  Using analytical and numerical Evans-function techniques, we examine the
spectral stability of strong-detonation-wave solutions of Majda's scalar model
for a reacting gas mixture with an Arrhenius-type ignition function. We
introduce an efficient energy estimate to limit possible unstable eigenvalues
to a compact region in the unstable complex half plane, and we use a numerical
approximation of the Evans function to search for possible unstable eigenvalues
in this region. Our results show, for the parameter values tested, that these
waves are spectrally stable. Combining these numerical results with the
pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [J.
Differential Equations 233 (2007), no. 2, 654-698.], we conclude that these
waves are nonlinearly stable. This represents the first demonstration of
nonlinear stability for detonation-wave solutions of the Majda model without a
smallness assumption. Notably, our results indicate that, for the simplified
Majda model, there does not occur, either in a normal parameter range or in the
limit of high activation energy, Hopf bifurcation to "galloping" or "pulsating"
solutions as is observed in the full reactive Navier-Stokes equations. This
answers in the negative a question posed by Majda as to whether the scalar
detonation model captures this aspect of detonation behavior.
