Page 151 - Computational Fluid Dynamics for Engineers

Problems 137

Fig. P4.2. The eigenvectors v/k,i = s i n ( ^ ) , 0 < i < N with wave numbers k = 1, 3, 6.

2w w
~ Wfc,i-1 + k,i ~ k,i+l = ^k™k,i 0 < i < N (P4.ll.7)

4-12. There are several iterative methods that can be used to solve Eq.
(P4.11.3). A simple one is the Jacobi iteration method, in which for an ini-
tial guess u\ , new values of u\ n ' are obtained from the previous values of
u\ n according to

2
u( n + 1 ) = i ( u W + u £ > 1 + Ai; /i) l<i<N-l (P4.12.1)
An important modification to Eq. (P4.12.1) results by introducing a weighted
factor u. First the intermediate values are calculated as in (P4.12.1)

2
i
v* = f a j j i + u^X + Ax fi) l<i<N-l (P4.12.2a)
and to obtain u\ n ) that is,

u(n+l) = u(n) + ^ _ (n)j 1 < z < A^ - 1 (P4.12.2b)
u

which generates an entire family of iterations called the weighted Jacobi iteration
method. These iteration sweeps are continued until convergence. Note that for
UJ — 1, the original Jacobi iteration is recoved
(a) Show that the Jacobi iteration method can be expressed as
n
D u (n+i) = ( L + u)u( ) + f (P4.12.3)

or
u (n+l) = (j _ D - l ) u (n) + D - l f (P4.12.4)
A
where I denotes the identity matrix,