Skip to main content
Springer Nature Link
Log in

High-Order Multi-resolution Central Hermite WENO Schemes for Hyperbolic Conservation Laws

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, a class of high-order multi-resolution central Hermite WENO (C-HWENO) schemes for solving hyperbolic conservation laws is proposed. Formulated in a central finite volume framework on staggered meshes, the methods adopt the multi-resolution HWENO reconstructions (Li et al. in J Comput Phys 446:110653, 2021; Li et al. in Commun Comput Phys 32(2): 364-400, 2022) in space and the natural continuous extension of Runge–Kutta methods in time. Based on the zeroth-order and first-order moments of the solution defined on a series of hierarchical central spatial stencils, the proposed methods are sixth-order while the C-HWENO methods by Tao et al. (J Comput Phys 318:222-251, 2016) are fifth-order in accuracy. The linear weights of such HWENO reconstructions can be any positive numbers as long as their sum equals one, which leads to much simpler implementation and better cost efficiency than the methods by Tao et al. (J Comput Phys 318:222-251, 2016). The first-order moments are modified and the HWENO reconstructions are applied in the troubled-cells, while the linear reconstructions are used for the rest. Meanwhile, our new methods have compact stencils in the reconstructions and require neither numerical fluxes nor flux splitting. Extensive one- and two-dimensional numerical examples are performed to illustrate the accuracy and high resolution of the new C-HWENO schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000)

    Article  MathSciNet  Google Scholar 

  2. Bianco, F., Puppo, G., Russo, G.: High-order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput. 21(1), 294–322 (1999)

    Article  MathSciNet  Google Scholar 

  3. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)

    Article  MathSciNet  Google Scholar 

  4. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  Google Scholar 

  5. Dumbser, M., Boscheri, W., Semplice, M., Russo, G.: Central WENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. SIAM J. Sci. Comput. 39(6), 2564 (2016)

    Article  Google Scholar 

  6. Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150(1), 97–127 (1999)

    Article  MathSciNet  Google Scholar 

  7. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  Google Scholar 

  8. Jiang, G.-S., Tadmor, E.: Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19(6), 1892–1917 (1998)

    Article  MathSciNet  Google Scholar 

  9. Krivodonova, L., Xin, J., Remacle, J.F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3), 323–338 (2004)

    Article  MathSciNet  Google Scholar 

  10. Kurganov, A., Levy, D.: A third-order semidiscrete central scheme for conservation laws and convection–diffusion equations. SIAM J. Sci. Comput. 22(4), 1461–1488 (2000)

    Article  MathSciNet  Google Scholar 

  11. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. Esaim Math. Model. Numer. Anal. 33(3), 547–571 (1999)

    Article  MathSciNet  Google Scholar 

  12. Levy, D., Puppo, G., Russo, G.: A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws. SIAM J. Sci. Comput. 24(2), 480–506 (2002)

    Article  MathSciNet  Google Scholar 

  13. Li, J., Shu, C.-W., Qiu, J.: Multi-resolution HWENO schemes for hyperbolic conservation laws. J. Comput. Phys. 446, 110653 (2021)

    Article  MathSciNet  Google Scholar 

  14. Li, J., Shu, C.-W., Qiu, J.: Moment-based multi-resolution HWENO scheme for hyperbolic conservation laws. Commun. Comput. Phys. 32(2), 364–400 (2022)

    Article  MathSciNet  Google Scholar 

  15. Liu, H., Qiu, J.: Finite difference Hermite WENO schemes for conservation laws. J. Sci. Comput. 63, 548–572 (2015)

    Article  MathSciNet  Google Scholar 

  16. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  Google Scholar 

  17. Liu, X.-D., Tadmor, E.: Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math. 79(3), 397–425 (1998)

    Article  MathSciNet  Google Scholar 

  18. Liu, Y.: Central schemes on overlapping cells. J. Comput. Phys. 209(1), 82–104 (2005)

    Article  MathSciNet  Google Scholar 

  19. Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MathSciNet  Google Scholar 

  20. Qiu, J., Shu, C.-W.: On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes. J. Comput. Phys. 183(1), 187–209 (2002)

    Article  MathSciNet  Google Scholar 

  21. Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 193(1), 115–135 (2004)

    Article  MathSciNet  Google Scholar 

  22. Qiu, J., Shu, C.-W.: A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput. 27(3), 995–1013 (2005)

    Article  MathSciNet  Google Scholar 

  23. Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method II: two dimensional case. Comput. Fluids 34, 642–663 (2005)

    Article  MathSciNet  Google Scholar 

  24. Shi, J., Hu, C., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175(1), 108–127 (2002)

    Article  Google Scholar 

  25. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

    Article  MathSciNet  Google Scholar 

  26. Tao, Z., Li, F., Qiu, J.: High-order central Hermite WENO schemes on staggered meshes for hyperbolic conservation laws. J. Comput. Phys. 281, 148–176 (2015)

    Article  MathSciNet  Google Scholar 

  27. Tao, Z., Li, F., Qiu, J.: High-order central Hermite WENO schemes: dimension-by-dimension moment-based reconstructions. J. Comput. Phys. 318, 222–251 (2016)

    Article  MathSciNet  Google Scholar 

  28. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)

    Article  MathSciNet  Google Scholar 

  29. Zahran, Y.H., Abdalla, A.H.: Seventh order Hermite WENO scheme for hyperbolic conservation laws. Comput. Fluids 131, 66–80 (2016)

    Article  MathSciNet  Google Scholar 

  30. Zennaro, M.: Natural continuous extensions of Runge–Kutta methods. Math. Comput. 46, 119–133 (1986)

    Article  MathSciNet  Google Scholar 

  31. Zhao, Z., Chen, Y., Qiu, J.: A hybrid Hermite WENO scheme for hyperbolic conservation laws. J. Comput. Phys. 405, 109175 (2020)

    Article  MathSciNet  Google Scholar 

  32. Zhao, Z., Qiu, J.: A Hermite WENO scheme with artificial linear weights for hyperbolic conservation laws. J. Comput. Phys. 417, 109583 (2020)

    Article  MathSciNet  Google Scholar 

  33. Zhu, J., Qiu, J.: A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes. Sci. China 51(08), 1549–1560 (2008)

    Article  MathSciNet  Google Scholar 

  34. Zhu, J., Shu, C.-W.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys. 375, 659–683 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Funding

Zhanjing Tao: Research is supported by NSFC Grant 12001231 and Fundamental Research Funds for the Central Universities. Jun Zhu: Research is supported by NSFC Grant 11872210. Jianxian Qiu: Research is supported by NSFC Grant 12071392.

Author information

Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, Jilin, People’s Republic of China

    Zhanjing Tao & Jinming Zhang

  2. State Key Laboratory of Mechanics and Control for Aerospace Structures, Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 210016, Jiangsu, People’s Republic of China

    Jun Zhu

  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

Authors
  1. Zhanjing Tao
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jinming Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Jun Zhu
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Jianxian Qiu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Zhanjing Tao.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tao, Z., Zhang, J., Zhu, J. et al. High-Order Multi-resolution Central Hermite WENO Schemes for Hyperbolic Conservation Laws. J Sci Comput 99, 40 (2024). https://doi.org/10.1007/s10915-024-02499-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02499-0

Keywords