# American Institute of Mathematical Sciences

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

 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.
Citation: Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
##### References:
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Google Scholar [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. doi: 10.1016/S0021-9991(05)80008-7. Google Scholar [13] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications,, Philadelphia, (1977). Google Scholar [14] B. Y. Guo, Spectral Methods and Their Applications,, World Scientific, (1998). doi: 10.1142/3662. Google Scholar [15] B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations,, Adv. Comput. Math., 30 (2009), 249. doi: 10.1007/s10444-008-9067-6. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.apnum.2008.08.007. Google Scholar [18] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems,, Springer-Verlag, (1987). Google Scholar [19] E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems,, Springer-Verlag, (1991). Google Scholar [20] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms,, 2nd edition, (2006). Google Scholar [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. Google Scholar [22] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,, SIAM, (1995). doi: 10.1137/1.9781611970944. Google Scholar [23] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem,, John Wiley & Sons, (1991). Google Scholar [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. doi: 10.1016/S0375-9601(01)00580-1. Google Scholar [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. doi: 10.1137/S0036142999352394. Google Scholar [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. doi: 10.1007/s100920070002. Google Scholar [27] J. Shen and T. Tang, Spectral and High-order Methods Methods with Application,, Science Press, (2006). Google Scholar [28] J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications,, Springer Series in Computational Mathematics, (2011). doi: 10.1007/978-3-540-71041-7. Google Scholar [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. doi: 10.1016/j.apnum.2006.07.012. Google Scholar [30] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis,, Cambridge University Press, (1996). Google Scholar [31] H. Tal-Ezer, Spectral methods in time for hyperbolic equations,, SIAM J. Numer. Anal., 23 (1986), 11. doi: 10.1137/0723002. Google Scholar [32] H. Tal-Ezer, Spectral methods in time for parabolic problems,, SIAM J. Numer. Anal., 26 (1989), 1. doi: 10.1137/0726001. Google Scholar [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. doi: 10.1023/A:1016273820035. Google Scholar [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. doi: 10.1016/j.apnum.2005.11.009. Google Scholar [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. doi: 10.1007/s10915-011-9538-7. Google Scholar [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. doi: 10.1007/s10915-004-4796-2. Google Scholar [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. doi: 10.1007/s10543-012-0379-z. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0045-7825(94)80017-0. Google Scholar

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. doi: 10.1002/num.1690050407. Google Scholar [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. doi: 10.1002/num.1690060406. Google Scholar [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. doi: 10.1006/jcph.1995.1116. Google Scholar [4] C. Bernardi and Y. Maday, Spectral Methods, in Handbook of Numerical Analysis,, (eds. P. G. Ciarlet and J. L. Lions), (1997). doi: 10.1016/S1570-8659(97)80003-8. Google Scholar [5] J. P. Boyd, Chebyshev and Fourier Spectral Methods,, 2nd edition, (2001). Google Scholar [6] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods,, John Wiley & Sons, (1987). Google Scholar [7] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains,, Springer-Verlag, (2006). Google Scholar [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, (2007). Google Scholar [9] J. M. Franco, Runge-Kutta-Nyström method adapted to the numerical integration of perturbed oscillators,, Comput. Phys. Comm., 147 (2002), 770. doi: 10.1016/S0010-4655(02)00460-5. Google Scholar [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. doi: 10.1023/A:1016629706668. Google Scholar [11] D. Funaro, Polynomial Approximations of Differential Equations,, Springer-Verlag, (1992). Google Scholar [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. doi: 10.1016/S0021-9991(05)80008-7. Google Scholar [13] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications,, Philadelphia, (1977). Google Scholar [14] B. Y. Guo, Spectral Methods and Their Applications,, World Scientific, (1998). doi: 10.1142/3662. Google Scholar [15] B. Y. Guo and Z. Q. Wang, Legendre-Gauss collocation methods for ordinary differential equations,, Adv. Comput. Math., 30 (2009), 249. doi: 10.1007/s10444-008-9067-6. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.apnum.2008.08.007. Google Scholar [18] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equation I: Nonstiff Problems,, Springer-Verlag, (1987). Google Scholar [19] E. Hairer and G. Wanner, Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems,, Springer-Verlag, (1991). Google Scholar [20] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms,, 2nd edition, (2006). Google Scholar [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. Google Scholar [22] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,, SIAM, (1995). doi: 10.1137/1.9781611970944. Google Scholar [23] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem,, John Wiley & Sons, (1991). Google Scholar [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. doi: 10.1016/S0375-9601(01)00580-1. Google Scholar [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. doi: 10.1137/S0036142999352394. Google Scholar [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. doi: 10.1007/s100920070002. Google Scholar [27] J. Shen and T. Tang, Spectral and High-order Methods Methods with Application,, Science Press, (2006). Google Scholar [28] J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications,, Springer Series in Computational Mathematics, (2011). doi: 10.1007/978-3-540-71041-7. Google Scholar [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. doi: 10.1016/j.apnum.2006.07.012. Google Scholar [30] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis,, Cambridge University Press, (1996). Google Scholar [31] H. Tal-Ezer, Spectral methods in time for hyperbolic equations,, SIAM J. Numer. Anal., 23 (1986), 11. doi: 10.1137/0723002. Google Scholar [32] H. Tal-Ezer, Spectral methods in time for parabolic problems,, SIAM J. Numer. Anal., 26 (1989), 1. doi: 10.1137/0726001. Google Scholar [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. doi: 10.1023/A:1016273820035. Google Scholar [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. doi: 10.1016/j.apnum.2005.11.009. Google Scholar [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. doi: 10.1007/s10915-011-9538-7. Google Scholar [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. doi: 10.1007/s10915-004-4796-2. Google Scholar [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. doi: 10.1007/s10543-012-0379-z. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1016/S0045-7825(94)80017-0. Google Scholar
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