  We present numerical hydrodynamical modeling of the effects of a giant planet
or brown dwarf companion orbiting within the extended atmosphere and wind
formation zone of an approximately solar-mass Mira variable star. The
large-scale, time-dependent accretion flows within the radially oscillating and
outflowing circumstellar gas around Miras are related to Bondi-Hoyle-Lyttleton
flows, but have not, to our knowledge, been previously modelled. The new models
presented in this paper illustrate the changes in accretion and wake dynamics
as the companion mass is varied over a range from 10 to 50 Jupiter masses
($M_J$). The character of the accretion onto the companion changes greatly as
the companion mass is increased. At the lowest companion masses considered
here, a low continuous rate of mass accretion is punctuated by large, nearly
periodic bursts of accretion. When the companion mass is large, the mass
accretion has both a continuous part, and a rapidly varying, nearly stochastic
part. These trends can be understood as the result of the interplay between the
shocks and radial oscillations in the circumstellar gas, and the wake flow
behind the companion. Models with accretion bursts may produce observable
optical brightenings, and may affect SiO maser emission. (Abridged)
