Secondary 65M12
- DOI: https://doi.org/10.1090/S0025-5718-06-01875-8
- MathSciNet review: 2240633