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Spectral Collocation Solutions to a Class of Pseudo-parabolic Equations

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

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Abstract

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.

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.

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References

  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). https://doi.org/10.1016/0022-0396(89)90176-9

    Article  MathSciNet  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). https://doi.org/10.1098/rsta.1972.0032

    Article  MathSciNet  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). https://doi.org/10.1016/0096-3003(95)00326-6

    Article  MathSciNet  MATH  Google Scholar 

  4. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993). https://doi.org/10.1103/PhysRevLett.71.1661

    Article  MathSciNet  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

    Article  MathSciNet  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). https://doi.org/10.1016/j.amc.2015.10.078

    Article  MathSciNet  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). https://doi.org/10.1137/S1064827502410633

    Article  MathSciNet  MATH  Google Scholar 

  10. Stenger, F.: Summary of sinc numerical methods. J. Comput. Appl. Math. 121, 379–420 (2000). https://doi.org/10.1016/S0377-0427(00)00348-4

    Article  MathSciNet  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). https://doi.org/10.1145/365723.365727

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Romanian Academy, “Tiberiu Popoviciu” Institute of Numerical Analysis, Str. Fantanele 57/67-68, 400320, Cluj-Napoca, Romania

    Călin-Ioan Gheorghiu

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  1. Călin-Ioan Gheorghiu
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Correspondence to Călin-Ioan Gheorghiu .

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Editors and Affiliations

  1. Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria

    Geno Nikolov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Natalia Kolkovska

  3. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir Georgiev

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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. https://doi.org/10.1007/978-3-030-10692-8_20

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  • DOI: https://doi.org/10.1007/978-3-030-10692-8_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10691-1

  • Online ISBN: 978-3-030-10692-8

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