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Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem

Author: Zhengfu Xu
Journal: Math. Comp. 83 (2014), 2213-2238
MSC (2010): Primary 65M08, 65M06
Published electronically: December 18, 2013
MathSciNet review: 3223330
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Abstract: In this paper, we present a class of parametrized limiters used to achieve strict maximum principle for high order numerical schemes applied to hyperbolic conservation laws computation. By decoupling a sequence of parameters embedded in a group of explicit inequalities, the numerical fluxes are locally redefined in consistent and conservative formulation. We will show that the global maximum principle can be preserved while the high order accuracy of the underlying scheme is maintained. The parametrized limiters are less restrictive on the CFL number when applied to high order finite volume scheme. The less restrictive limiters allow for the development of the high order finite difference scheme which preserves the maximum principle. Within the proposed parametrized limiters framework, a successive sequence of limiters are designed to allow for significantly large CFL number by relaxing the limits on the intermediate values of the multistage Runge-Kutta method. Numerical results and preliminary analysis for linear and nonlinear scalar problems are presented to support the claim. The parametrized limiters are applied to the numerical fluxes directly. There is no increased complexity to apply the parametrized limiters to different kinds of monotone numerical fluxes.

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  • [1] Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357-393. MR 701178 (84g:65115),
  • [2] 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 (90a:65199),
  • [3] Xu-Dong Liu and Stanley Osher, Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes. I, SIAM J. Numer. Anal. 33 (1996), no. 2, 760-779. MR 1388497 (97h:65110),
  • [4] Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217-235. MR 736327 (86d:65119),
  • [5] Stanley Osher and Sukumar Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), no. 5, 955-984. MR 760626 (86a:65086),
  • [6] A. M. Rogerson, E. Meiburg, A numerical study of the convergence properties of ENO schemes, Journal of Scientific Computing, 5 (1990), 151-167.
  • [7] Richard Sanders, A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws, Math. Comp. 51 (1988), no. 184, 535-558. MR 935073 (89c:65104),
  • [8] Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439-471. MR 954915 (89g:65113),
  • [9] Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32-78. MR 1010162 (90i:65167),
  • [10] 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 (85m:65085),
  • [11] Zhengfu Xu and Chi-Wang Shu, Anti-diffusive flux corrections for high order finite difference WENO schemes, J. Comput. Phys. 205 (2005), no. 2, 458-485. MR 2134990 (2005m:65186),
  • [12] Xiangxiong Zhang and Chi-Wang Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys. 229 (2010), no. 9, 3091-3120. MR 2601091 (2010k:65181),
  • [13] Xiangxiong Zhang and Chi-Wang Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys. 229 (2010), no. 23, 8918-8934. MR 2725380 (2012c:76066),
  • [14] Xiangxiong Zhang and Chi-Wang Shu, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM J. Numer. Anal. 48 (2010), no. 2, 772-795. MR 2670004 (2012b:65126),
  • [15] Xiangxiong Zhang and Chi-Wang Shu, Positivity-preserving high order finite difference WENO schemes for compressible Euler equations, J. Comput. Phys. 231 (2012), no. 5, 2245-2258. MR 2876636,

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Additional Information

Zhengfu Xu
Affiliation: Department of Mathematical Science, Michigan Tech University, Houghton, Michigan 49931

Keywords: Hyperbolic conservation laws, maximum principle preserving, parametrized flux limiters, high order scheme
Received by editor(s): July 27, 2012
Received by editor(s) in revised form: November 14, 2012, and January 5, 2013
Published electronically: December 18, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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