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

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 Violaʹs eigenvalue problem is considered as a benchmark one. A conjecture is stated for the first eigenvalue of this problem.