Page 144 - Computational Fluid Dynamics for Engineers

130 4. Numerical Methods for Model Parabolic and Elliptic Equations

Once U is computed according to the above equation, then step 4 is carried
out on Q^. Similarly in F and G the corrections are

2
U 2h = U 2h + I £u 4h

and
\jh = U^ + l2h u2h -
Once a V-cycle is completed, the whole procedure, called the multigrid V-
cycle method (MV) and designated as

h
h
h
h
U ^MV (U ,F ),
is repeated until convergence. An algorithm that performs the above tasks is
given below.
V-Cycle Method (MV)

h
h
h
U h ^MV {V ,F )
h h h
1. Relax n\ times on A^\J — F on fi^ with initial guess U
2. If f2h = coarsest grid, then go to 4.
2h
Else F 2h = I {F h - A^U*)
U 2 / l = 0
2h
2h
2h
\j2h <-MV (U ,F )
3. Correct U h = U^ + I^ hU 2h
4. Relax n^ times on A^U^ 1 = F h on i?^ with initial guess U^
A second popular procedure, called the /i-Cycle method (M/x), is shown
below.

/Li-Cycle Method (Mfi)
/ l
/ l
U h ^ M / / ( U , F )
1. Relax n\ times on A^U^ 1 = F h on Q^ with initial guess XJ h
2. If £2^ = coarsest grid, then go to 4.
1
2h
Else F 2h = I (F h - A/.U' )
U 2 / l = 0
2h
2h
2h
U 2 / l <- Mfi {V , F ) fi times.
3. Correct V h = U h + I$ hU 2h
4. Relax ri2 times on A^U^ = F^ on Q^ with initial guess U^
The /i- Cycle method becomes the V- Cycle method for /i = 1. If fj, — 2, the
/i-Cycle method is the W-Cycle method (see Fig. 4.12). In practice, only /x = 1
and n = 2 are used.