The compound matrix method for ordinary differential systems

Ng, B. S.; Reid, W. H.

doi:10.1016/0021-9991(85)90177-9

Cited by 68 publications

(57 citation statements)

References 5 publications

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“…Although this method is in general superior to shooting techniques [7], this method too does not provide the overall picture of the eigenvalue spectrum. Building upon the paper by ٞ 2k Ϫ 3Ͱ 2 Ј 2k Ϫ 2Ͱ 2 UЈ 2k 2k /(c Ϫ U 2k ) Su and Khomami [9], as well as on our own experience [10], we therefore solve the eigenvalue problem by means…”

Section: Numerical Proceduresmentioning

confidence: 96%

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A Chebyshev Collocation Method for Solving Two-Phase Flow Stability Problems

Boomkamp

Boersma

Miesen

et al. 1997

Journal of Computational Physics

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Section: Numerical Proceduresmentioning

confidence: 96%

“…) and Reid [6,7] and Yiantsios and Higgins [8]. Although this method is in general superior to shooting techniques [7], this method too does not provide the overall picture of the eigenvalue spectrum.…”

Section: Numerical Proceduresmentioning

confidence: 99%

A Chebyshev Collocation Method for Solving Two-Phase Flow Stability Problems

Boomkamp

Boersma

Miesen

et al. 1997

Journal of Computational Physics

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“…Its numerical solution is not straightforward due to stiffness caused by extreme (too large or too small) values of the governing nondimensional parameters (large Reynolds and Peclet numbers, and small capillary number). We used two numerical methods to solve the problem, the shooting method supplemented with the Gram-Schmidt orthogonalization [48][49][50][51] and the compound matrix method [52][53][54]. The fourth-order adaptive Runge-Kutta technique was used for numerical integration and the Muller's method was employed for root finding.…”

Section: B Basic State Linear Stability Analysismentioning

confidence: 99%

Linear stability analysis of a horizontal phase boundary separating two miscible liquids

Kheniene

Vorobev

2013

Phys. Rev. E

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The evolution of small disturbances to a horizontal interface separating two miscible liquids is examined. The aim is to investigate how the interfacial mass transfer affects development of the Rayleigh-Taylor instability and propagation and damping of the gravity-capillary waves. The phase-field approach is employed to model the evolution of a miscible multiphase system. Within this approach, the interface is represented as a transitional layer of small but nonzero thickness. The thermodynamics is defined by the Landau free energy function. Initially, the liquid-liquid binary system is assumed to be out of its thermodynamic equilibrium, and hence, the system undergoes a slow transition to its thermodynamic equilibrium. The linear stability of such a slowly diffusing interface with respect to normal hydro-and thermodynamic perturbations is numerically studied. As a result, we show that the eigenvalue spectra for a sharp immiscible interface can be successfully reproduced for long-wave disturbances, with wavelengths exceeding the interface thickness. We also find that thin interfaces are thermodynamically stable, while thicker interfaces, with the thicknesses exceeding an equilibrium value, are thermodynamically unstable. The thermodynamic instability can make the configuration with a heavier liquid lying underneath unstable. We also find that the interfacial mass transfer introduces additional dissipation, reducing the growth rate of the Rayleigh-Taylor instability and increasing the dissipation of the gravity waves. Moreover, mutual action of diffusive and viscous effects completely suppresses development of the modes with shorter wavelengths.

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“…Therefore, it might be necessary to employ special techniques. For example we employed the method of compound matrices in [19] (refer to Ng and Reid [25] for a thorough description).…”

Section: Solution Of U (2)mentioning

confidence: 99%

“…This has been previously done in the numerical solution of the incremental equations of nonlinear elasticity by (amongst others) Haughton and Orr [13], Goriely et al [12] and the authors of this paper [19]. The method is sourced from Ng and Reid [25], to which the reader is referred for a more detailed explanation.…”

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confidence: 99%

The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis

MacLaurin

Chapman

Jones

et al. 2013

Journal of the Mechanics and Physics of Solids

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Fine sinusoidal wrinkling on the surfaces of mechanically compressed objects has been observed in many contexts over many years. In this paper we investigate such wrinkling through the application of a boundary layer analysis to an elastostatic problem in nonlinear elasticity. We determine the onset of buckling using a linear-stability analysis, and the leading-order postbuckling behaviour through consideration of higher-order terms of the energy. The object is assumed to (initially) 'preserve' its shape, so that these equations reduce to ordinary differential equations. We then apply a boundary-layer analysis to this problem, determining (in the asymptotic limit of large wavenumbers) the leading order behaviours of the eigenmode, the critical parameter, and the magnitude of the buckle. We find that the magnitude of a buckle with wavenumbers ςγ * 2 and ςγ * 3 (for fixed γ * 2 and γ * 3 ) has leading asymptotic order ς λ (2) , for an increment λ (2) of the critical parameter beyond the critical time of buckling. We provide electronic supplementary material which extends this analysis to that of incompressible elasticity. Finally we confirm the accuracy of our ansatz on a compressed NeoHookean ring.

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The compound matrix method for ordinary differential systems

Cited by 68 publications

References 5 publications

A Chebyshev Collocation Method for Solving Two-Phase Flow Stability Problems

A Chebyshev Collocation Method for Solving Two-Phase Flow Stability Problems

Linear stability analysis of a horizontal phase boundary separating two miscible liquids

The study of asymptotically fine wrinkling in nonlinear elasticity using a boundary layer analysis

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