Numerical Solution of a Functional Equation on a Circle

Morlet, Anne C.; Lorenz, Jens

doi:10.1137/0729098

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…Usually, these spline schemes will lose certain properties of the continuous problem. As is shown in numerical results, spurious overshoots may happen for schemes with order of accuracy higher than 2 (see [7]). …”

Section: Introductionmentioning

confidence: 90%

“…For comparison, these examples are taken from [5] and [7], where they are numerically solved by means of difference methods or spline schemes. The main purpose of our experiments is to study the numerical behavior of the Fourier method for both the FE system (1.4) and the PDE system (1.5), and to confirm the theoretical results proved in the previous sections numerically.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…where ξ : T p × R q → T p , Q : T p × R q → R q are given functions and (1.4) appears after reduction via a Poincaré map [7]. In the context of numerical methods, one solves (1.3) or (1.4) by linearization and discretization.…”

Section: Introductionmentioning

confidence: 99%

“…It is known [7] that the function R(ϑ) is nowhere differentiable on (0, 1) if ab > 1. (iii) If the solution is smooth enough, the Fourier method may give results of any order of accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php However, only first-order accuracy can be obtained. Concerning the FE model, several schemes based on spline interpolation have been introduced and investigated by Morlet and Lorenz in [7]. Usually, these spline schemes will lose certain properties of the continuous problem.…”

Section: Introductionmentioning

confidence: 99%

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Computation of Invariant Tori by the Fourier Methods

Ming-you¹,

Küpper²,

Masbaum³

1997

SIAM J. Sci. Comput.

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In this paper, we study systems of functional equations (FEs) and first-order partial differential equations (PDEs) suggested in [SIAM ] as approximations for the computation of invariant tori. The main new ideas of this paper are, first, to investigate these systems in the setting of Hilbert spaces rather than in the setting of Banach spaces and, then, to employ Fourier methods instead of difference methods for a numerical solution. Based on the setting of Sobolev spaces H s (T p ), proper conditions for the PDE and FE systems to be dissipative are described, and some regularity results for the FE system are proved. We studied two kinds of Fourier methods, the spectral method and the pseudospectral method, in detail under dissipativity conditions. Convergence and optimal error estimates are shown theoretically for these Fourier methods in the case of general linear systems. Numerical results for three examples provided in the last section indicate that the Fourier method behaves very well not only for smooth solutions but also for nonsmooth solutions.

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Section: Introductionmentioning

confidence: 90%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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