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Journal of Scientific Computing

, Volume 67, Issue 1, pp 299–323 | Cite as

Modified Non-linear Weights for Fifth-Order Weighted Essentially Non-oscillatory Schemes

  • Chang Ho Kim
  • Youngsoo Ha
  • Jungho Yoon
Article

Abstract

This paper is concerned with fifth-order weighted essentially non-oscillatory (WENO) scheme with a new smoothness indicator. As the so-called WENO-JS scheme (Jiang and Shu in J Comput Phys 126:202–228, 1996) provides the third-order accuracy at critical points where the first and third order derivatives do not becomes zero simultaneously, several fifth-order WENO scheme have been developed through modifying the known smoothness indicators of WENO-JS. Recently a smoothness indicator based on \(L^1\)-norm has been proposed by Ha et al. (J Comput Phys 232:68–86, 2013) (denoted by WENO-NS). The aim of this paper is twofold. Firstly, we further improve the smoothness indicator of WENO-NS and secondly, using this measurement, we suggest new nonlinear weights by simplifying WENO-NS weights. The proposed WENO scheme provides the fifth-order accuracy, even at critical points. Some numerical experiments are provided to demonstrate that the present scheme performs better than other WENO schemes of the same order.

Keywords

Hyperbolic conservation laws Euler equation WENO scheme  Approximation order Smoothness indicator 

Mathematics Subject Classification

65M12 65M70 41A10 

Notes

Acknowledgments

Jungho Yoon was supported by the Grant 2015-R1A5A1009350 through the National Research Foundation of Korea (NRF). Youngsoo Ha was supported by NRF-2013R1A1A2013793 and Chang Ho Kim was by NRF-2014M1A7A1A03029872 through the National R&D Program funded by the Ministry of Education, Science and Technology.

References

  1. 1.
    Adams, N.A., Shariff, K.: A high-resolution hybrid compact-ENO scheme for shock–turbulence interaction problems. J. Comput. Phys. 127, 27–51 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aràndiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49(2), 893–915 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Balsara, D.S., Shu, C.W.: Monotonicity prserving WENO schemes with increasingly high-order of accuracy. J. Comput. Phys. 160, 405–452 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Balsara, D.S., Meyer, C., Dumbser, M., Du, H., Xu, Z.: Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes—speed comparisons with Runge–Kutta methods. J. Comput. Phys. 235, 934–969 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved WENO scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Costa, B., Don, W.S.: High order hybrid central-WENO finite difference scheme for conservation laws. J. Comput. Appl. Math. 204, 209–218 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gerolymos, G.A., Sénéchal, D., Vallet, I.: Very-high-order WENO schemes. J. Comput. Phys. 228, 8481–8524 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Glimm, J., Grove, J., Li, X., Oh, W., Tan, D.C.: The dynamics of bubble growth for Rayleigh–Taylor unstable interfaces. Phys. Fluids 31, 447–465 (1988)CrossRefGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Harten, A., Osher, S.: Uniformly high-order accurate non-oscillatory schemes. I. SIAM J. Numer. Anal. 24(2), 279–309 (1987)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high-order accurate non-oscillatory schemes. III. J. Comput. Phys. 71, 231–303 (1987)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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)CrossRefMATHGoogle Scholar
  14. 14.
    Jiang, G., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock–turbulence interaction. J. Comput. Phys. 178, 81–117 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ren, Y., Liu, M., Zhang, H.: A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 192, 365–386 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Shu, C.W. : Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin/New York (1998)Google Scholar
  21. 21.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. II. J. Comput. Phys. 83, 32–78 (1989)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, New York (1997)CrossRefMATHGoogle Scholar
  25. 25.
    Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Young, Y.-N., Tufo, H., Dubey, A., Rosner, R.: On the miscible Rayleigh–Taylor instability: two and three dimensions. J. Fluid Mech. 447, 377–408 (2001)CrossRefMATHGoogle Scholar
  27. 27.
    Zhou, Q., Yao, Z., He, F., Shen, M.Y.: A new family of high-order compact upwind difference schemes with good spectral resolution. J. Comput. Phys. 227, 1306–1339 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computer EngineeringKonkuk UniversityChungjuSouth Korea
  2. 2.Department of MathematicsSeoul National UniversitySeoulSouth Korea
  3. 3.Department of MathematicsEwha Womans UniversitySeoulSouth Korea

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