5.3 Iterative Methods
113
(prolonging) the update or correction from the coarse grid to the fine one;
the words in parentheses are special terms that are in common use in the
multigrid literature. Many choices are available for each item; they affect the
behavior of the method but, within the range of good choices, the differences
are not great. We shall thus present just one good choice for each item.
In a finite difference scheme, the coarse grid normally consists of every
second line of the fine grid. In a finite volume method, one usually takes the
coarse grid CVs to be composed of 2 (in two dimensions, 4, and in three
dimensions, 8) fine grid CVs; the coarse grid nodes then lie between the fine
grid nodes.
Although there is no reason to use the multigrid method in one dimension
(because the TDMA algorithm is very effective), it is easy to illustrate the
principles of the multigrid method and to derive some of the procedures used
in the general case. Thus consider the problem:
for which the standard FD approximation on a uniform grid is:
After performing n iterations on the grid with Ax spacing, we obtain an
approximate solution dn, and the above equation is satisfied to within the
residual pn:
Subtracting this equation from Eq. (5.61) gives
which is Eq. (5.15) for node i. This is the equation we want to iterate on the
coarse grid.
Fig. 5.2. The grids used in the multigrid technique in one dimension