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Parametrized Positivity Preserving Flux Limiters for the High Order Finite Difference WENO Scheme Solving Compressible Euler Equations

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Abstract

In this paper, we develop parametrized positivity satisfying flux limiters for the high order finite difference Runge–Kutta weighted essentially non-oscillatory scheme solving compressible Euler equations to maintain positive density and pressure. Negative density and pressure, which often leads to simulation blow-ups or nonphysical solutions, emerges from many high resolution computations in some extreme cases. The methodology we propose in this paper is a nontrivial generalization of the parametrized maximum principle preserving flux limiters for high order finite difference schemes solving scalar hyperbolic conservation laws (Liang and Xu in J Sci Comput 58:41–60, 2014; Xiong et al. in J Comput Phys 252:310–331, 2013; Xu in Math Comput 83:2213–2238, 2014). To preserve the maximum principle, the high order flux is limited towards a first order monotone flux, where the limiting procedures are designed by decoupling linear maximum principle constraints. High order schemes with such flux limiters are shown to preserve the high order accuracy via local truncation error analysis and by extensive numerical experiments with mild CFL constraints. The parametrized flux limiting approach is generalized to the Euler system to preserve the positivity of density and pressure of numerical solutions via decoupling some nonlinear constraints. Compared with existing high order positivity preserving approaches (Zhang and Shu in Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011; J Comput Phys 230:1238–1248, 2011; J Comput Phys 231:2245–2258, 2012), our proposed algorithm is positivity preserving by the design; it is computationally efficient and maintains high order spatial and temporal accuracy in our extensive numerical tests. Numerical tests are performed to demonstrate the efficiency and effectiveness of the proposed new algorithm.

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Acknowledgments

Tao Xiong and Jing-Mei Qiu are supported by Air Force Office of Scientific Computing YIP Grant FA9550-12-0318, NSF DMS-1217008 and DMS-152277. Zhengfu Xu is supported by NSF Grant DMS-1316662. We would like to thank Xiangxiong Zhang from Purdue University for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Tao Xiong & Jing-Mei Qiu

  2. Department of Mathematical Science, Michigan Tech University, Houghton, MI, 49931, USA

    Zhengfu Xu

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  1. Tao Xiong
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  3. Zhengfu Xu
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Correspondence to Zhengfu Xu.

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Xiong, T., Qiu, JM. & Xu, Z. Parametrized Positivity Preserving Flux Limiters for the High Order Finite Difference WENO Scheme Solving Compressible Euler Equations. J Sci Comput 67, 1066–1088 (2016). https://doi.org/10.1007/s10915-015-0118-0

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