Skip to main content
SpringerLink
Log in

A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

The purpose of this paper is to develop a hybridized discontinuous Galerkin (HDG) method for solving the Ito-type coupled KdV system. In fact, we use the HDG method for discretizing the space variable and the backward Euler explicit method for the time variable. To linearize the system, the time-lagging approach is also applied. The numerical stability of the method in the sense of the \(L_2\) norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet boundary conditions. Numerical experiments confirm that the HDG method is capable of solving the system efficiently. It is observed that the best possible rate of convergence is achieved by the HDG method. Also, it is being illustrated numerically that the corresponding conservation laws are satisfied for the approximate solutions of the Ito-type coupled KdV system. Thanks to the numerical experiments, it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.

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

Similar content being viewed by others

References

  1. Akbari, R., Mokhtari, R.: A new compact finite difference method for solving the generalized long wave equation. Numer. Funct. Anal. Optim. 35(2), 133–152 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkinson, K.E., Han, W.: Numerical Solution of Ordinary Differential Equations. John Wiley and Sons, Inc., New Jersey (2009)

    Book  MATH  Google Scholar 

  3. Başhan, A.: An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods. Comput. Appl. Math. 39(80), 1–23 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Bona, J.L., Chen, H., Karakashian, O., Wise, M.M.: Finite element methods for a system of dispersive equations. J. Sci. Comput. 77(3), 1371–1401 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buffa, A., Ortner, C.: Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 29(4), 827–855 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, W., Fei, J., Ma, Z., Liu, Q.: Bosonization and new interaction solutions for the coupled Korteweg-de Vries system. Waves Random Complex Media 30(1), 130–141 (2020)

    Article  MathSciNet  Google Scholar 

  7. Castillo, P., Gomez, S.: Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrödinger equations. Appl. Math. Comput. 371, 124950 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castillo, P., Gomez, S.: Conservative local discontinuous Galerkin method for the fractional Klein-Gordon-Schrödinger system with generalized Yukawa interaction. Numer. Algor. 84(1), 407–425 (2020)

    Article  MATH  Google Scholar 

  9. Chegini, N., Stevenson, R.: An adaptive wavelet method for semi-linear first-order system least squares. J. Comput. Meth. Appl. Math. 15(4), 439–463 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, Y., Song, S., Zhu, H.: Multi-symplectic methods for the Ito-type coupled KdV equation. Appl. Math. Comput. 218(9), 5552–5561 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. 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 

  13. Dong, B.: Optimally convergent HDG method for third-order Korteweg-de Vries type equations. J. Sci. Comput. 73(2/3), 712–735 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Drinfel'd, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. J. Sov. Math. 30, 1975–2036 (1975)

    Article  MATH  Google Scholar 

  15. Gear, J.A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70(3), 235–258 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guha-Roy, C.: Solution of coupled KdV-type equations. Int. J. Theor. Phys. 29(8), 863–866 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hirota, R., Satsuma, J.: Soliton solutions of a coupled Korteweg-de Vries equation. Phys. Lett. A 85(8/9), 407–408 (1981)

    Article  MathSciNet  Google Scholar 

  18. Ito, M.: Symmetries and conservation laws of a coupled nonlinear wave equation. Phys. Lett. A 91(7), 335–338 (1982)

    Article  MathSciNet  Google Scholar 

  19. Kirby, R.M., Sherwin, S.J., Cockburn, B.: To CG or to HDG: a comparative study. J. Sci. Comput. 51(1), 183–212 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Leng, H.T., Chen, Y.P.: Adaptive hybridizable discontinuous Galerkin methods for nonstationary convection diffusion problems. Adv. Comput. Math. 46(50), 1–23 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Lou, S.Y., Tong, B., Hu, H.C., Tang, X.Y.: Coupled KdV equations derived from two-layer fluids. J. Phys. A 39(3), 513–527 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Luo, D.M., Huang, W.Z., Qiu, J.X.: An hybrid LDG-HWENO scheme for KdV-type equations. J. Comput. Phys. 313, 754–774 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mirza, A., ul Hassan, M.: Bilinearization and soliton solutions of the supersymmetric coupled KdV equation. Theor. Math. Phys. 202(1), 11–16 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mogorosi, E.T., Muatjetjeja, B., Khalique, C.M.: Conservation laws for a generalized Ito-type coupled KdV system. Bound. Value Probl. 2012(1), 1–7 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mohammadi, M., Mokhtari, R., Schaback, R.: A meshless method for solving the 2D Brusselator reaction-diffusion system. Comput. Model. Eng. Sci. 101(2), 113–138 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Mokhtari, R., Isvand, D., Chegini, N.G., Salaripanah, A.: Numerical solution of the Schrödinger equations by using Delta-shaped basis functions. Nonlinear Dyn. 74(1/2), 77–93 (2013)

    Article  MATH  Google Scholar 

  27. Mokhtari, R., Mohseni, M.: A meshless method for solving mKdV equation. Comput. Phys. Commun. 183(6), 1259–1268 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Qiu, L., Deng, W., Hesthaven, J.S.: Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes. J. Comput. Phys. 298, 678–694 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Saad, Y., Sosonkina, M.: Distributed Schur complement techniques for general sparse linear systems. SIAM J. Sci. Comput. 21(4), 1337–1356 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Sánchez, M.A., Ciuca, C., Nguyen, N.C., Peraire, J., Cockburn, B.: Symplectic Hamiltonian HDG methods for wave propagation phenomena. J. Comput. Phys. 350, 951–973 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, S., Yuan, J., Deng, W., Wu, Y.: A hybridized discontinuous Galerkin method for 2D fractional convection-diffusion equations. J. Sci. Comput. 68(2), 826–847 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  34. Xie, S.S., Yi, S.C.: A conservative compact finite difference scheme for the coupled Schrödinger-KdV equations. Adv. Comput. Math. 46(1), 1–22 (2020)

    Article  MATH  Google Scholar 

  35. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for three class of nonlinear wave equations. J. Comput. Math. 22, 250–274 (2004)

    MathSciNet  MATH  Google Scholar 

  36. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Comput. Methods Appl. Mech. Eng. 195(25/26/27/28), 3430–3447 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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 

  38. Yeganeh, S., Mokhtari, R., Hesthaven, J.S.: A local discontinuous Galerkin method for two-dimensional time fractional diffusion equations. Commun. Appl. Math. Comput. 2, 689–709 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yeganeh, S., Mokhtari, R., Hesthaven, J.S.: Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method. BIT Numer. Math. 57(3), 685–707 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yu, J.P., Sun, Y.L., Wang, F.D.: N-soliton solutions and long-time asymptotic analysis for a generalized complex Hirota-Satsuma coupled KdV equation. Appl. Math. Lett. 106, 106370 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    Shima Baharlouei & Reza Mokhtari

  2. Department of Mathematics, Tafresh University, 39518-79611, Tafresh, Iran

    Nabi Chegini

Authors
  1. Shima Baharlouei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reza Mokhtari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nabi Chegini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reza Mokhtari.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baharlouei, S., Mokhtari, R. & Chegini, N. A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System. Commun. Appl. Math. Comput. 4, 1351–1373 (2022). https://doi.org/10.1007/s42967-021-00178-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-021-00178-7

Keywords

Mathematics Subject Classification

Search

Navigation