An Adapted Symplectic Integrator for Hamiltonian Problems

Vigo‐Aguiar, Jesús; Simos, T. E.; Tocino, Ángel

doi:10.1142/s0129183101001626

Cited by 53 publications

(17 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The floristic and syntaxonomic characterisation of plant communities has been performed according to the concepts and methods of classical phytosociology, thoroughly revised in the literature for the last two decades (Gehu & Rivas-Martinez, 1981;Alcaraz, 1996;Aguiar & Honrado, 2001;Rivas-Martinez, 2002;Honrado, 2003). In short, phytosociolo- gical releves were performed in representative formations with widely repeatable (i.e.…”

Section: Methodsmentioning

confidence: 99%

A new association of perennial nitrophilous vegetation from North-Western Iberian Peninsula

Honrado

Alves

Lomba

et al. 2004

Acta Botanica Gallica

View full text Add to dashboard Cite

Abstract.-A new phytosociological association of perennial nitrophilous vegetation is described from submediterranean mountain areas of North-Western Iberian Peninsula. Geranio lusitanici-Scrophularietum henninii ass. nova (class Artemisietea vulgaris) is physiognomically predominated by Scrophularia henninii and its individuality is based on the two nominal taxa (Scrophularia henninii and Geranium pyrenaicum subsp. lusitanicum), both endemic to the NorthWestern Iberian Peninsula. In this paper, the new association is fully characterised and distinguished from other perennial nitrophilous vegetation types in the territory in terms of its floristic composition, synecologic requirements, syntaxonomy, chorology and biogeography.

show abstract

Section: Methodsmentioning

confidence: 99%

A new association of perennial nitrophilous vegetation from North-Western Iberian Peninsula

Honrado

Alves

Lomba

et al. 2004

Acta Botanica Gallica

View full text Add to dashboard Cite

show abstract

“…Surveys of the literature with numerous references and useful bibliography and a review of these methods can be found in detail in [1,2,25,27,32,34]. These nonlinear equations can be also solved using an exponential fitting method proposed by Vigo-Aguiar et al [46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Analytical approximate solutions for conservative nonlinear oscillators by modified rational harmonic balance method

Gimeno

Álvarez

Yebra

et al. 2010

International Journal of Computer Mathematics

View full text Add to dashboard Cite

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have a rational form. This approach gives us not only a truly periodic solution but also the frequency of motion as a function of the amplitude of oscillation. Three truly nonlinear oscillators including the cubic Duffing oscillator, fractional-power restoring force and anti-symmetric quadratic nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. We find that this method works very well for the cubic oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation, we have shown that the relative error in the analytical approximate frequency is as low as 0.0046%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This has allowed us to compare the coefficients for the different harmonic terms in these solutions. For the other two nonlinear oscillators considered, the relative errors in the analytical approximate frequencies are 0.098 and 0.066%, respectively. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values, and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems.

show abstract

“…, c q the Gauss nodes in the interval [0, 1] we have the super convergent methods with maximum order p = 2q which are both symplectic and symmetric (see [5], p. 28 and p. 179). Methods that are exact for other spaces of functions F such as trigonometrical polynomials, exponential functions or mixed algebraic and trigonometrical polynomials suitable for special types of differential equations have been considered in [1,3,4,6,7,10,11,[13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

et al. 2010

View full text Add to dashboard Cite

The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν = ωh (h is the step size), and for ω → 0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255-262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421-434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.

show abstract

An Adapted Symplectic Integrator for Hamiltonian Problems

Cited by 53 publications

References 5 publications

A new association of perennial nitrophilous vegetation from North-Western Iberian Peninsula

A new association of perennial nitrophilous vegetation from North-Western Iberian Peninsula

Analytical approximate solutions for conservative nonlinear oscillators by modified rational harmonic balance method

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

Contact Info

Product

Resources

About