An Improvement of Third Order WENO Scheme for Convergence Rate at Critical Points with New Non-linear Weights

Abstract

In this paper, we construct and implement a new improvement of third order weighted essentially non-oscillatory (WENO) scheme in the finite difference framework for hyperbolic conservation laws. In our approach, a modification in the global smoothness measurement is reported by applying all three points on global stencil \((i-1,i,i+1)\) which is used for convergence of non-linear weights towards the optimal weights at critical points and achieves the desired order of accuracy for third order WENO scheme. We use the third order accurate total variation diminishing (TVD) Runge-Kutta time stepping method. The major advantage of the proposed scheme is its better numerical accuracy in smooth regions. The computational performance of the proposed WENO scheme with this global smoothness measurement is verified in several benchmark one- and two-dimensional test cases for scalar and vector hyperbolic equations. Extensive computational results confirm that the new proposed scheme achieves better performance as compared with WENO-JS3, WENO-Z3 and WENO-F3 schemes.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  1. 1.

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

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Harten, A.: On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21(1), 1–23 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Harten, A., Osher, S., Engquist, B., Chakravarthy, S.R.: Some results on uniformly high-order accurate essentially nonoscillatory schemes. Appl. Numer. Math. 2(3–5), 347–377 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes. i. SIAM J. Numer. Anal. 24(2), 279–309 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

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

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    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(2), 542–567 (2005)

    MATH  Article  Google Scholar 

  9. 9.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Yamaleev, N.K., Carpenter, M.H.: Third-order energy stable weno scheme. J. Comput. Phys. 228(8), 3025–3047 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Don, W.-S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Xiaoshuai, W., Yuxin, Z.: A high-resolution hybrid scheme for hyperbolic conservation laws. Int. J. Numer. Methods Fluids 78(3), 162–187 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Wu, X., Liang, J., Zhao, Y.: A new smoothness indicator for third-order weno scheme. Int. J. Numer. Methods Fluids 81(7), 451–459 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Balsara, D.S., Garain, S., Shu, C.-W.: An efficient class of weno schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Zhu, J., Qiu, J.: A new fifth order finite difference weno scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Kumar, R., Chandrashekar, P.: Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws. J. Comput. Phys. 375, 1059–1090 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Kumar, R., Chandrashekar, P.: Efficient seventh order WENO schemes of adaptive order for hyperbolic conservation laws. Comput. Fluids. (2019). https://doi.org/10.1016/j.compfluid.2019.06.003

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    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(1), 68–86 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Fan, P., Shen, Y., Tian, B., Yang, C.: A new smoothness indicator for improving the weighted essentially non-oscillatory scheme. J. Comput. Phys. 269, 329–354 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Acker, F., Borges, R.R., Costa, B.: An improved WENO-z scheme. J. Comput. Phys. 313, 726–753 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Xu, W., Wu, W.: Improvement of third-order WENO-Z scheme at critical points. Comput. Math. Appl. 75(9), 3431–3452 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Xu, W., Wu, W.: An improved third-order weighted essentially non-oscillatory scheme achieving optimal order near critical points. Comput. Fluids 162, 113–125 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Gande, N.R., Rathod, Y., Rathan, S.: Third-order weno scheme with a new smoothness indicator. Int. J. Numer. Methods Fluids 85(2), 90–112 (2017)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335–362 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

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

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7(1), 159–193 (1954)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Lax, P.D., Liu, X.-D.: Solution of two-dimensional riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319–340 (1998)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to Center for Fundamental Research in Space Dynamics and Celestial Mechanics (CFRSC), New Delhi, Delhi, India for providing research facilities. We also express gratitude to CSIR, Govt. of India for the grant reference no. 09/045(1438)/2016-EMR-I.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bhavneet Kaur.

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

Verify currency and authenticity via CrossMark

Cite this article

Kumar, A., Kaur, B. An Improvement of Third Order WENO Scheme for Convergence Rate at Critical Points with New Non-linear Weights. Differ Equ Dyn Syst 28, 539–557 (2020). https://doi.org/10.1007/s12591-019-00508-5

Download citation

Keywords

  • Smoothness measurement
  • WENO
  • Critical points
  • Sufficient condition
  • Convergence analysis
  • Accuracy