Evolving past instabilities on the thermally pulsing- (super) asymptotic giant branch

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 Mᵢ/2) at the end of the TP-(S)AGB in our models with initial mass Mᵢ 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 convective envelope and is prevented by boosting the convective energy transport in this region. We provide 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 (Mₑnv 1.0~M_). Consequently, our Mᵢ=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.