Page 133 - Computational Fluid Dynamics for Engineers

4.5 Finite-Difference Methods for Elliptic Equations 119

Whether the boundary conditions are given in terms of u or its derivatives,
the solution of Eq. (4.5.6) can be obtained by the block-elimination method
as discussed in subsection 4.4.3 or Gauss' elimination method discussed below.
Since the coefficient matrix A has large blocks of zero elements, it is more ef-
ficient to solve Eq. (4.5.6) with the block-elimination method than with the
Gaussian elimination method. As we shall see shortly, however, it is still neces-
sary to make partial use of the Gaussian elimination in the block elimination
for the solution of elliptic equations.
The block elimination method for this problem is identical to the one dis-
cussed before except for the difference of the indices in the coefficient matrix,
Eq. (4.4.30). So for convenience the two steps in this method are repeated below.
In the first step of the forward sweep, Fj and Aj are computed from

(4.5.20a)
A x = A x

3
rjAj-i =Bj j = 2, , . . . J (4.5.20b)
2
Aj = Aj - TjCj-x j = , 3 , . . . J (4.5.20c)
In the second part of the forward sweep, the Wj are computed from

(4.5.21a)
w 1 = F x

Wj = Fj - FjWj_ x 2<j<J (4.5.21b)
In the backward sweep, the Uj are computed from

Ajuj = wj (4.5.22a)

AjUj = Wj - CjU j+1 j = J - l , J - 2 , . . . , 1 (4.5.22b)

In the application of the block elimination method to solve the Laplace
difference equations, the Aj matrix in Eqs. (4.5.20b) and (4.5.22) is a full matrix
of order / and is not a tridiagonal matrix except for J — 1. Thus, the inversion
of Aj is not a trivial task. On the other hand, in the application of this method
to solve the difference equations for boundary layers (Chapter 7), the order of
Aj matrix is generally small. For this reason, the inversion of the Aj matrix is
relatively simple.
To solve Eqs. (4.5.20b) and (4.5.22), we use the Gaussian elimination method
and write both equations in the form

Ax = b (4.5.23)

Here A = [aij] is a square matrix of order n, that is,