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Dirichlet series for dynamical systems of first-order ordinary differential equations
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 |
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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