Skip to main content
SpringerLink
Log in

A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations

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

Abstract

A hybrid staggered discontinuous Galerkin method is developed for the Korteweg–de Vries equation. The equation is written into a system of first order equations by introducing auxiliary variables. Two sets of finite element functions are introduced to approximate the solution and the auxiliary variables. The staggered continuity of the two finite element function spaces gives a natural flux condition and trace value on the element boundaries in the derivation of Galerkin approximation. On the other hand, to deal with the third order derivative term an hybridization idea is used and additional flux unknowns are introduced. The auxiliary variables can be eliminated in each element and the resulting algebraic system on the solution and the additional flux unknowns is solved. Stability of the semi discrete form is proven for various boundary conditions. Numerical results present the optimal order of \(L^2\)-errors of the proposed method for a given polynomial order.

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

Similar content being viewed by others

References

  1. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272(1220), 47–78 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  3. Bona, J.L., Dougalis, V.A., Karakashian, O.A.: Fully discrete Galerkin methods for the Korteweg–de Vries equation. Comput. Math. Appl. Ser. A 12(7), 859–884 (1986)

    MathSciNet  MATH  Google Scholar 

  4. Chan, H.N., Chung, E.T.: A staggered discontinuous Galerkin method with local TV regularization for the Burgers equation. Numer. Math. Theory Methods Appl. 8(4), 451–474 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chertock, A., Levy, D.: A particle method for the KdV equation. In: Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), vol. 17, pp. 491–499 (2002)

  6. Cheung, S.W., Chung, E., Kim, H.H., Qian, Y.: Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 302, 251–266 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chung, E., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. Part II: The Stokes flow. J. Sci. Comput. 66(2), 870–887 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chung, E., Lee, C.S.: A staggered discontinuous Galerkin method for the convection–diffusion equation. J. Numer. Math. 20(1), 1–31 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for wave propagation. SIAM J. Numer. Anal. 44(5), 2131–2158 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47(5), 3820–3848 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chung, E.T., Kim, H.H.: A deluxe FETI-DP algorithm for a hybrid staggered discontinuous Galerkin method for H(curl)-elliptic problems. Internat. J. Numer. Methods Eng. 98(1), 1–23 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chung, E.T., Kim, H.H., Widlund, O.B.: Two-level overlapping Schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51(1), 47–67 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Debussche, A., Printems, J.: Numerical simulation of the stochastic Korteweg–de Vries equation. Phys. D 134(2), 200–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feng, B.F., Mitsui, T.: A finite difference method for the Korteweg–de Vries and the Kadomtsev–Petviashvili equations. J. Comput. Appl. Math. 90(1), 95–116 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Holmer, J.: The initial-boundary value problem for the Korteweg–de Vries equation. Commun. Partial Differ. Equ. 31(7–9), 1151–1190 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51(6), 3327–3350 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. (5) 39(240), 422–443 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee, J.J., Kim, H.H.: Analysis of a staggered discontinuous Galerkin method for linear elasticity. J. Sci. Comput. 66(2), 625–649 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Levy, D., Shu, C.W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196(2), 751–772 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, H., Yan, J.: A local discontinuous Galerkin method for the Korteweg–de Vries equation with boundary effect. J. Comput. Phys. 215(1), 197–218 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Samii, A., Panda, N., Michoski, C., Dawson, C.: A hybridized discontinuous Galerkin method for the nonlinear Korteweg–de Vries equation. J. Sci. Comput. 68(1), 191–212 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Süli, E., Mayers, D.F.: An Introduction to Numerical Analysis. Cambridge University  Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  25. Yan, J., Shu, C.W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Prof. Sunmi Lee at Kyung Hee University for introducing KdV equations to the authors and suggesting development of staggered discontinuous Galerkin methods for the KdV equations.

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyung Hee University, Seoul, Korea

    Hee Jun Yang

  2. Department of Applied Mathematics, Kyung Hee University, Yongin, Korea

    Hyea Hyun Kim

Authors
  1. Hee Jun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyea Hyun Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyea Hyun Kim.

Additional information

The authors were supported by NRF-2015R1A5A1009350.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, H.J., Kim, H.H. A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations. J Sci Comput 77, 502–523 (2018). https://doi.org/10.1007/s10915-018-0714-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0714-x

Keywords

Search

Navigation