Page 153 - Computational Fluid Dynamics for Engineers

Problems 139

(a) By using Eqs. (P4.12.4-5) and (P4.15.1), show that the error at the nth
iteration can be expressed as

N-l N-l r
2
e ( n ) = E Ag(P w )cgw fc = £ l - 2 o ; s i n ( - ^ ) cgw, (P4.15.2)

and thus high and low frequency errors can be identified.
(b) Show that for 0 < u < 1

A /c (P^) < 1 1 < f c < 7 V - l

(c) Show that the weighted Jacobi iteration method converges with the rate
/ 7T AT\ 7T^Z\T
max |Afc(Pa;)| = Ai(Po;) = l - 2 u ; s i n 2 I - — J « 1 - w—-— (P4.15.3)

which implies that the eigenvalue associated with the lowest frequency A^P^)
will always be close to 1 for any value u.
(d) Show that, by requiring

Ayv/ 2(Pu;) = -A N (Pa;), (P4.15.4)

the optimal value of uu — 2/3. This means that |Afc(Po;)| < 1/3 for all high fre-
quency wave numbers N'/2 < k < N — 1.
Note uo = 2/3 is independent of grid size Ax, and the errors corresponding to
high frequency wave number are reduced much faster than those corresponding
to low-frequency numbers.

4-16. Develop a 4-level multigrid program to solve

n
—u (x) — IT sin7rx 0 < x < 1
subject to the boundary conditions

u(0) = u(l) = 0.
Also show that the convergence rate is independent of grid size.