Numerical aspects of searching convective/absolute instability transition

An overview of various numerical techniques used to determine the spatio-temporal character of instabilities in fluid flows is given. The advantageous features of various previously known individual techniques are discussed and a practical procedure combining them is suggested for a specific task of determining the complete boundary between linearly convectively and absolutely unstable regimes in a multi-parameter space in problems with a fully numerical dispersion relation. Special attention is paid to aspects of automatization of computations as this is a crucial condition for their efficiency. The suggested procedure is successfully used and is shown to provide a high degree of automatism in the physical example of non-Boussinesq mixed convection in a vertical channel. This example comprises most of the major numerical difficulties found in various spatio-temporal instability studies of two-dimensional fluid flows which previously could not be handled without frequent human intervention and visual inspection of intermediate results. This paper focuses on the general numerical aspects of the computations leaving the detailed discussion of the obtained physical results for a separate publication.