  The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.
