Numerical Solution of Instability Problems

Abstract

The aim of this chapter is to provide the description of a numerical solution procedure for the differential eigenvalue problems encountered in the analysis of convective instability or absolute instability. The technique presented is a combination of a solver for initial value problems, based on a system of ordinary differential equations, and the shooting method. The solution procedure is illustrated starting from a specific convective instability problem, namely the Rayleigh–Bénard problem for a fluid layer bounded by a pair of impermeable rigid walls kept at different uniform temperatures. A specific numerical code for the implementation of the solution algorithm is developed, by using the open-source software environment Octave. A description of how the described numerical technique can be adapted for the solution of absolute instability problems is also provided.