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A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

Part of: Approximations and expansions Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  17 November 2016

Zhong-Qing Wang  and
Jun Mu
Zhong-Qing Wang*
Affiliation:
School of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China
Jun Mu*
Affiliation:
School of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China
*
*Corresponding author. Email addresses:zqwang@usst.edu.cn (Z.-Q. Wang), mujun06@163.com (J. Mu)
*Corresponding author. Email addresses:zqwang@usst.edu.cn (Z.-Q. Wang), mujun06@163.com (J. Mu)
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Abstract

We introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain the hp-version bound on the numerical error of the multiple interval collocation method under H 1-norm. Numerical experiments confirm the theoretical expectations.

Keywords

Multiple interval Chebyshev-Gauss-Lobatto spectral collocation methodnonlinear ordinary differential equationserror analysis

MSC classification

Secondary: 65L05: Initial value problems 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 41A10: Approximation by polynomials 65L70: Error bounds
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 4 , November 2016 , pp. 619 - 639
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Bar-Yoseph, P. Z., Fisher, D. and Gottlieb, O., Spectral element methods for nonlinear spatio-temporal dynamics of an Euler-Bernoulli beam, Comput. Mech., vol. 19, no. 2 (1996), pp. 136–151.Google Scholar
[2] Bar-Yoseph, P. Z. and Moses, E., Space-time spectral element methods for unsteady convection-diffusion problems, Int. J. Numer. Methods for Heat & Fluid Flow, vol. 7, no. 2-3 (1997), pp. 215–235.Google Scholar
[3] Bernardi, C. and Maday, Y., Spectral method, in Handbook of Numerical Analysis, vol. 5, 209–485, edited by Ciarlet, P. G. and Lions, J. L., North-Holland, Amsterdam, 1997.Google Scholar
[4] Boyd, J. P., Chebyshev and Fourier Spectral Methods, Springer-Verlag, Berlin, 1989.Google Scholar
[5] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.CrossRefGoogle Scholar
[6] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007.CrossRefGoogle Scholar
[7] Dutt, P., Spectral methods for initial-boundary value problems-an alternative approach, SIAM J. Numer. Anal., vol. 27, no. 4 (1990), pp. 885–903.Google Scholar
[8] Funaro, D., Polynomial Approximations of Differential Equations, Springer-Verlag, Berlin, 1992.Google Scholar
[9] Glenn, I., Brian, S. and Rodney, W., Spectral methods in time for a class of parabolic partial differential equations, J. Comput. Phys., vol. 102, no. 1 (1992), pp. 88–97.Google Scholar
[10] Guo, B.-Y., Spectral Methods and Their Applications, World Scientific, Singapore, 1998.Google Scholar
[11] Guo, B.-Y. and Wang, Z.-Q., Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., vol. 30, no. 3 (2009), pp. 249–280.Google Scholar
[12] Guo, B.-Y. and Wang, Z.-Q., A spectral collocationmethod for solving initial value problems of first order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, vol. 14, no. 3 (2010), pp. 1029–1054.Google Scholar
[13] Guo, B.-Y. and Wang, Z.-Q., Numerical integration based on Laguerre-Gauss interpolation, Comput. Methods Appl. Mech. Engrg., vol. 196, no. 37-40 (2007), pp. 3726–3741.Google Scholar
[14] Guo, B.-Y., Wang, Z.-Q., Tian, H.-J. and Wang, L.-L., Integration processes of ordinary differential equations based on Laguerre-Radau interpolations, Math. Comp., vol. 77, no. 261 (2008), pp. 181–199.Google Scholar
[15] Guo, B.-Y. and Yan, J.-P., Legendre-Gauss collocation methods for initial value problems of second order ordinary differential equations, Appl. Numer. Math., vol. 59, no. 6 (2009), pp. 1386–1408.Google Scholar
[16] Guo, B.-Y. and Zhang, K.-J., On non-isotropic Jacobi pseudospectral method, J. Comput. Math., vol. 26, no. 4 (2008), pp. 511–535.Google Scholar
[17] Ierley, G., Spencer, B. and Worthing, R., Spectral methods in time for a class of parabolic partial differential equations, J. Comput. Phys., vol. 102, no. 1 (1992), pp. 88–97.Google Scholar
[18] Kanyamee, N. and Zhang, Z.-M., Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems, Int. J. Numer. Anal. Model., vol. 8, no. 1 (2011), pp. 86–104.Google Scholar
[19] Ma, H.-P., Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., vol. 35, no. 3 (1998), pp. 869–892.Google Scholar
[20] De Maerschalck, B. and Gerritsma, M. I., The use of Chebyshev polynomials in the space-time least-squares spectral element method, Numer. Algorithms, vol. 38, (2005), pp. 173–196.Google Scholar
[21] Shen, J., Tang, T. and Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011.Google Scholar
[22] Tal-Ezer, H., Spectral methods in time for hyperbolic equations, SIAM J. Numer. Anal., vol. 23, no. 1 (1986), pp. 11–26.Google Scholar
[23] Tal-Ezer, H., Spectral methods in time for parabolic problems, SIAM J. Numer. Anal., vol. 26, no. 1 (1989), pp. 1–11.Google Scholar
[24] Tang, J.-G. and Ma, H.-P., Single and multi-interval Legendre τ-methods in time for parabolic equations, Adv. Comput. Math., vol. 17, no. 4 (2002), pp. 349–367.Google Scholar
[25] Tang, J.-G. and Ma, H.-P., A Legendre spectral method in time for first-order hyperbolic equations, Appl. Numer. Math., vol. 57, no. 1 (2007), pp. 1–11.Google Scholar
[26] Wang, Z.-Q. and Guo, B.-Y., Legendre-Gauss-Radau collocation methods for solving initial value problems of first order ordinary differential equations, J. Sci. Comput., vol. 52, no. 1 (2012), pp. 226–255.Google Scholar
[27] Wang, Z.-Q. and Wang, L.-L., A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete Contin. Dyn. Syst. Ser. B, vol. 13, no. 3 (2010), pp. 685–708.Google Scholar
[28] Werner, H. and Arndt, H., Gewöhnliche Differentialgleichungen, Springer Verlag, Heidelberg, 1986.Google Scholar
[29] Wihler, T. P., An a priori error analysis of the hp-version of the continuous Galerkin FEM for nonlinear initial value problems, J. Sci. Comput., vol. 25, no. 3 (2005), pp. 523–549.Google Scholar
[30] Yang, X. and Wang, Z.-Q., A Chebyshev-Gauss spectral collocation method for ordinary differential equations, J. Comput. Math., vol. 33, no. 1 (2015), pp. 59–85.Google Scholar
[31] Yi, L.-J., Liang, Z.-Q. and Wang, Z.-Q., Legendre-Gauss-Lobatto spectral collocation method for nonlinear delay differential equations, Math. Methods Appl. Sci., vol. 36, no. 18 (2013), pp. 2476–2491.Google Scholar
[32] Yi, L.-J. and Wang, Z.-Q., Legendre-Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations, Int. J. Comput. Math., vol. 91, no. 7 (2014), pp. 1434–1460.Google Scholar
[33] Yi, L.-J. and Wang, Z.-Q., Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations, Discrete Contin. Dyn. Syst. Ser. B, vol. 19, no. 1 (2014), pp. 299–322.Google Scholar
[34] Yi, L.-J. and Wang, Z.-Q., Legendre-Gauss spectral collocation method for second order nonlinear delay differential equations, Numer. Math. Theory Methods Appl., vol. 7, no. 2 (2014), pp. 149–178.Google Scholar