Maximum principle and convergence of central schemes based on slope limiters
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- by Orhan Mehmetoglu and Bojan Popov PDF
- Math. Comp. 81 (2012), 219-231 Request permission
Abstract:
A maximum principle and convergence of second order central schemes is proven for scalar conservation laws in dimension one. It is well known that to establish a maximum principle a nonlinear piecewise linear reconstruction is needed and a typical choice is the minmod limiter. Unfortunately, this implies that the scheme uses a first order reconstruction at local extrema. The novelty here is that we allow local nonlinear reconstructions which do not reduce to first order at local extrema and still prove maximum principle and convergence.References
- Paul Arminjon and Marie-Claude Viallon, Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d’espace, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 1, 85–88 (French, with English and French summaries). MR 1320837
- Ivan Christov and Bojan Popov, New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws, J. Comput. Phys. 227 (2008), no. 11, 5736–5757. MR 2414928, DOI 10.1016/j.jcp.2008.02.007
- 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
- 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
- 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
- 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
- Stanley Osher and Eitan Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), no. 181, 19–51. MR 917817, DOI 10.1090/S0025-5718-1988-0917817-X
- Bojan Popov and Ognian Trifonov, One-sided stability and convergence of the Nessyahu-Tadmor scheme, Numer. Math. 104 (2006), no. 4, 539–559. MR 2249677, DOI 10.1007/s00211-006-0015-4
Additional Information
- Orhan Mehmetoglu
- Affiliation: Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, Texas 77843
- Email: omehmet@math.tamu.edu
- Bojan Popov
- Affiliation: Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, Texas 77843
- Email: popov@math.tamu.edu
- Received by editor(s): May 13, 2010
- Received by editor(s) in revised form: September 9, 2010, and December 3, 2010
- Published electronically: May 16, 2011
- Additional Notes: This material is based on work supported by the National Science Foundation grant DMS-0811041. This publication is based on work partially supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST)
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 219-231
- MSC (2010): Primary 65M12; Secondary 65M08
- DOI: https://doi.org/10.1090/S0025-5718-2011-02514-7
- MathSciNet review: 2833493