Other unusual features are the use of i) an automatic location of the numerical mesh with a variable total number of points and adjustment of grid points in close neighborhoods of each limit of convective zones, ii) a stiffly stable implicit Runge-Kutta scheme with orders up to four for the computation of the chemicals change, iii) finite elements approach for the calculation of diffusion, using Petrov-Galerkin formalism with a large turbulent diffusion for mixing the chemicals in convective zones, iv) precise reconstitution of the atmosphere with the stellar radius located at the level where the local temperature is equal to the effective temperature, v) mass changes due to the mass loss and mass defect are also allowed for.
The linearized equations are solved using the damped Newton-Raphson that allows to follow evolution with good time steps and reduced nets, even with shell sources. Although the set of lagrangian variables used so far allows to reach the best numerical accuracy, it does not permit to pursue the evolution towards more advanced phases if negative values of the local luminosity are expected; in such cases the calculations are possible with a numerically, less accurate set of eulerian variables.
The most constraining features are i) the use of the pressure, instead of the density, as state variable and ii) the necessity, for all the physical quantities, of numerically consistent derivatives with respect to the variables of state, i.e., pressure, temperature and chemical composition.
The extension of this code towards more advanced stages of stellar evolution is under investigations.
Acknowledgements
I acknowledge the suggestion by E. Schatzman and A. Baglin to undertake the development of this code; J. Provost and G. Berthomieu for advices, debugging and constructive criticisms of the results; J. Christensen-Dalsgaard, M. Gabriel, A. Noels, J. Reiter for stimulating and helpful discussions; C. van't Veer, Y. Lebreton, M.J. Goupil and my colleagues of GDR CNRS 131 for various contributions to CESAM. I wish to express my thanks to the anonymous referee whose comments and remarks greatly helped to improve the presentation of this paper.