Spectral methods in linear stability. Applications to thermal convection with variable gravity field

Gheorghiu, Călin-Ioan; Dragomirescu, F.I.

doi:10.1016/j.apnum.2008.07.004

Cited by 23 publications

(19 citation statements)

References 21 publications

(35 reference statements)

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“…5 Discretization of Mathieu's system as a MEP Our previous numerical experiments concerning non-standard, high order and singularly perturbed eigenvalue problems reported in [21,22,23] proved that the Chebyshev collocation method is fairly accurate, flexible and implementable. It turned out to be superior to the spectral Galerkin or tau method also based on the Chebyshev polynomials.…”

Section: Overview Of Jacobi-davidson Methodsmentioning

confidence: 99%

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Spectral collocation solutions to multiparameter Mathieu’s system

Gheorghiu¹,

Hochstenbach

Plestenjak

et al. 2012

Applied Mathematics and Computation

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Section: Overview Of Jacobi-davidson Methodsmentioning

confidence: 99%

“…Unfortunately, the matrices e,π D 2 n and e,π D 2 nd are dense, non-symmetric and have high condition numbers (see, for instance, [23]). …”

Section: Overview Of Jacobi-davidson Methodsmentioning

confidence: 99%

Spectral collocation solutions to multiparameter Mathieu’s system

Gheorghiu¹,

Hochstenbach

Plestenjak

et al. 2012

Applied Mathematics and Computation

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“…In spite of this fact,up to our knowledge, the classical QZ method was used in quasitotality of such singular generalized eigenvalue problems encountered in linear hydrodynamic stability; see for one of the first applications of QZ in this context. Our papers and are not an exception to this situation.…”

Section: Introductionmentioning

confidence: 76%

Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems

Gheorghiu¹,

Rommes

2012

Numerical Methods in Fluids

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SUMMARY The aim of this paper is to show that the Jacobi–Davidson (JD) method is an accurate and robust method for solving large generalized algebraic eigenvalue problems with a singular second matrix. Such problems are routinely encountered in linear hydrodynamic stability analysis of flows that arise in various areas of continuum mechanics. As we use the Chebyshev collocation as a discretization method, the first matrix in the pencil is nonsymmetric, full rank, and ill‐conditioned. Because of the singularity of the second matrix, QZ and Arnoldi‐type algorithms may produce spurious eigenvalues. As a systematic remedy of this situation, we use two JD methods, corresponding to real and complex situations, to compute specific parts of the spectrum of the eigenvalue problems. Both methods overcome potentially severe problems associated with spurious unstable eigenvalues and are fairly stable with respect to the order of discretization. The real JD outperforms the shift‐and‐invert Arnoldi method with respect to the CPU time for large discretizations. Three specific flows are analyzed to advocate our statements, namely a multicomponent convection–diffusion in a porous medium, a thermal convection in a variable gravity field, and the so‐called Hadley flow. Copyright © 2012 John Wiley & Sons, Ltd.

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“…[22,23,26,37,109,110,115,116,154,155,164,179,206,213,222,223,267,275,385,487,488,498]. This class of problem is very difficult to handle mathematically due to the fact that the mathematical operators which arise in the instability analysis are non-symmetric and the resulting eigenfunctions are close to being linearly dependent.…”

Section: Poiseuille Flow Slip Boundary Conditionsmentioning

confidence: 99%

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Straughan

2015

Advances in Mechanics and Mathematics

111

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In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

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Spectral methods in linear stability. Applications to thermal convection with variable gravity field

Cited by 23 publications

References 21 publications

Spectral collocation solutions to multiparameter Mathieu’s system

Spectral collocation solutions to multiparameter Mathieu’s system

Application of the Jacobi–Davidson method to accurate analysis of singular linear hydrodynamic stability problems

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

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