Numerical computation of nonlinear forced oscillations by Galerkin's procedure

Urabe, Minoru; Reiter, Allen

doi:10.1016/0022-247x(66)90066-7

Cited by 162 publications

(50 citation statements)

References 2 publications

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“…Thus, it is difficult to examine to what extent the approximation solution by using one periodic component only is close to the exact one. We can, of course, expect to get more exact solutions if we take as many time-spectral components as possible (Urabe and Reiter 1966). Thus, in Appendix D, we will illustrate only an example for a comparison between the result based on the time-spectral method and that of time-integration of the original equations (2.3).…”

Section: Discussionmentioning

confidence: 99%

“…When we study an atmospheric system with a clear periodic forcing, such as the annual cycle, we can employ such an alternative approach as will be presented in this work instead of time-integrating method: that is a model with variables expanded in the timespectral domain besides in the space-domain. This time spectral method in the case of one variable is well known as Galerkin-Urabe procedure (Urabe and Reiter, 1966). We are going to construct a space-and time-spectral general circulation model for the purpose of investigating the sensitivity or the shift of the annual change in response to variations of parameters that can be regarded as external forcing, e.g.…”

Section: Introductionmentioning

confidence: 99%

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Time-Space Spectral General Circulation Model I. Time-Space Spectral Model of Low-Order Barotropic System with Periodic Forcing

Iwashima

Yamamoto

1986

Journal of the Meteorological Society of Japan

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For the purpose of studying the atmospheric response (or sensitivity) to the seasonal forcing, we propose a time-space spectral general circulation model. In the present work we use a low-order equivalent Barotropic vorticity equation with forcing and dissipation terms. The original low-order equation consists of three-component in the space-spectral domain: one purely zonal component and two wave components with the same zonal wavenumber but different mode numbers. Further expanding this equation in the space-spectral domain into that in the time-spectral domain, we finally obtain a time-space spectral model, which is a simultaneous nonlinear algebraic equations system.The model used in the present work consists of one steady and one periodic components in the time-domain. The nonlinear equations system is solved by using the "revised Marquardt method" (Levenberg-Marquardt-Morrison method) and a "Continuation method". Addition of the periodic forcing besides the steady forcing brings about a new stable solution together with the stable solutions corresponding to those under the steady forcing only. This multiplicity of solutions with the moderate intensity of steady and periodic forcings is very interesting for understanding climatic change, and suggests that we can expect to obtain useful results, by extending our model to a further larger timespace spectral general circulation model based on the above-mentioned numerical methods.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time-Space Spectral General Circulation Model I. Time-Space Spectral Model of Low-Order Barotropic System with Periodic Forcing

Iwashima

Yamamoto

1986

Journal of the Meteorological Society of Japan

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“…The harmonic balance method [13,14] is very commonly used to ÿnd periodic solutions for nonlinear dynamical systems. In the harmonic balance method, the Fourier coe cients of the residuals are required to be zero.…”

Section: Harmonic Balance and Trigonometric Collocationmentioning

confidence: 99%

Generalized Lagrange functions and weighting coefficient formulae for the harmonic differential quadrature method

Fung

2003

Numerical Meth Engineering

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SUMMARYIn the harmonic di erential quadrature method, truncated Fourier series comprising the trigonometric functions are used to approximate the solutions. The generalized Lagrange functions composed of trigonometric functions are constructed as interpolation functions so that the unknowns are the function values, rather than the Fourier coe cients. In the spirit of the di erential quadrature method, the derivatives at a sampling grid point are expressed as weighted linear sums of function values at all the sampling grid points. It is shown that the corresponding weighting coe cients of higher order derivatives can be evaluated recursively and the general explicit formulae are given in this paper. The di erential quadrature analog of the governing equations can then be established easily at the sampling grid points. For the periodic boundary value problems, the periodic boundary conditions are satisÿed automatically and no other boundary conditions are required in general. It is also shown that the harmonic di erential quadrature method is related to the trigonometric collocation method and the harmonic balance method. Numerical examples are given to illustrate the validity and e ciency of the present method.

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“…The method used by Wood is a variation of that used by Urabe and Reiter [1966]. Predergast and Wood [1982] Having obtained these approximate orbits we may determine their stability by calculating the Floquet multipliers of each orbit.…”

Section: The Numerical Methodsmentioning

confidence: 99%

Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

Holmes¹

1985

Proc. R. Soc. Lond. A

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Numerical computation of nonlinear forced oscillations by Galerkin's procedure

Cited by 162 publications

References 2 publications

Time-Space Spectral General Circulation Model I. Time-Space Spectral Model of Low-Order Barotropic System with Periodic Forcing

Time-Space Spectral General Circulation Model I. Time-Space Spectral Model of Low-Order Barotropic System with Periodic Forcing

Generalized Lagrange functions and weighting coefficient formulae for the harmonic differential quadrature method

Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

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