Spectral methods in computing invariant tori

Applied Numerical Mathematics

Volumn 34, Issue 2, 2000, Pages 275-292

  Trummer, Manfred R ^a  

^a SIMON FRASER UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; BOUNDARY CONDITIONS; ITERATIVE METHODS; MATRIX ALGEBRA; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; TWO DIMENSIONAL;

DYNAMICAL SYSTEMS; INVARIANT TORUS; PSEUDOSPECTRAL METHOD; SPECTRAL METHODS;

NONLINEAR EQUATIONS;

EID: 0034227333     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(99)00133-6     Document Type: Article

References (21)

1. [1] U.M. Ascher, R.M.M. Mattheij, R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1988.
2. [2] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Series of Computational Physics, Springer-Verlag, Heidelberg, 1988.
3. [3] L. Dieci, G. Bader, Solution of the Systems associated to Invariant Tori Approximation. II: Multigrid Methods, SIAM J. Sci. Comput. 15 (1994) 1375-1400.
4. [4] L. Dieci, G. Bader, Block iterations and compactification for periodic block dominant systems associated to invariant tori approximation, Appl. Numer. Math. 17 (1995) 251-274.
5. [5] L. Dieci, J. Lorenz, Block M-matrices and computation of invariant tori, SIAM J. Sci. Statist. Comput. 13 (1992) 885-903.
6. [6] L. Dieci, J. Lorenz, R.D. Russell, Numerical calculation of invariant tori, SIAM J. Sci. Statist. Comput. 12 (1991) 607-647.
7. [7] J.E. Dennis, R.D. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ, 1983.
8. [8] W.S. Don, A. Solomonoff, Accuracy and speed in computing the Chebyshev collocation derivative, SIAM J. Sci. Comput. 6 (1994) 1253-1268.
9. [9] R. Fletcher, Conjugate gradient methods for indefinite systems, in: G.A. Watson (Ed.), Numerical Analysis, Lecture Notes in Mathematics, Vol. 506, Springer, Berlin, 1976, pp. 73-89.
10. [10] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations in Vector Fields, Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New York, 1983.
11. [11] M.H. Gutknecht, Variants of BiCGStab for matrices with complex spectrum, SIAM J. Sci. Statist. Comput. 14 (1993) 1020-1033.
12. [12] R.D. Haynes, Invariant manifolds of dynamical systems: theory and computation, M.Sc. thesis, Simon Fraser University, Canada, 1998.
13. [13] C. Lanczos, Solution of systems of linear equations by minimized iterations, J. Res. Nat. Bureau Standards 49 (1952) 33-53.
14. [14] H. Mingyou, T. Küpper, N. Masbaum, Computation of invariant tori by the Fourier method, SIAM J. Sci. Comput. 18 (1997) 918-942.
15. [15] N.M. Nachtigal, S.C. Reddy, L.N. Trefethen, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl. 13 (1992) 778-795.
16. [16] G.L.G. Sleijpen, D.R. Fokkema, BiCGStab(ℓ) for linear equations involving unsymmetric matrices with complex spectrum, Electronic Trans. Numer. Anal. 1 (1993) 11-32.
17. [17] P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10 (1989) 36-52.
18. [18] L.N. Trefethen, Spectra and Pseudospectra: The Behavior of Non-normal Matrices and Operators, book in preparation.
19. [19] M.R. Trummer, Unsymmetric systems arising in the computation of invariant tori, in: Copper Mountain Conference on Iterative Methods, 1994.
20. [20] H.A. van der Vorst, Bi-CGSTAB: A fast and smoothly convergent variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631-644.
21. [21] M. van Veldhuizen, A new algorithm for the numerical approximation of an invariant curve, SIAM J. Sci. Statist. Comput. 8 (1987) 951-962.