Skip to main content

Spectral Collocation Solutions to a Class of Pseudo-parabolic Equations

  • 710 Accesses

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11189)


In this paper we solve by method of lines (MoL) a class of pseudo-parabolic PDEs defined on the real line. The method is based on the sinc collocation (SiC) in order to discretize the spatial derivatives as well as to incorporate the asymptotic behavior of solution at infinity. This MoL casts an initial value problem attached to these equations into a stiff semi-discrete system of ODEs with mass matrix independent of time. A TR-BDF2 finite difference scheme is then used in order to march in time.

The method does not truncate arbitrarily the unbounded domain to a finite one and does not assume the periodicity. These are two omnipresent, but non-natural, ingredients used to handle such problems.

The linear stability of MoL is proved using the pseudospectrum of the discrete linearized operator. Some numerical experiments are carried out along with an estimation of the accuracy in conserving two invariants. They underline the efficiency and robustness of the method. The convergence order of MoL is also established.


  • Pseudo-parabolic equation
  • Infinite domain
  • Camassa-Holm
  • Peakon
  • Sinc collocation
  • TR-BDF2
  • Linear stability
  • Pseudospectrum

This work was supported by a mobility grant of the Romanian Ministery of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-MC-2018-1869, within PNCDI III.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-10692-8_20
  • Chapter length: 8 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-10692-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.


  1. Amick, C.J., Bona, J.L., Schonbek, M.E.: Decay of solutions of some nonlinear wave equations. J. Differ. Equations 81, 1–49 (1989).

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Boyd, J.P.: Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation. Appl. Math. Comput. 81, 173–187 (1997).

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993).

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Camassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. In: Wu, T.-Y., Hutchinson, J.W. (eds.) Advances in Applied Mechanics, vol. 31, pp. 1–31. Academic Press, New York (1994)

    Google Scholar 

  6. Constantin, A., Strauss, A.W.: Stability of Peakons. Commun. Pure Appl. Math. 53, 603–610 (2000). 10.1002/(SICI)1097-0312(200005)53:5\(<\)603::AID-CPA3\(>\)3.0.CO;2-L

    MathSciNet  CrossRef  Google Scholar 

  7. Gheorghiu, C.I.: Stable spectral collocation solutions to a class of Benjamin Bona Mahony initial value problems. Appl. Math. Comput. 273, 1090–1099 (2016).

    MathSciNet  CrossRef  Google Scholar 

  8. Gheorghiu, C.I.: Spectral Collocation Solutions to Problems on Unbounded Domains. Casa Cărţii de Ştiinţă Publishing House, Cluj-Napoca (2018)

    MATH  Google Scholar 

  9. Kassam, A.-K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005).

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Stenger, F.: Summary of sinc numerical methods. J. Comput. Appl. Math. 121, 379–420 (2000).

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM Philadelphia (2000)

    Google Scholar 

  12. Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM T. Math. Softw. 26, 465–519 (2000).

    MathSciNet  CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Călin-Ioan Gheorghiu .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Gheorghiu, CI. (2019). Spectral Collocation Solutions to a Class of Pseudo-parabolic Equations. In: Nikolov, G., Kolkovska, N., Georgiev, K. (eds) Numerical Methods and Applications. NMA 2018. Lecture Notes in Computer Science(), vol 11189. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10691-1

  • Online ISBN: 978-3-030-10692-8

  • eBook Packages: Computer ScienceComputer Science (R0)