Page 138 - Computational Fluid Dynamics for Engineers

124 4. Numerical Methods for Model Parabolic and Elliptic Equations

The convergence rate of the Gauss-Seidel method may be improved by in-
troducing an appropriate acceleration parameter. That is, set

(
v# +1) = e „£# + «&,,) + e y („<£!> + u% ) - *%• (4.5.29)
+1
x
and then, at the point (xi,yj), take
1
0+1) T/(^+ ) , n \ 0)
u. • ' = UJV- • - M l — UJ)U- '
i,3 i,3 ^ V } 1,3
)+w
1) u )
4 5 3
= «£ ^!r - S ) ( - - °)
Here uo is the acceleration parameter to be determined. Note that for UJ — 1 this
scheme reduces to that in Eq. (4.5.27), i.e. to the ordinary Gauss-Seidel method.
The order in which the components of the new iterates are to be computed is
just as in the previous successive scheme.
As discussed by Isaacson and Keller [3], the iterations in this scheme converge
if the acceleration parameter u lies in the interval 0 < u < 2, and its optimal
value, u; opt, is
2
^opt = r- ^ (4.5.31a)
1 + V 1 - V
where r\ = cos nAx for Ax = Ay and the rate of convergence is
2
2
= 26ir^2(l/a 2 + l/b ) + 0{8 ) (4.5.31b)
R AGS
By comparing Eq. (4.5.31b) with (4.5.28), it is seen that the power of 6 in the
rate of convergence for the optimal accelerated Gauss-Seidel (AGS) method is
lower than the power of 6 appearing in the ordinary Gauss-Seidel method. For
UJ > 1, the accelerated Gauss-Seidel is called successive over-relaxation (SOR)
and for UJ < 1, the method is called under-relaxation.
In the block iteration method, unlike the point iteration method in which
the next iterate is determined at each grid point (see Fig. 4.8a) the iterations
are performed at each column or row, as indicated in (b) and (c) of Fig. 4.8.
In either choice (column or row), one must solve a tridiagonal matrix with the
Thomas algorithm discussed in the previous subsection. To examine this in more
detail, consider the choice of column iterations; in this case, at each value of i,
Eq. (4.5.4b), can be written as
2
O y(uij+i + Uij-i) - Uij = 6 fij - 6 x(u i+ij + Ui-ij) (4.5.32)
and is solved for all values of j subject to the boundary conditions. Eq. (4.5.32)
is of the form given by Eq. (4.5.13) with coefficients aj, 6j, Cj and rj given by

aj = 8y, bj = -l, Cj = 9 y ^ ^
r = u
3 <5 fi,j ~ Ox(Ui+l,j + i-l,j)