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A Moment-Based Hermite WENO Scheme with Unified Stencils for Hyperbolic Conservation Laws

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Abstract

In this paper, a fifth-order moment-based Hermite weighted essentially non-oscillatory scheme with unified stencils (termed as HWENO-U) is proposed for hyperbolic conservation laws. The main idea of the HWENO-U scheme is to modify the first-order moment by a HWENO limiter only in the time discretizations using the same information of spatial reconstructions, in which the limiter not only overcomes spurious oscillations well, but also ensures the stability of the fully-discrete scheme proved by the von-Neumann analysis. Benefited by this new framework, the HWENO-U scheme involves only a single HWENO reconstruction throughout the entire spatial discretizations, while previous HWENO schemes have to bring additional procedures. Meanwhile, the HWENO-U scheme can use the artificial linear positive weights (the sum is one), but a normalization is made for the original definition of non-linear weights to achieve scale-invariance, which can reduce problem-specific dependencies especially for simulating the problems with sharp scale variations. Compared with previous HWENO schemes, the HWENO-U scheme is simpler and more efficient for utilizing the same candidate stencils, reconstructed polynomials, and nonlinear weights both in the limiter and the spatial reconstruction. Besides, the HWENO-U scheme has more compact stencils, higher resolutions near discontinuities, and smaller numerical errors in smooth regions than a fifth-order WENO scheme with the same framework. Extensive numerical tests are carried out to validate the efficiency, robustness, accuracy, and resolution of the proposed scheme.

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Data availability

All datasets generated during the current study are available from the corresponding author upon reasonable request.

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Funding

This work was partially supported by National Key R &D Program of China [Grant Number 2022YFA1004500], National Natural Science Foundation of China [Grant Number 12401541, 12471390], Postdoctoral Science Foundation of China [Grant Number 2024M751284], and Fundamental Research Funds for the Central Universities [Grant Number 20720240132].

Author information

Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, Guangdong, People’s Republic of China

    Chuan Fan

  2. 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

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

    Zhuang Zhao

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  1. Chuan Fan
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  2. Jianxian Qiu
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Correspondence to Zhuang Zhao.

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The research was partially supported by National Key R & D Program of China [Grant Number 2022YFA1004500], National Natural Science Foundation of China [Grant Number 12401541, 12471390], Postdoctoral Science Foundation of China [Grant Number 2024M751284], and Fundamental Research Funds for the Central Universities [Grant Number 20720240132].

Appendix

Appendix

In the one-dimensional case, the coefficients of the reconstructed polynomials \(\{p_m(x)\}^2_{m=0}\) in (2.9) are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} c_{0,0}= -{\frac{43\,\bar{u}_{i-1}}{384}}+{\frac{235\,\bar{u}_{i}}{192}}-{\frac{43\,\bar{u}_{i+1}}{384}}-{\frac{27\,\bar{v}_{i-1}}{64}}+{\frac{27\,\bar{v}_{i+1}}{64}},\\ c_{0,1}= -{\frac{63\,\bar{u}_{i-1}}{76}}+{\frac{63\,\bar{u}_{i+1}}{76}}-{\frac{75\,\bar{v}_{i-1}}{19}}-{\frac{75\,\bar{v}_{i+1}}{19}},\\ c_{0,2}= {\frac{23\,\bar{u}_{i-1}}{16}}-{\frac{23\,\bar{u}_{i}}{8}}+{\frac{23\,\bar{u}_{i+1}}{16}}+{\frac{45\,\bar{v}_{i-1}}{8}}-{\frac{45\,\bar{v}_{i+1}}{8}},\\ c_{0,3}= {\frac{5\,\bar{u}_{i-1}}{19}}-{\frac{5\,\bar{u}_{i+1}}{19}}+{\frac{60\,\bar{v}_{i-1}}{19}}+{\frac{60\,\bar{v}_{i+1}}{19}},\\ c_{0,4}= -\frac{5\,\bar{u}_{i-1}}{8}+\frac{5\,\bar{u}_{i}}{4}-\frac{5\,\bar{u}_{i+1}}{8}-{\frac{15\,\bar{v}_{i-1}}{4}}+{\frac{15\,\bar{v}_{i+1}}{4}};\\ c_{1,0}= \bar{u}_{i},\ c_{1,1}= \bar{u}_{i}-\bar{u}_{i-1};\\ c_{2,0}= \bar{u}_{i},\ c_{2,1}= \bar{u}_{i+1}-\bar{u}_{i}. \end{array}\right. } \end{aligned}$$

In the two-dimensional case, the coefficients of the reconstructed polynomials \(\{p_m(x,y)\}^4_{m=0}\) in (2.20) are given as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} c_{0,0} = {\frac{{\bar{{u}}_1}}{576}}-{\frac{133\,{\bar{{u}}_2}}{1152}}+{\frac{{\bar{{u}}_3} }{576}}-{\frac{133\,{\bar{{u}}_4}}{1152}}+{\frac{419\,{\bar{{u}}_5}}{288}}-{ \frac{133\,{\bar{{u}}_6}}{1152}}+{\frac{{\bar{{u}}_7}}{576}}-{\frac{133\,{ \bar{{u}}_8}}{1152}}+{\frac{{\bar{{u}}_9}}{576}}-{\frac{27\,{\bar{{v}}_4}}{64}}\\ \quad \quad \,\, +{ \frac{27\,{\bar{{v}}_6}}{64}}-{\frac{27\,{\bar{{w}}_2}}{64}}+{\frac{27\,{\bar{{w}}_8}}{64}}, \\ c_{0,1} =\frac{\bar{{u}}_1}{48}-\frac{\bar{{u}}_3}{48}-{\frac{397\,{\bar{{u}}_4}}{456}}+{\frac{397\,{ \bar{{u}}_6}}{456}}+\frac{\bar{{u}}_7}{48}-\frac{\bar{{u}}_9}{48}-{\frac{75\,{\bar{{v}}_4}}{19}}-{ \frac{75\,{\bar{{v}}_6}}{19}} , \\ c_{0,2} = \frac{\bar{{u}}_1}{48} - \frac{397\,{\bar{{u}}_2}}{456} +\frac{\bar{{u}}_3}{48}-\frac{\bar{{u}}_7}{48}+{ \frac{397\,{\bar{{u}}_8}}{456}}-\frac{\bar{{u}}_9}{48}-{\frac{75\,{\bar{{w}}_2}}{19}}-{ \frac{75\,{\bar{{w}}_8}}{19}} , \\ c_{0,3} = -\frac{\bar{{u}}_1}{48}+\frac{\bar{{u}}_2}{24}-\frac{\bar{{u}}_3}{48}+{\frac{71\,{\bar{{u}}_4}}{48}}-{ \frac{71\,{\bar{{u}}_5}}{24}}+{\frac{71\,{\bar{{u}}_6}}{48}}-\frac{\bar{{u}}_7}{48}+\frac{\bar{{u}}_8}{24} -\frac{\bar{{u}}_9}{48}+{\frac{45\,{\bar{{v}}_4}}{8}}-{\frac{45\,{\bar{{v}}_6}}{8}} , \\ c_{0,4} = -{\frac{7\,{\bar{{u}}_1}}{22}}+{\frac{7\,{\bar{{u}}_3}}{22}}+{\frac{7\,{\bar{{u}}_7}}{22}}-{\frac{7\,{\bar{{u}}_9}}{22}}-{\frac{75\,{\bar{{v}}_2}}{11}}+{\frac{75\,{\bar{{v}}_8}}{11}}-{\frac{75\,{\bar{{w}}_4}}{11}}+{\frac{75\,{\bar{{w}}_6}}{ 11}} , \\ c_{0,5} = -\frac{\bar{{u}}_1}{48}+{\frac{71\,{\bar{{u}}_2}}{48}}-\frac{\bar{{u}}_3}{48}+\frac{\bar{{u}}_4}{24}-{ \frac{71\,{\bar{{u}}_5}}{24}}+\frac{\bar{{u}}_6}{24}-\frac{\bar{{u}}_7}{48}+{\frac{71\,{\bar{{u}}_8} }{48}}-\frac{\bar{{u}}_9}{48}+{\frac{45\,{\bar{{w}}_2}}{8}}-{\frac{45\,{\bar{{w}}_8}}{8}} , \\ c_{0,6} = {\frac{5\,{\bar{{u}}_4}}{19}}-{\frac{5\,{\bar{{u}}_6}}{19}}+{\frac{60\,{\bar{{v}}_4}}{19}}+{\frac{60\,{\bar{{v}}_6}}{19}} , \\ c_{0,7} = -\frac{\bar{{u}}_1}{4}+\frac{\bar{{u}}_2}{2}-\frac{\bar{{u}}_3}{4}+\frac{\bar{{u}}_7}{4}-\frac{\bar{{u}}_8}{2}+\frac{\bar{{u}}_9}{4} , \\ c_{0,8} = -\frac{\bar{{u}}_1}{4}+\frac{\bar{{u}}_3}{4}+\frac{\bar{{u}}_4}{2}-\frac{\bar{{u}}_6}{2}-\frac{\bar{{u}}_7}{4}+\frac{\bar{{u}}_9}{4} , \\ c_{0,9} = {\frac{5\,{\bar{{u}}_2}}{19}}-{\frac{5\,{\bar{{u}}_8}}{19}}+{\frac{60\,{\bar{{w}}_2}}{19}}+{\frac{60\,{\bar{{w}}_8}}{19}} , \\ c_{0,10} = -\frac{5\,{\bar{{u}}_4}}{8}+\frac{5\,{\bar{{u}}_5}}{4}-\frac{5\,{\bar{{u}}_6}}{8}-{\frac{15\,{\bar{{v}}_4}}{4}}+{ \frac{15\,{\bar{{v}}_6}}{4}} , \\ c_{0,11} = {\frac{5\,{\bar{{u}}_1}}{22}}-{\frac{5\,{\bar{{u}}_3}}{22}}-{\frac{5\,{\bar{{u}}_7 }}{22}}+{\frac{5\,{\bar{{u}}_9}}{22}}+{\frac{60\,{\bar{{v}}_2}}{11}}-{\frac{ 60\,{\bar{{v}}_8}}{11}} , \\ c_{0,12} = \frac{\bar{{u}}_1}{4}-\frac{\bar{{u}}_2}{2}+\frac{\bar{{u}}_3}{4}-\frac{\bar{{u}}_4}{2}+{\bar{{u}}_5}-\frac{\bar{{u}}_6}{2}+\frac{\bar{{u}}_7}{4}-\frac{\bar{{u}}_8}{2}+\frac{\bar{{u}}_9}{4} , \\ c_{0,13} ={\frac{5\,{\bar{{u}}_1}}{22}}-{\frac{5\,{\bar{{u}}_3}}{22}}-{\frac{5\,{\bar{{u}}_7 }}{22}}+{\frac{5\,{\bar{{u}}_9}}{22}}+{\frac{60\,{\bar{{w}}_4}}{11}}-{\frac{ 60\,{\bar{{w}}_6}}{11}} , \\ c_{0,14} = -\frac{5\,{\bar{{u}}_2}}{8}+\frac{5\,{\bar{{u}}_5}}{4}-\frac{5\,{\bar{{u}}_8}}{8}-{\frac{15\,{\bar{{w}}_2}}{4}}+{ \frac{15\,{\bar{{w}}_8}}{4}}; \\ c_{1,0}={\bar{{u}}_5}, \ c_{1,1}={\bar{{u}}_5}-{\bar{{u}}_4}, \ c_{1,2}={\bar{{u}}_5}-{\bar{{u}}_2};\\ c_{2,0}={\bar{{u}}_5},\ c_{2,1}={\bar{{u}}_6}-{\bar{{u}}_5},\ c_{2,2}={\bar{{u}}_5}-{\bar{{u}}_2};\\ c_{3,0}={\bar{{u}}_5},\ c_{3,1}={\bar{{u}}_5}-{\bar{{u}}_4},\ c_{3,2}={\bar{{u}}_8}-{\bar{{u}}_5};\\ c_{4,0}={\bar{{u}}_5},\ c_{4,1}={\bar{{u}}_6}-{\bar{{u}}_5},\ c_{4,2}={\bar{{u}}_8}-{\bar{{u}}_5}.\\ \end{array}\right. } \end{aligned}$$

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Fan, C., Qiu, J. & Zhao, Z. A Moment-Based Hermite WENO Scheme with Unified Stencils for Hyperbolic Conservation Laws. J Sci Comput 102, 9 (2025). https://doi.org/10.1007/s10915-024-02732-w

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