Skip to main content
SpringerLink
Log in

Modified Stencil Approximations for Fifth-Order Weighted Essentially Non-oscillatory Schemes

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

Abstract

In this paper, a modified fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme is presented. The quadratic polynomial approximation of numerical flux on each candidate stencil of the traditional WENO-JS scheme is modified by adding a form of cubic terms such that the resulting stencil approximation achieves fourth-order accuracy. And the corresponding smoothness indicators are calculated. The modified candidate fluxes and local smoothness indicators, when used in the WENO-JS scheme, can make the resulting new scheme (called WENO-MS) achieve fifth-order convergence in smooth regions including first-order critical points. A series of one- and two-dimensional numerical examples are presented to demonstrate the performance of the new scheme. The numerical results show that the proposed WENO-MS scheme provides a comparable or higher resolution of fine structures compared with the WENO-M, WENO-Z and P-WENO schemes, while it increases only 7% of CPU time compared with the traditional WENO-JS scheme.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

    MathSciNet  MATH  Google Scholar 

  2. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high-order accurate non-oscillatory schemes III. J. Comput. Phys. 71, 231–303 (1987)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted-essentially-non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Ha, Y., Kim, C.H., Lee, Y.J., Yoon, J.: An improved weighted essentially non-oscillatory scheme with a new smoothness indicator. J. Comput. Phys. 232, 68–86 (2013). https://doi.org/10.1016/j.jcp.2012.06.016

    Article  MathSciNet  MATH  Google Scholar 

  8. Kim, C.H., Ha, Y., Yoon, J.: Modified non-linear weights for fifth-order weighted essentially non-oscillatory schemes. J. Sci. Comput. 67, 299–323 (2016). https://doi.org/10.1007/s10915-015-0079-3

    Article  MathSciNet  MATH  Google Scholar 

  9. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys 230, 1766–92 (2011)

    Article  MathSciNet  Google Scholar 

  10. Arandiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49, 893–915 (2011)

    Article  MathSciNet  Google Scholar 

  11. Don, W.S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–72 (2013)

    Article  MathSciNet  Google Scholar 

  12. Zeng, F.J., Shen, Y.Q., Liu, S.P.: A perturbational weighted essentially non-oscillatory scheme. Comput. Fluids 172, 196–208 (2018). https://doi.org/10.1016/j.compfluid.2018.07.003

    Article  MathSciNet  MATH  Google Scholar 

  13. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  Google Scholar 

  14. Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA/CR–97–206253, ICASE report no. 97–65

  15. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  Google Scholar 

  16. Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)

    Article  MathSciNet  Google Scholar 

  17. Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)

    Article  MathSciNet  Google Scholar 

  18. 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 

  19. Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J. Comput. Phys. 178, 81–117 (2002)

    MathSciNet  MATH  Google Scholar 

  20. Schulz-Rinne, C.W., Collins, J.P., Glaz, H.M.: Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput. 14(6), 1394–1414 (1993)

    Article  MathSciNet  Google Scholar 

  21. Glimm, J., Grove, J., Li, X.L., Oh, W., Tan, D.C.: The dynamics of bubble growth for Rayleigh–Taylor unstable interfaces. Phys. Fluids 31, 447–465 (1988)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by Natural Science Foundation of China (11261160486, 91641107, 91852116), Fundamental Research of Civil Aircraft (MJ-F-2012-04).

Author information

Authors and Affiliations

  1. ICMSEC and LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yahui Wang, Kunlei Zhao & Li Yuan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yahui Wang, Kunlei Zhao & Li Yuan

  3. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, People’s Republic of China

    Yulong Du

Authors
  1. Yahui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yulong Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kunlei Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Li Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Yuan.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Du, Y., Zhao, K. et al. Modified Stencil Approximations for Fifth-Order Weighted Essentially Non-oscillatory Schemes. J Sci Comput 81, 898–922 (2019). https://doi.org/10.1007/s10915-019-01042-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01042-w

Keywords

Mathematics Subject Classification

Search

Navigation