A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws
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A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws Zhuang Zhao1 · Yong-Tao Zhang2 · Jianxian Qiu3 Received: 28 June 2020 / Revised: 6 September 2020 / Accepted: 12 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 twodimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed novel HWENO scheme. Keywords Hermite WENO scheme · Finite difference method · Hyperbolic conservation laws · Modification for derivative · Hermite interpolation Mathematics Subject Classification 65M60 · 35L65
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). Extended author information available on the last page of the article 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 85:29
1 Introduction In this paper, we propose a modified fifth order Hermite weighted essentially nonoscillatory (HWENO) scheme in the finite difference framework for one-dimensional and two-dimensional problems. HWENO scheme was derived from essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) schemes, and they have been widely applied for solving nonlinear hyperbolic conservation laws in recent decades. As it is known, the solution of the nonlinear hyperbolic conserv
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