Figure 4: Panels a), b), c), e):
relative accuracy on the closure for,
respectively, the first four differential equations of stellar evolution; panel
d): enlargement for outer layers for the pressure gradient; panel f): relative
difference on number of baryons.
These plots are computed, using the improved solution, for more than 3500
points, randomly distributed in the model; the radius of the innermost point is
.
The accuracy of the solution is better than 
except in the
neighbourhood of the LMR and in the atmosphere
.
cycles chain (Clayton 1968): 1H,
2D, 3He, 4He, 7Li, 8Be, 12C, 13C,
14N, 15N, 16O, 17O; in some models, the species 2D, 7Li, 8Be are
assumed to be at equilibrium; the relevant nuclear reactions are interpolated
from the tabulated rates of Caughlan & Fowler (1988); weak screening is
assumed; the initial abundances are taken from Anders & Grevesse (1989)
meteoric abundances.
law derived from Kurucz's model atmospheres
(Morel et al. 1994).
by
superconvergence (see Appendix C2).
This nice numerical property avoids the Richarson extrapolation
(Christensen-Dalsgaard 1982) which is, with stiff problems, sometimes
unstable (Hairer & Wanner 1991).
That improvement is illustrated Fig. 3 (click here): for a
calibrated standard solar model, evolved from the PMS to the present age with
"standard accuracy" (sa), but with a constant number of shells fixed to
300, the
relative accuracy on the pressure gradient 
:
;
as seen, the relative precision, taking advantage of the super convergence, is
improved by one order of magnitude.
Figure 3 (click here) illustrates also the stability of the numerical solution near
the center, the possibility of recovering, with full accuracy, the
numerical solution everywhere and, the significant
improvement of the accuracy in the envelope due to the superconvergence.
, evolved from
ZAMS to 4550My, with "Solar accuracy" (Sa), the relative differences
between left and right hand sides of the four first differential equations of
Eq. (1 (click here)), namely the derivatives , 
,
and 
, have been computed from the improved solution of
fourth order; the results, plotted Fig. 4 (click here), allow an estimate of the
internal accuracy achieved. That model includes an atmosphere reconstructed (see
Sect. 3.2.4 (click here)) and smooth opacity data have been
provided using the bivariate rational spline interpolation scheme of
Houdek & Rogl (1996).
As seen, the differential equations are fulfilled within a relative
precision better than except in the neighbourhood of
the LMR at 
, and in the atmosphere for
.
At LMR the local drop of the precision is a consequence of the small
discontinuity of the density, caused by the mixing in the convective zone
where the nuclear reactions, though weak, are active; there is
discontinuous, the solution is only of class 
, the fourth order
accuracy cannot be reached; the effect is more sensitive for
than for the other gradients, Fig. 4 (click here) panel (c). The cause of
the low accuracy 
in the restored atmosphere, panel (d),
is a consequence of the non-linear dependence of the natural variable
, used for the restoration of the atmosphere, with respect to R through
the law and the opacity.
In the model, the closest grid
point from the center is located at 
;
in the plots, deliberately, the first point was located
at , i.e., quite near to the central singularities;
the stability of the numerical solution there is clearly illustrated
Fig. 4 (click here), panels (a), (b) (c) and (e).
Figure 5: Plots, with respect to the normalized radius, of
discontinuous quantities 
(dots) and 
(full) - 
is the mean molecular weight - in the neighbourhood of the LMR
of a solar model with microscopic diffusion; on the curve,
between two grid points 
, each dot corresponds to a
calculated and not to an interpolated value of
As demonstrated Appendix B4 and Appendix C4, the algorithm employed for the
calculation of the chemical composition are conservative; that is illustrated
Fig. 4 (click here) panel (f), where the local departure from the mean value of the
number of baryons,
is plotted; ni and Ai are, respectively, the baryonic number and the
atomic weight of Xi. As seen, the conservation of the number of baryon is
ensured within the machine accuracy except at LMR, due to the mixing which is
not made with so high numerical accuracy (see Sect. 3.3.2 (click here)) and, also to the
displacement of the LMR.
Figure 6: Evolutionary tracks of two 
models computed with low
(dotted) and super (full) accuracy from PMS initial conditions to the onset of
the cycle. Large time steps are responsible of the angular behavior
of the solution of low accuracy; the two tracks do not significantly
differ, they are superimposed on the main sequence
) and the quantity:
, difference between two gradients,
is equivalent to a second derivative - 
is the adiabatic exponent. In
the neighbourhood of the LMR of a model with microscopic diffusion (Michaud & Proffitt
1993), the above discontinuous functions have been derived from the
solution, taking advantage of the superconvergence; in Fig. 5 (click here) more
than 20 point are inserted between two adjacent grid points.
Emphasize is made on the following facts: i) the discontinuity is well marked,
ii) in its vicinity, even with jumps,
for the diffusion coefficients, as large as thirteen magnitudes, the solution has
no oscillation, iii) , a second
derivative, is smooth, iv) a grid point is placed right on the LMR.
have been evolved starting from PMS to the onset of
the 4He burning phase; these two models differ only by the numerical
accuracy achieved. The first one, is computed with low accuracy (la), the
second with
standard accuracy (sa), see Table 1 (click here); the numbers
of, time steps, shells at the onsets of PMS, ZAMS, POST and cycle
burning phase, are given Table 2 (click here). The
evolutionary tracks are notably stable, even for the rugged solution obtained
with (la); notice also that, on the main sequence,
the two tracks are superimposed.
| (la) | (sa) | |
 | 103 | 137 |
 | 250 | 270 |
 | 420 | 460 |
 | 430 | 480 |
 | 480 | 530 |