Iterative eigenmode tracking for computing the saddle points of general dispersion relationships

This article presents a numerical approach which overcomes the notorious difficulty of identifying the eigenmodes when evaluating a fully numerical multivalued dispersion relationship. The utility of this approach, referred to as the SIVP-approach, is illustrated by means of a sample absolute/convective flow instability problem. Following an analysis on the numerical issue of eigenmode tracking and a synopsis of previous methods, of which none were successful in alleviating the issue for the sample problem, we propose an eigenmode-tracking approach that proves to be successful and computationally efficient. This primarily consists of making an estimation of the eigenmode of interest via the solution of a set of successive initial value problems (SIVPs), together with a correction step. The SIVP-approach is applicable for general numerical dispersion relationships and successfully enabled the saddle points of the dispersion relationship for the sample problem to be located and thereby for the absolute instability characteristics of this flow to be established. We show how our approach is readily incorporated into an iterative scheme which automates the computation of the absolute/convective instability characteristics. •Track the eigenmodes of a general dispersion relationship over the wavenumber space via the novel SIVP-approach.•Address the notorious numerical issue of eigenvalue collision due to intersections of different Riemann sheets.•Search for the saddle points of a dispersion relationship in an automated manner without traditional human intervention.•First allow the absolute/convective instability of an arbitrary instead of a specific eigenmode to be analysed.