## 3. Discretisation and numerical resolutionThe spherical nucleus with the radius is subdivided into concentric shells. The radial depth of the shells increases from surface to centre by an exponentially growing step. Each meridian is cut into pieces unequal in size. The corresponding latitudinal surface belts are chosen so that each belt receives an identical amount of solar flux for a zero obliquity of the nucleus rotation axis. To each belt corresponds the energetically averaged mean latitude The sphere is divided into segments of equal size. The subvolumes are discretised using a centred spatial grid. The set of diffusion equations can be solved by applying a finite
difference method. An efficient scheme for parallel computing is the
(with and the differential operator ) is differenced implicitly in two time steps weighted with the constant value Rearranging and writing in matrix vector notation, we have with . is a unit matrix. The right hand side can be evaluated readily as it contains only "old" values. The operators on the left hand side of Eq. (42) are tridiagonal matrices that can be solved by applying the Thomas algorithm. In order to represent derivatives accurately at the boundaries by a central difference formula we used the standard introduction of a fictitious grid point beyond the boundary. Stability is given for . We have chosen
( The computation is performed with the message-passing interface MPI
(Message-Passing Interface Forum 1995) on a CRAY T3D parallel
supercomputer. The used partition has to account for the stepwise
integration in the radial and meridional dimensions. We haven chosen a
regular domain decomposition for each integration step. After each
integration step the entire data matrix is updated by means of a
collective communication. One could reduce the number of transferred
data by using point to point communications with a structured data
type, but, the communication would be more time expensive because of a
non-contiguous data access. Various mathematical functions are
vetorised by using their corresponding subroutines of the
© European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |