Page 134 - Computational Fluid Dynamics for Engineers

120 4. Numerical Methods for Model Parabolic and Elliptic Equations

an a\2 aij • • a in
Q>21 «22 • Cl2n
A = (4.5.24)
an &i2 a ij ' din
a„A a„A • • a„„

(
and x = # i , . . . , Xi,..., x n) T and b= (&i,..., b{,..., b n) T with T denoting the
transpose. According to the Gaussian elimination method, the elements of x are
given by

k('-l) (i-l)
n
Xi — E i == , . . . , 1 (4.5.25)
,(*-!)
.7=2+1
where
fc = 1, .. ,71 — 1
.
(fc-i)
(fc) _ (fc-i) _ <hk (k-i) j — k + 1,..., n
a
,
% ~% (k-l) kj ' i — k + 1,... n (4.5.26a)
a kk
(o) _
,
(fc-i) k — 1,... n — 1
, (fc) _ , (*-l) _ <hk , (fc-i)
,
i — k + 1,... n (4.5.26b)
a l {0)
^ kk b = 6.
Table 4.2 gives the FORTRAN listing based an the Gaussian elimination. Thus,
the block-elimination method together with the Gaussian elimination method
Table 4.2. FORTRAN Listing of Subroutine GAUSS

SUBROUTINE GAUSS (N.M.A.B)
DIMENSION A(100,100),B(100,100)
DO 100 K = l.N-1
KP = K + 1
DO 100 I = KP,N
R = A(I,K)/A(K,K)
DO 200 J = KP,N
200 A(I,J) = A(I,J) - R*A(K,J)
DO 100 J = 1,M
100 B(I,J) = B(I,J) - R*B(K,J)
DO 300 K = 1,M
B(N,K) = B(N,X)/A(N,N)
DO 300 I = N-1,1,-1
IP = 1 + 1
DO 400 J = IP,N
400 B(I,K) = B(I,K) - A(I,J)*B(J,K)
300 B(I,K) = B(I,K)/A(I,I)
RETURN
END