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A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws

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Abstract

In this paper, we develop a modified fifth order accuracy finite difference Hermite WENO (HWENO) scheme for solving hyperbolic conservation laws. The main idea is that we first modify the derivatives of the solution by Hermite WENO interpolations, then we discretize the original and derivative equations in the spatial directions by the same approximation polynomials. Comparing with the original finite difference HWENO scheme of Liu and Qiu (J Sci Comput 63:548–572, 2015), one of the advantages is that the modified HWENO scheme is more robust than the original one since we do not need to use the additional positivity-preserving flux limiter methodology, and larger CFL number can be applied. Another advantage is that higher order numerical accuracy than the original scheme can be achieved for two-dimensional problems under the condition of using the same approximation stencil and information. Furthermore, the modified scheme preserves the nice property of compactness shared by HWENO schemes, i.e., only immediate neighbor information is needed in the reconstruction, and it has smaller numerical errors and higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu (J Comput Phys 126:202–228, 1996). Various benchmark numerical tests of both one-dimensional and two-dimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed novel HWENO scheme.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Zhuang Zhao

  2. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Yong-Tao Zhang

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianxian Qiu

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  1. Zhuang Zhao
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  2. Yong-Tao Zhang
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  3. Jianxian Qiu
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Correspondence to Jianxian Qiu.

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Z. Zhao and J. Qiu were supported partly by Science Challenge Project (China), No. TZ 2016002 and National Natural Science Foundation-Joint Fund (China) Grant U1630247. Y.-T. Zhang was partially supported by NSF Grant DMS-1620108 (USA). This work was carried out while Z. Zhao was visiting the Department of Applied and Computational Mathematics and Statistics, the University of Notre Dame under the support by the China Scholarship Council (CSC: 201906310075).

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Zhao, Z., Zhang, YT. & Qiu, J. A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws. J Sci Comput 85, 29 (2020). https://doi.org/10.1007/s10915-020-01347-1

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  • DOI: https://doi.org/10.1007/s10915-020-01347-1

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