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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
DOI: https://doi.org/10.1090/S0025-5718-2013-02788-3
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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Additional Information

Zhengfu Xu
Affiliation: Department of Mathematical Science, Michigan Tech University, Houghton, Michigan 49931
Email: zhengfux@mtu.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02788-3
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.