On the convergence of Hill’s method

Curtis, Christopher W.; Deconinck, Bernard

doi:10.1090/s0025-5718-09-02277-7

Cited by 35 publications

(25 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted that (1.9), being a generalized spectral problem, is outside the realm of problems for which the convergence properties of Hill's method were analyzed in [12]. As remarked above, since the diagonal matrix diag(L, K, K) is nonsingular for b > 0, d > 0, the problem may be rewritten as a standard spectral problem, to which the techniques of [12] may be applied. It follows that our results are trustworthy, provided a sufficient number of Fourier modes are used in Hill's method.…”

Section: Numerical Methods For Studying Cnoidal Wavesmentioning

confidence: 99%

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

Chen¹,

Curtis²,

Deconinck³

et al. 2010

SIAM J. Appl. Dyn. Syst.

Self Cite

View full text Add to dashboard Cite

Abstract. We study the spectral (in)stability of one-dimensional solitary and cnoidal waves of various Boussinesq systems. These systems model three-dimensional water waves (i.e., the surface is two-dimensional) with or without surface tension. We present the results of numerous computations examining the spectra related to the linear stability problem for both stationary solitary and cnoidal waves with various amplitudes, as well as multipulse solutions. The one-dimensional nature of the wave forms allows us to separate the dependence of the perturbations on the spatial variables by transverse wave number. The compilation of these results gives a full view of the two-dimensional stability problem of these one-dimensional solutions. We demonstrate that line solitary waves with elevated profiles are spectrally stable with respect to one-dimensional perturbations and long transverse perturbations. We show that depression solitary waves are spectrally stable with respect to one-dimensional perturbations, but unstable with respect to transverse perturbations. We also discuss the instability of multipulse solitary waves and cnoidal-wave solutions of the Boussinesq system.

show abstract

Section: Numerical Methods For Studying Cnoidal Wavesmentioning

confidence: 99%

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

Chen¹,

Curtis²,

Deconinck³

et al. 2010

SIAM J. Appl. Dyn. Syst.

Self Cite

View full text Add to dashboard Cite

show abstract

“…When the periodic system matrix has a finite Fourier series (Eq. 29), the periodic eigenvectors can also be represented by a finite Fourier series (Curtis, 2010) in the homogeneous solution…”

Section: Periodic Mode Shapes and Hill's Truncated Eigenvalue Problemmentioning

confidence: 99%

Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors

Hansen

2016

Wind Energ. Sci.

View full text Add to dashboard Cite

Abstract. The modal dynamics of structures with bladed isotropic rotors is analyzed using Hill's method. First, analytical derivation of the periodic system matrix shows that isotropic rotors with more than two blades can be represented by an exact Fourier series with 3/rev (three per rotor revolution) as the highest order. For two-bladed rotors, the inverse mass matrix has an infinite Fourier series with harmonic components of decreasing norm; thus, the system matrix can be approximated by a truncated Fourier series of predictable accuracy. Second, a novel method for automatically identifying the principal solutions of Hill's eigenvalue problem is introduced. The corresponding periodic eigenvectors can be used to compute symmetric and antisymmetric components of the two-bladed rotor motion, as well as the additional forward and backward whirling components for rotors with more than two blades. To illustrate the use of these generic methods, a simple wind turbine model is set up with three degrees of freedom for each blade and seven degrees of freedom for the nacelle and drivetrain. First, the model parameters are tuned such that the low-order modal dynamics of a three-bladed 10 MW turbine from previous studies is recaptured. Second, one blade is removed, leading to larger and higher harmonic terms in the system matrix. These harmonic terms lead to modal couplings for the two-bladed turbine that do not exist for the three-bladed turbine. A single mode of a two-bladed turbine will also have several resonance frequencies in both the ground-fixed and rotating frames of reference, which complicates the interpretation of simulated or measured turbine responses.

show abstract

“…The convergence of this method as N → ∞ is proven in [14]. This method is spectrally accurate for differential eigenvalue problems with periodic coefficients and an almost-uniform approximation to the entire spectrum is obtained [15].…”

Section: Nontrivial-phase Solutionsmentioning

confidence: 89%

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

Deconinck

Sheils

Nguyen

et al. 2013

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Abstract. We consider a system of coupled cubic nonlinear Schrödinger (NLS) equationswhere the interaction coefficients α jk are real. The spectral stability of solitary wave solutions (both bright and dark) is examined both analytically and numerically. Our results build on preceding work by Nguyen et al. and others. Specifically, we present closed-form solitary wave solutions with trivial and non-trivial phase profiles. Their spectral stability is examined analytically by determining the locus of their essential spectrum. Their full stability spectrum is computed numerically using a large-period limit of Hill's method. We find that all nontrivial-phase solutions are unstable while some trivial-phase solutions are spectrally stable. To our knowledge, this paper presents the first investigation of the stability of the solitary waves of the coupled cubic NLS equation without the restriction that all components ψ j are proportional to sech.

show abstract

On the convergence of Hill’s method

Cited by 35 publications

References 24 publications

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors

On the spectral stability of solitary wave solutions of the vector nonlinear Schrödinger equation

Contact Info

Product

Resources

About