  Generalizing similar results for viscous shock and relaxation waves, we
establish sharp pointwise Green function bounds and linearized and nonlinear
stability for traveling wave solutions of an abstract viscous combustion model
including both Majda's model and the full reacting compressible Navier--Stokes
equations with artificial viscosity with general multi-species reaction and
reaction-dependent equation of state, % under the necessary conditions of
strong spectral stability, i.e., stable point spectrum of the linearized
operator about the wave, transversality of the profile as a connection in the
traveling-wave ODE, and hyperbolic stability of the associated Chapman--Jouguet
(square-wave) approximation. Notably, our results apply to combustion waves of
any type: weak or strong, detonations or deflagrations, reducing the study of
stability to verification of a readily numerically checkable Evans function
condition. Together with spectral results of Lyng and Zumbrun, this gives
immediately stability of small-amplitude strong detonations in the small
heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending
previous results obtained by energy methods by Liu--Ying and Tesei--Tan for
Majda's model and the reactive Navier--Stokes equations, respectively.
