Page 152 - Computational Fluid Dynamics for Engineers

138 4. Numerical Methods for Model Parabolic and Elliptic Equations

1 =

P j , is called the Jacobi iteration matrix

P j = (I - D _ 1 A ) = D ^ L + U) (P4.12.5)

(b) Show that the weighted Jacobi iteration method can be written as
l
u ( n + 1 ) = [(1 - CJ)I - wPjJuW + uoT> f (P4.12.6)
and the weighted Jacobi iteration matrix is
p ^ = [(l - )l - UJVJ] = I - C J D 1 A (P4.12.7)
u

4-13. The convergence rate of the weighted Jacobi iteration method can be
determined from the eigenvalues A/ C(P CJ) of P w . Show that

W . , . x . ~ . 2 k7T
A fc(Pu 1 - -X k(A) = 1 - 2u;siiT —- 1 < k < N - 1 (P4.13.1)

and the corresponding eigenvectors are the same as those given by Eq. (P4.11.6b)
and
P^W/e = A/ C(P U;)W/ C (P4.13.2)

4-14. The relation between the errors
(n)
U — U (P4.14.1)
in each iteration can be examined by
(a) Show that
u = (I - C J D 1 A ) U + w D ^ f (P4.14.2)

where u denotes the exact solution of Eq. (P4.11.4).
(b) Show that

e ( n + 1 ) = (I - CJD" 1 A)eW = P ^ e W (P4.14.3)

4-15. The initial error, denoted by,
e(o) = u _ (o)
u
can be represented by the eigenvectors of matrix P ^ in the form

(P4.15.1)
k=l