Data from the Global Oscillation Network Group (GONG) project and other helioseismic experiments provide a test for models of stellar interiors and for the thermodynamic and radiative properties, on which the models depend, of matter under the extreme conditions found in the sun. Current models are in agreement with the helioseismic inferences, which suggests, for example, that the disagreement between the predicted and observed fluxes of neutrinos from the sun is not caused by errors in the models. However, the GONG data reveal subtle errors in the models, such as an excess in sound speed just beneath the convection zone. These discrepancies indicate effects that have so far not been correctly accounted for; for example, it is plausible that the sound-speed differences reflect weak mixing in stellar interiors, of potential importance to the overall evolution of stars and ultimately to estimates of the age of the galaxy based on stellar evolution calculations.

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Abstract. The code CESAM is a consistent set of programs and routines which performs calculations of 1D quasi-static stellar evolution including diffusion and rotation. The principal new feature is the solution of the quasi-static equilibrium by collocation method based on piecewise polynomials approximations projected on their B-spline basis; that allows stable and robust calculations and the exact restitution of the solution not only at grid points. Another advantage is the monitoring by only one parameter of the accuracy and its improvement by superconvergence. An automatic mesh refinement has been designed for adjusting the localizations of grid points according to the changes of unknowns, each limit between a radiative and a convective zones being shifted to the closest grid point. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; for non-standard models the solution of the diffusion equation employs the Petrov-Galerkin scheme, with the mixing of chemicals in convective zones performed by strong turbulent diffusion. A precise restoration of the atmosphere is allowed for. CESAM computes evolution of stars from the pre-main sequence to the beginning of the 4 He burning 3α cycle. In this paper a detailed description of the algorithms is presented.