Abstract
We address the challenge of running thermally pulsing-(super)asymptotic giant branch [TP-(S)AGB] models, with a 1D hydrostatic stellar evolution code, without suffering instabilities that terminate the evolution. We investigate two instabilities that usually occur during the luminosity peak following a thermal pulse: the hydrogen recombination instability and the Fe-peak instability. Both instabilities occur when the stellar mass is significantly reduced (M M i / 2) at the end of the TP-(S)AGB in our models with initial mass M i 2 M. The hydrogen recombination instability occurs due to the difficulty of modelling a thermally and dynamically unstable envelope in a 1D hydrostatic code, and is prevented by damping the energy released by hydrogen recombination in the outer envelope. The Fe-peak instability occurs when the radiation pressure drops at the base of the conv ectiv e env elope and is prev ented by boosting the conv ectiv e energy transport in this re gion. We pro vide custom routines to prevent these instabilities in the stellar evolution code MESA. The impact of these routines on the stellar structure is minimized so as to not affect the efficiency of third dredge-up, hot-bottom burning, or the wind mass-loss rate. We find only a modest reduction in third dredge-up efficiency at small envelope masses (M env 1. 0 M). Consequently, our M i = 5 M star, with hot-bottom burning, becomes a carbon star for the last ∼ 10 per cent of its thermally pulsing lifetime. The largest stellar radii are reached during the final thermal pulses, which may have important consequences for binary–star interactions.