Observations and theory suggest that line-driven winds from hot stars and luminous accretion disks adopt a unique, critical solution that corresponds to maximum mass-loss rate. We analyze the numerical stability of the infinite family of shallow wind solutions, which resemble solar wind breezes, and their transition to the critical wind. Shallow solutions are subcritical with respect to radiative (or Abbott) waves. These waves can propagate upstream through shallow winds at high speeds. If the waves are not accounted for in the Courant time step, numerical runaway results. The outer boundary condition is equally important for wind stability. Pure outflow conditions, as assumed in the literature, trigger runaway of shallow winds to the critical solution or to accretion flow.
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