Proper use of Schwarzschild Ledoux criteria in stellar evolution computations

The era of detailed asteroseismic analyses opened by space missions such as CoRoT and Kepler has highlighted the need for stellar models devoid of numerical inaccuracies, in order to be able to diagnose which physical aspects are being ignored or poorly treated in standard stellar modeling. We tackle here the important problem of fixing convective zone boundaries in the frame of the local mixing length theory. First we show that the only correct way to locate a convective zone boundary is to find, at each iteration step, through interpolations or extrapolations from points within the convective zone, the mass where the radiative luminosity is equal to the total luminosity. We then discuss two misuses of the boundary condition and the ways they affect stellar modeling and stellar evolution. The first consists in applying the neutrality condition for convective instability on the radiative side of the convective boundary. The second way of misusing the boundary condition comes from the process of fixing the convective boundary through the search for a change of sign of a possibly discontinuous function. We show that these misuses can lead to completely wrong estimates of convective core sizes with important consequences for the following evolutionary phases. We point out the advantages of using a double mesh point at each convective zone boundary. The specific problem of a convective shell is discussed and some remarks concerning overshooting are given.