Legendre-Gauss collocation methods for ordinary differential equations

Advances in Computational Mathematics

Volumn 30, Issue 3, 2009, Pages 249-280

  Guo, Ben Yu ^a,b,c   Wang, Zhong Qing ^a,b,c  

^a SHANGHAI NORMAL UNIVERSITY   (China)

^b Shanghai Universities   (China)

^c Scientific Computing Key Laboratory   (China)

Author keywords

Initial value problems of ordinary differential equations; Legendre Gauss collocation methods; Spectral accuracy

Indexed keywords

EID: 60749106103     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-008-9067-6     Document Type: Article

References (28)

1. Babuška, L. Guo, B.Q.: Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spacecs, part I, approximabiiity of functions in weighted Besov spaces. SIAM J. Numer. Anal. 39,1512-1538 (2001)
2. Babuska, I., Janik, T.: The h-p version of the finite element method for parabolic equations, Part 1, The p- version in time. Numer. Methods Partial Differential Equations 5,363-399 (1989)
3. Bernardi, C, Maday, Y.: Spectral method. In: Ciarlet, P.G., Lions, J.L.(eds.) Handbook of Numerical Analysis, part 5. North-Holland (1997)
4. Boyd. J.P.: Chebyshev and Fourier Spectral Methods. Springer-Verlag, Berlin (1989)
5. Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal. 16, 46-57 (1979)
6. Butcher, J.C.: Implicit Runge-Kutta processes. Math. Comp. 18, 50-64 (1964)
7. Butcher, J.C.: Integration processes based on Radau quadrature formulas. Math. Comp. 18, 233-244 (1964)
8. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. John Wiley & Sons, Chichester (1987)
9. Camito, C, Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin (1988)
10. Donchev, T.. Farkhi. E.: Stability and Euler approximations of one sided Lipschitz convex differential inclusions. SIAM J. Control Optim. 36, 780-796 (1998)
11. Feng, K.: Difference schemes for Hamiltonian formulism and symplectic geometry. J. Comput. Math 4,279-289 (1986)
12. Feng. K.. Qin, M.Z.: Sympletic Geometric Algorithms for Hamiltonian Systems. Zhejiang Science and Technology Press, Hangzhou (2003)
13. Funaro, D.: Polynomial Approximations of Differential liquations. Springer-Verlag (1992)
14. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS, Philadelphia (1977)
15. Guo, B.-y.: Spectral Methods and their Applications. World Scietific, Singapore (1998)
16. Guo. B.-y.: Jacobi approximations in certain Hilbert spaces and their applications to singu¡ar differential equations. J. Math. Anal. Appl. 243, 373-408 (2000)
17. Guo, B.-y., Wang, L.-1: Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math. 14, 227-276 (2001)
18. Guo, B.-y., Wang, L.-1: Jacobi approximations in non-uniformiy Jacobi-weighted Sobolev spaces. J. Approx. Theory 128,1-41 (2004)
19. Guo. B.-y.. Wang Z.-q.: Numerical intergration based on Laguerre-Gauss interpolation. Com-put. Methods Appl. Mech. Engrg. 196, 3726-3741 (2007)
20. Guo, B.-y., Wang Z.-q.. Hong-Jiong, T., Wang, L.-L: Integration processes of ordinarny differen-tial equations based on Laguerre-Radau interpolations. Math. Comp. 77, 181-199 (2008)
21. Hairer, E., Lubich, C, Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Comput. Mathematics, vol. 31. Springer-Verlag, Berlin (2002)
22. Hairer. E.. Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equation I: Nonstiff Problems. Springer-Verlag, Berlin (1987)
23. Hairer, E., Wanner, G.: Solving Ordinary Differential Equation II: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin (1991)
24. Higham, DJ.: Analysis of the Enright-Kamel partitioning method for stiff ordinary differential equations. IMA J. Numer. Anal. 9, 1-14 (1989)
25. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems, The Initial Value Problem. John Wiley and Sons, Chichester (1991)
26. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems, AMMC7. Chapman and Hall, London (1994)
27. Stuart, A.M., Humphries, A.R.: Dynamical systems and Numerical Analysis. Cambridge Univer-sity Press, Cambridge (1996)
28. Wright, K.: Some relationship between implicit Runge-Kutta, collocation and -methods and their stability properties. BIT l0, 217-227 (1970)