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

## Abstract

The onset of convection in a horizontal layer of fluid heated from below in the presence of a gravity field varying across the layer is numerically investigated. The eigenvalue problem governing the linear stability of the mechanical equilibria of the fluid, in the case of free boundaries, is a sixth order differential equation with Dirichlet and hinged boundary conditions. It is transformed into a system of second order differential equations supplied only with Dirichlet boundary conditions. Then it is solved using two distinct classes of spectral methods namely, weighted residuals (Galerkin type) methods and a collocation (pseudospectral) method, both based on Chebyshev polynomials. The methods provide a fairly accurate approximation of the lower part of the spectrum without any scale resolution restriction. The Viola’s eigenvalue problem is considered as a benchmark one. A conjecture is stated for the first eigenvalue of this problem.

## Authors

Călin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Florica-Ioana Dragomirescu

## Keywords

Bénard convection; Variable gravity field; Hydrodynamic stability; High order two-point boundary value problem; Hinged boundary conditions; Spectral methods

### References

See the expanding block below.

## Cite this paper as

C.I. Gheorghiu, F.-I. Dragomirescu, Spectral methods in linear stability. Applications in thermal convection with variable gravity field, Appl. Numer. Math., 59 (2009) 1290-1302
doi: 10.1016/j.apnum.2008.07.004

## PDF

https://www.romai.ro/documente_poze/PremiulAG/2010/DragomirescuFI/DragomirescuL2.pdf

##### Journal

Applied Numerical Mathematics

Elsevier

0168-9274