  We carry out a systematic numerical stability analysis of ZND detonations of
Majda's model with Arrhenius-type ignition function, a simplified model for
reacting flow, as heat release and activation energy are varied. Our purpose
is, first, to answer a question of Majda whether oscillatory instabilities can
occur for high activation energies as in the full reacting Euler equations,
and, second, to test the efficiency of various versions of a numerical
eigenvalue-finding scheme suggested by Humpherys and Zumbrun against the
standard method of Lee and Stewart. Our results suggest that instabilities do
not occur for Majda's model with Arrhenius-type ignition function, nor with a
modified Arrhenius-type ignition function suggested by Lyng--Zumbrun, even in
the high-activation energy limit. We find that the algorithm of
Humpherys--Zumbrun is in the context of Majda's model $100$-$1,000$ times
faster than the one described in the classical work of Lee and Stewart and
$1$-$10$ times faster than an optimized version of the Lee--Stewart algorithm
using an adaptive-mesh ODE solver
