11.5 Parallel Computing in CFD
359
send UE(ie, j ) , UN(ie, j ) t o e a s t neighbor ;
end j;
f o r m = 1 t o M d o :
f o r j = 2 t o N j - l d o :
r e c e i v e R(i, - 1, j) from west neighbor;
f o r i = i , t o i , do:
c a l c u l a t e p(i, j), R(i, j);
end i;
send R(ie, j) t o e a s t neighbor ;
end j ;
f o r j = N j - 1 t o 2 s t e p - 1 do:
r e c e i v e S(ie + 1, j) from e a s t neighbor;
f o r i = ze to is s t e p -1 do:
c a l c u l a t e S(i, j);
update v a r i a b l e ;
end i;
send S(i,,j) t o west neighbor;
end j;
end m.
The problem is that this parallelization technique requires a lot of (fine
grain) communication and there are idle times at the beginning and end
of each iteration; these reduce the efficiency. Also, the approach is limited
to structured grids. Bastian and Horton (1989) obtained good efficiency on
transputer-based machines, which have a favorable ratio of communication
t o computation speed. With a less favorable ratio, the method would be less
efficient.
The conjugate gradient method (without preconditioning) can be parallelized straightforwardly. The algorithm involves some global communication
(gathering of partial scalar products and broadcasting of the final value),
but the performance is nearly identical to that on a single processor. However, t o be really efficient, the conjugate gradient method needs a good preconditioner. Since the best pre-conditioners are of the ILU-type (SIP is a very
good pre-conditioner), the problems described above come into play again.
The above development shows that parallel computing environments require redesign of algorithms. Methods that are excellent on serial machines
may be almost impossible to use on parallel machines. Also, new standards
have to be used in assessing the effectiveness of a method. Good parallelization of implicit methods requires modification of the solution algorithm. The
performance in terms of the number of numerical operations may be poorer
than on a serial computer, but if the load carried by the processors is equalized
and the communication overhead and computing time are properly matched,
the modified method may be more efficient overall.
359
send UE(ie, j ) , UN(ie, j ) t o e a s t neighbor ;
end j;
f o r m = 1 t o M d o :
f o r j = 2 t o N j - l d o :
r e c e i v e R(i, - 1, j) from west neighbor;
f o r i = i , t o i , do:
c a l c u l a t e p(i, j), R(i, j);
end i;
send R(ie, j) t o e a s t neighbor ;
end j ;
f o r j = N j - 1 t o 2 s t e p - 1 do:
r e c e i v e S(ie + 1, j) from e a s t neighbor;
f o r i = ze to is s t e p -1 do:
c a l c u l a t e S(i, j);
update v a r i a b l e ;
end i;
send S(i,,j) t o west neighbor;
end j;
end m.
The problem is that this parallelization technique requires a lot of (fine
grain) communication and there are idle times at the beginning and end
of each iteration; these reduce the efficiency. Also, the approach is limited
to structured grids. Bastian and Horton (1989) obtained good efficiency on
transputer-based machines, which have a favorable ratio of communication
t o computation speed. With a less favorable ratio, the method would be less
efficient.
The conjugate gradient method (without preconditioning) can be parallelized straightforwardly. The algorithm involves some global communication
(gathering of partial scalar products and broadcasting of the final value),
but the performance is nearly identical to that on a single processor. However, t o be really efficient, the conjugate gradient method needs a good preconditioner. Since the best pre-conditioners are of the ILU-type (SIP is a very
good pre-conditioner), the problems described above come into play again.
The above development shows that parallel computing environments require redesign of algorithms. Methods that are excellent on serial machines
may be almost impossible to use on parallel machines. Also, new standards
have to be used in assessing the effectiveness of a method. Good parallelization of implicit methods requires modification of the solution algorithm. The
performance in terms of the number of numerical operations may be poorer
than on a serial computer, but if the load carried by the processors is equalized
and the communication overhead and computing time are properly matched,
the modified method may be more efficient overall.