Unformatted text preview: T = 1 +2Ch
where h is some measure of the grid spacing, e.g., h = max( x y). The value of the
constant C can be determined by calculating !OPT on some coarse grids and then extrapolating to ner grids. The values of !OPT on a coarse grid is determined experimentally
by making several computations on the grid with di erent values of !. The value that
produces the fewest iterations for a given level of accuracy is assumed to be !OPT .
Some common variations of SOR iteration follow:
1. Red-Black (Checkerboard) ordering. We presented this ordering in Example 9.2.7.
As usual, let a point in a rectangular mesh be denoted as (j k). With red-black
ordering, we number all equations and unknowns at, e.g., odd values of j + k before
those with even values of j + k (Figure 9.2.3). Recall, that this gave us a system
of the form
F x1 = b1
Ax = D1 D
2 2 2 where D1 and D2 are diagonal and x1 and x2 correspond to unknowns at oddand even-numbered points, respectively. SOR iteration is performed rst on the
odd points and then on the even points. Note that the updating of an unknown
at each odd point is independent of that at any other odd point hence, they may
be done in parallel without a need for synchronization. Similarly, unknowns at all
even points may be updated in parallel.
2. Symmetric ordering (SSOR). Generally, the matrix M! is not symmetric even when
the original matrix A is. There are instances when it is important to maintain
symmetry, e.g., when using SOR iteration as a preconditioning for the conjugate
gradient method (Section 9.3). A symmetric iteration matrix can be obtained by 28 Solution Techniques for Elliptic Problems
performing a standard SOR sweep with, say, row ordering followed by one with the
reverse of this ordering.
3. Line or block procedures. Order the unknowns by rows, but gather all of the unknowns in a row into a vector to obtain a \block tridiagonal" system. For Poisson's
equation, this system was given as (8.1.4). It is partially reproduced here for con...
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