In this paper, we present a parallel across time and space algorithm, based
on an implicit collocation method, for the heat transfer equation.
The solution is approximated by polynomials,
the coefficients of which are computed from a block-tridiagonal linear system.
The optimal degree of the polynomials is the highest degree for which all the
coefficients are significant.
It is obtained with the use of the CADNA library.
If the time step and the space step increase,
this optimal degree may increase and the time interval where the solution can
be computed in parallel becomes larger.
Once the polynomials have been determined for a given time interval,
the solution can be accurately computed in parallel at any point of this interval.
If the number of points where the solution is computed
in the considered time interval is sufficient, it appears that
the run-time performances of the
implicit collocation method on a parallel machine are better than
those of an explicit finite difference method.
Numerical experiments on an MIMD architecture are presented.

Keywords:
partial differential equations, collocation methods, finite
difference methods, numerical validation, CESTAC method, CADNA library.