American Institute of Mathematical Sciences

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January  2014, 19(1): 299-322. doi: 10.3934/dcdsb.2014.19.299

Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations

Lijun Yi 1, and  Zhongqing Wang 1,

1. 

Department of Mathematics, Shanghai Normal University, Division of Computational Science of E-institute of Shanghai Universities, Shanghai, 200234, China, China

Received  August 2012 Revised  September 2013 Published  December 2013

We propose an efficient Legendre-Gauss collocation algorithm for second-order nonlinear ordinary differential equations (ODEs). We also design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partial differential equations (PDEs), which can be implemented in a synchronous parallel fashion. Numerical results indicate the high accuracy and effectiveness of the suggested algorithms.
Keywords: second-order ordinary differential equations, Legendre spectral collocation method, time-dependent second-order partial differential equations..
Mathematics Subject Classification: Primary: 65M70, 41A10, 65L05, 35L2.
Citation: Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
References:
[1]

I. Babuška and T. Janik, The $h$-$p$ version of the finite element method for parabolic equations: I. The $p$-version in time, Numer. Methods Partial Differential Equations, 5 (1989), 363-399. doi: 10.1002/num.1690050407.

[2]

I. Babuška and T. Janik, The $h$-$p$ version of the finite element method for parabolic equations: II. The $h$-$p$ version in time, Numer. Methods Partial Differential Equations, 6 (1990), 343-369. doi: 10.1002/num.1690060406.

[3]

P. Bar-Yoseph, E. Moses, U. Zrahia and A. L. Yarin, Space-time spectral element methods for one-dimensional nonlinear advection-diffusion problems, J. Comput. Phys., 119 (1995), 62-74. doi: 10.1006/jcph.1995.1116.

[4]

C. Bernardi and Y. Maday, Spectral Methods, in Handbook of Numerical Analysis, (eds. P. G. Ciarlet and J. L. Lions), North-Holland, Amsterdam, 1997. doi: 10.1016/S1570-8659(97)80003-8.

[5]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edition, Dover Publications, New York, 2001.

[6]

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987.

[7]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.

[8]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007.

[9]

J. M. Franco, Runge-Kutta-Nyström method adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm., 147 (2002), 770-787. doi: 10.1016/S0010-4655(02)00460-5.

[10]

J. M. Franco, I. Gómez and L. Rández, Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order, Numer. Algor., 26 (2001), 347-363. doi: 10.1023/A:1016629706668.

[11]

D. Funaro, Polynomial Approximations of Differential Equations, Springer-Verlag, Berlin, 1992.

[12]

I. Glenn, S. Brian and W. Rodney, Spectral methods in time for a class of parabolic partial differential equations, J. Comput. Phys., 102 (1992), 88-97. doi: 10.1016/S0021-9991(05)80008-7.

[13]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Philadelphia, Pa., 1977.

[14]

B. Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662.

[15]

B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., 30 (2009), 249-280. doi: 10.1007/s10444-008-9067-6.

[16]

B. Y. Guo and Z. Q. Wang, A spectral collocation method for solving initial value problems of first order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1029-1054.

[17]

B. Y. Guo and J. P. Yan, Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations, Appl. Numer. Math., 59 (2009), 1386-1408. doi: 10.1016/j.apnum.2008.08.007.

[18]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems, Springer-Verlag, Berlin, 1987.

[19]

E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.

[20]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms, 2nd edition, Springer, Berlin, Heidelberg, 2006.

[21]

N. Kanyamee and Z. Zhang, Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems, Int. J. Numer. Anal. Model., 8 (2011), 86-104.

[22]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944.

[23]

J. D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem, John Wiley & Sons, Chichester, 1991.

[24]

S. Liu and Z. Fu, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69-74. doi: 10.1016/S0375-9601(01)00580-1.

[25]

D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), 837-875. doi: 10.1137/S0036142999352394.

[26]

D. Schötzau and C. Schwab, An $hp$ a-priori error analysis of the DG time-stepping method for initial value problems, Calcolo, 37 (2000), 207-232. doi: 10.1007/s100920070002.

[27]

J. Shen and T. Tang, Spectral and High-order Methods Methods with Application, Science Press, Beijing, 2006.

[28]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[29]

J. Shen and L. L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), 710-720. doi: 10.1016/j.apnum.2006.07.012.

[30]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.

[31]

H. Tal-Ezer, Spectral methods in time for hyperbolic equations, SIAM J. Numer. Anal., 23 (1986), 11-26. doi: 10.1137/0723002.

[32]

H. Tal-Ezer, Spectral methods in time for parabolic problems, SIAM J. Numer. Anal., 26 (1989), 1-11. doi: 10.1137/0726001.

[33]

J. G. Tang and H. P. Ma, Single and multi-interval Legendre $\tau$-methods in time for parabolic equations, Adv. Comput. Math., 17 (2002), 349-367. doi: 10.1023/A:1016273820035.

[34]

J. G. Tang and H. P. Ma, A Legendre spectral method in time for first-order hyperbolic equations, Appl. Numer. Math., 57 (2007), 1-11. doi: 10.1016/j.apnum.2005.11.009.

[35]

Z. Q. Wang and B. Y. Guo, Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput., 52 (2012), 226-255. doi: 10.1007/s10915-011-9538-7.

[36]

T. P. Wihler, An a priori error analysis of the $hp$-version of the continuous Galerkin FEM for nonlinear initial value problems, J. Sci. Comput., 25 (2005), 523-549. doi: 10.1007/s10915-004-4796-2.

[37]

X. Y. Wu, B. Wang and J. L. Xia, Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods, BIT Numer. Math., 52 (2012), 773-795. doi: 10.1007/s10543-012-0379-z.

[38]

L. J. Yi and Z. Q. Wang, Legendre-Gauss-type collocation algorithms for nonlinear ordinary/ partial differential equations,, Int. J. Comput. Math.., ().  doi: 10.1080/00207160.2013.841901.

[39]

S. S. Zhang, S. Y. Chen and H. P. Ma, Legendre spectral methods for initial-boundary value problem of Klein-Gordon-Zakharov equations, Commun. Appl. Math. Comput., 26 (2012), 223-238.

[40]

U. Zrahia and P. Bar-Yoseph, Space-time spectral element method for solution of second-order hyperbolic equations, Comput. Methods Appl. Mech. Engrg., 116 (1994), 135-146. doi: 10.1016/S0045-7825(94)80017-0.

show all references

References:
[1]

I. Babuška and T. Janik, The $h$-$p$ version of the finite element method for parabolic equations: I. The $p$-version in time, Numer. Methods Partial Differential Equations, 5 (1989), 363-399. doi: 10.1002/num.1690050407.

[2]

I. Babuška and T. Janik, The $h$-$p$ version of the finite element method for parabolic equations: II. The $h$-$p$ version in time, Numer. Methods Partial Differential Equations, 6 (1990), 343-369. doi: 10.1002/num.1690060406.

[3]

P. Bar-Yoseph, E. Moses, U. Zrahia and A. L. Yarin, Space-time spectral element methods for one-dimensional nonlinear advection-diffusion problems, J. Comput. Phys., 119 (1995), 62-74. doi: 10.1006/jcph.1995.1116.

[4]

C. Bernardi and Y. Maday, Spectral Methods, in Handbook of Numerical Analysis, (eds. P. G. Ciarlet and J. L. Lions), North-Holland, Amsterdam, 1997. doi: 10.1016/S1570-8659(97)80003-8.

[5]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edition, Dover Publications, New York, 2001.

[6]

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987.

[7]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.

[8]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007.

[9]

J. M. Franco, Runge-Kutta-Nyström method adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm., 147 (2002), 770-787. doi: 10.1016/S0010-4655(02)00460-5.

[10]

J. M. Franco, I. Gómez and L. Rández, Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order, Numer. Algor., 26 (2001), 347-363. doi: 10.1023/A:1016629706668.

[11]

D. Funaro, Polynomial Approximations of Differential Equations, Springer-Verlag, Berlin, 1992.

[12]

I. Glenn, S. Brian and W. Rodney, Spectral methods in time for a class of parabolic partial differential equations, J. Comput. Phys., 102 (1992), 88-97. doi: 10.1016/S0021-9991(05)80008-7.

[13]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Philadelphia, Pa., 1977.

[14]

B. Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662.

[15]

B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., 30 (2009), 249-280. doi: 10.1007/s10444-008-9067-6.

[16]

B. Y. Guo and Z. Q. Wang, A spectral collocation method for solving initial value problems of first order ordinary differential equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1029-1054.

[17]

B. Y. Guo and J. P. Yan, Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations, Appl. Numer. Math., 59 (2009), 1386-1408. doi: 10.1016/j.apnum.2008.08.007.

[18]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems, Springer-Verlag, Berlin, 1987.

[19]

E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.

[20]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms, 2nd edition, Springer, Berlin, Heidelberg, 2006.

[21]

N. Kanyamee and Z. Zhang, Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems, Int. J. Numer. Anal. Model., 8 (2011), 86-104.

[22]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944.

[23]

J. D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem, John Wiley & Sons, Chichester, 1991.

[24]

S. Liu and Z. Fu, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69-74. doi: 10.1016/S0375-9601(01)00580-1.

[25]

D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), 837-875. doi: 10.1137/S0036142999352394.

[26]

D. Schötzau and C. Schwab, An $hp$ a-priori error analysis of the DG time-stepping method for initial value problems, Calcolo, 37 (2000), 207-232. doi: 10.1007/s100920070002.

[27]

J. Shen and T. Tang, Spectral and High-order Methods Methods with Application, Science Press, Beijing, 2006.

[28]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[29]

J. Shen and L. L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), 710-720. doi: 10.1016/j.apnum.2006.07.012.

[30]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.

[31]

H. Tal-Ezer, Spectral methods in time for hyperbolic equations, SIAM J. Numer. Anal., 23 (1986), 11-26. doi: 10.1137/0723002.

[32]

H. Tal-Ezer, Spectral methods in time for parabolic problems, SIAM J. Numer. Anal., 26 (1989), 1-11. doi: 10.1137/0726001.

[33]

J. G. Tang and H. P. Ma, Single and multi-interval Legendre $\tau$-methods in time for parabolic equations, Adv. Comput. Math., 17 (2002), 349-367. doi: 10.1023/A:1016273820035.

[34]

J. G. Tang and H. P. Ma, A Legendre spectral method in time for first-order hyperbolic equations, Appl. Numer. Math., 57 (2007), 1-11. doi: 10.1016/j.apnum.2005.11.009.

[35]

Z. Q. Wang and B. Y. Guo, Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput., 52 (2012), 226-255. doi: 10.1007/s10915-011-9538-7.

[36]

T. P. Wihler, An a priori error analysis of the $hp$-version of the continuous Galerkin FEM for nonlinear initial value problems, J. Sci. Comput., 25 (2005), 523-549. doi: 10.1007/s10915-004-4796-2.

[37]

X. Y. Wu, B. Wang and J. L. Xia, Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods, BIT Numer. Math., 52 (2012), 773-795. doi: 10.1007/s10543-012-0379-z.

[38]

L. J. Yi and Z. Q. Wang, Legendre-Gauss-type collocation algorithms for nonlinear ordinary/ partial differential equations,, Int. J. Comput. Math.., ().  doi: 10.1080/00207160.2013.841901.

[39]

S. S. Zhang, S. Y. Chen and H. P. Ma, Legendre spectral methods for initial-boundary value problem of Klein-Gordon-Zakharov equations, Commun. Appl. Math. Comput., 26 (2012), 223-238.

[40]

U. Zrahia and P. Bar-Yoseph, Space-time spectral element method for solution of second-order hyperbolic equations, Comput. Methods Appl. Mech. Engrg., 116 (1994), 135-146. doi: 10.1016/S0045-7825(94)80017-0.

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