Page 137 - Computational Fluid Dynamics for Engineers

4.5 Finite-Difference Methods for Elliptic Equations 123

DO 30 L=1,II
ELEMENTS OF DELTA MATRIX, SEE EQ. (4.4.20C)
DELTA(K,L,J) = DELTA(K,L,J)-BM(L,K)*E(L,J-1)
ELEMENTS OF W-VECTOR, SEE EQ. (4.4.21B)
W(K,J) = W(K,J) -BM(L,K) *W(L,J-1)
30 CONTINUE

BACKWARD SWEEP
DO 80 K=l,II
UT(K) = W(K,JJ)
DO 80 L=1,II
DM(K,L) = DELTA(K,L,JJ)
80 CONTINUE
SEE EQ. (4.4.22A)
CALL GAUSS(II,1,DM,UT)
SEE EQ. (4.4.22B) ", "
DO 90 J=JJ-1,1,-1
DO 100 K=l,II
U(K, J+l) = UT(K)
UT(K) = W(K,J)-E(K,J)*U(K,J+1)
DO 100 L=l, II
DM(K,L) = DELTA(K,L,J)
100 CONTINUE
CALL GAUSS(II,1,DM,UT)
90 CONTINUE
DO 110 K=1,II
U(K,1) = UT(K)
110 CONTINUE
RETURN
END

be taken to ensure that sufficient accuracy is obtained in a "reasonable" number
of iterations.
A useful point-iteration method is the Gauss-Seidel or successive iteration
method; the Laplace difference equations are written as

U Ux U U U U + U h
hj ~ \ i-lJ + i+lj) + V \ iJ-l hj+l) ° ^ (4 5 27)
/
1 < i < , 1 < j < J
In this scheme the evaluation of the new iterates is not completely arbitrary.
First compute u ^ and then, in order, the other elements on the coordi-
nate lines with j = 1. Next, u^ 2 is determined, etc. By slight changes in
the scheme, the calculations could start at either of the other three corners of
the rectangle (Fig. 4.7). However, all of these methods have the same rate of
convergence.
The rate of convergence for the Gauss-Seidel scheme is

2
4
= 26 TT 2 ( 1 + 1 ) + 0(6 ) (4.5.28)
R GS