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

  • Authors:

  • C. I. Gheorghiu;Florica-Ioana Dragomirescu

  • Affiliations:

  • “T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca 400110, Romania;Department of Mathematics, University “Politehnica” of Timisoara, P-ta Victoriei, No. 2, 300006, Romania

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2009

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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.