Porous Convection, the Chebyshev Tau Method, and Spurious Eigenvalues

Abstract

Highly efficient numerical techniques are discussed for solving eigenvalue problems which arise in convection driven instability problems in porous media. The differential equations are written as a system of second order or first order equations and boundary conditions are incorporated naturally into the generalised matrix eigenvalue problem which arises in order that the problem of suppression of spurious eigenvalues is addressed. The methods easily give high resolution in boundary layers, yield all the eigenvalues and eigenfunctions, deal with complex coefficients, and can handle spatially dependent coefficients in a very efficient manner. The numerical techniques are illustrated by application to two very practical instability problems, namely convective motion of brine in a layer of salty sediments off the coast of Alaska (Hutter and Straughan, 1997, 1999) and inclined temperature gradient convection (Nield, 1994, 1998a). The methods are applicable to many, many more practical porous convection problems and some are mentioned.