Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability |
| |
Authors: | E H Twizell A Boutayeb K Djidjeli |
| |
Institution: | (1) Department of Mathematics and Statistics, Brunel University, UB8 3PH Uxbridge, Middlesex, UK;(2) Department of Ship Science, University of Southampton, S09 5NH Southampton, UK;(3) Present address: Département de Mathématiques, Université Mohamed I, Oujda, Maroc, (Morocco) |
| |
Abstract: | Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order
eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability
setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field
also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled
by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential
equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations
and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from
the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems
of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical
results obtained using the two approaches are compared with estimates appearing in the literature. |
| |
Keywords: | High-order eigenvalue problems rotation magnetic field instabilities finite difference methods |
本文献已被 SpringerLink 等数据库收录! |
|