Spectral Collocation Solutions to a Class of Pseudo-parabolic Equations

  • Călin-Ioan GheorghiuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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 


  1. 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). Scholar
  2. 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). Scholar
  3. 3.
    Boyd, J.P.: Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation. Appl. Math. Comput. 81, 173–187 (1997). Scholar
  4. 4.
    Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993). Scholar
  5. 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. 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-LMathSciNetCrossRefGoogle Scholar
  7. 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). Scholar
  8. 8.
    Gheorghiu, C.I.: Spectral Collocation Solutions to Problems on Unbounded Domains. Casa Cărţii de Ştiinţă Publishing House, Cluj-Napoca (2018)zbMATHGoogle Scholar
  9. 9.
    Kassam, A.-K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005). Scholar
  10. 10.
    Stenger, F.: Summary of sinc numerical methods. J. Comput. Appl. Math. 121, 379–420 (2000). Scholar
  11. 11.
    Trefethen, L.N.: Spectral Methods in MATLAB. SIAM Philadelphia (2000)Google Scholar
  12. 12.
    Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM T. Math. Softw. 26, 465–519 (2000). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Romanian Academy“Tiberiu Popoviciu” Institute of Numerical AnalysisCluj-NapocaRomania

Personalised recommendations