5.8 Examples
131
Table 5.2. Numbers of iterations required by various solvers to reduce the normalized L1 error norm below lop5 for the 2D Laplace equation with Dirichlet boundary
conditions on a square domain X x Y = 1 x 1 with uniform grid in both directions
Grid
GS LGS-AD1 AD1 SIP ICCG MG-GS MG-SIP
8 x 8
74
22
16
8
7
12
7
16 x 16
292
77
31
20
13
10
6
32 x 32 1160
294
64
67
23
10
6
64 x 64 4622
1160
132
254
46
10
6
128 x 128
-
274 1001
9 1
10
6
256 x 256
-
181
10
6
Table 5.3. Numbers of iterations required by AD1 solver as a function of the time
step size (uniform grid in both directions, 64 x 64 CV)
1 /At
80
68
64
60
32
16
8
No.Iter. 152 134 132 134 234 468 936
ICCG is substantially faster than SIP; the number of iterations doubles
when the grid is refined, so its advantage is bigger on fine grids. Multigrid
solvers are very efficient; with SIP as the smoother, only 6 iterations on the
finest grid are required. In the MG solvers, the coarsest level was 2 x 2 CV,
so there were three levels on the 8 x 8 CV grid and eight levels on the 256 x
256 CV grid. One iteration was performed on the finest grid and on all grids
after prolongation, while 4 iterations were performed during the restriction
phase. In SIP, the parameter a was set to 0.92. No attempt was made t o find
optimum values of the parameters; the results obtained are representative
enough to show the trends and the relative performance of the various solvers.
Note also that the computing effort per iteration is different for each solver.
Using the cost of a GS iteration as a reference, we find the following relative
costs: LGS-AD1 - 2.5, AD1 - 3.0, SIP - 4.0 for the first iteration and 2.0
afterwards, ICCG - 4.5 for the first iteration and 3.0 afterwards. For MG
methods, one needs to multiply the number of iterations on the finest grid
by roughly 1.5 to account for the cost of iterations on coarse grids. MG-GS
is therefore computationally the most efficient solver in this case.
Since the rate of convergence is different for each solver, the relative cost
depends on how accurately we want to solve the equations. To analyze this
issue we have plotted the variation of the sum of absolute residuals, and the
variation of iteration error with iterations in Fig. 5.5. Two observations can
be made:
0 The fall of the residual sum is irregular initially, but after a certain number
of iterations, the convergence rate becomes constant. An exception is the
ICCG solver, which becomes faster as iterations go on. When very accurate
solution is needed, MG solvers and ICCG are the best choice. If - as is the
131
Table 5.2. Numbers of iterations required by various solvers to reduce the normalized L1 error norm below lop5 for the 2D Laplace equation with Dirichlet boundary
conditions on a square domain X x Y = 1 x 1 with uniform grid in both directions
Grid
GS LGS-AD1 AD1 SIP ICCG MG-GS MG-SIP
8 x 8
74
22
16
8
7
12
7
16 x 16
292
77
31
20
13
10
6
32 x 32 1160
294
64
67
23
10
6
64 x 64 4622
1160
132
254
46
10
6
128 x 128
-
274 1001
9 1
10
6
256 x 256
-
181
10
6
Table 5.3. Numbers of iterations required by AD1 solver as a function of the time
step size (uniform grid in both directions, 64 x 64 CV)
1 /At
80
68
64
60
32
16
8
No.Iter. 152 134 132 134 234 468 936
ICCG is substantially faster than SIP; the number of iterations doubles
when the grid is refined, so its advantage is bigger on fine grids. Multigrid
solvers are very efficient; with SIP as the smoother, only 6 iterations on the
finest grid are required. In the MG solvers, the coarsest level was 2 x 2 CV,
so there were three levels on the 8 x 8 CV grid and eight levels on the 256 x
256 CV grid. One iteration was performed on the finest grid and on all grids
after prolongation, while 4 iterations were performed during the restriction
phase. In SIP, the parameter a was set to 0.92. No attempt was made t o find
optimum values of the parameters; the results obtained are representative
enough to show the trends and the relative performance of the various solvers.
Note also that the computing effort per iteration is different for each solver.
Using the cost of a GS iteration as a reference, we find the following relative
costs: LGS-AD1 - 2.5, AD1 - 3.0, SIP - 4.0 for the first iteration and 2.0
afterwards, ICCG - 4.5 for the first iteration and 3.0 afterwards. For MG
methods, one needs to multiply the number of iterations on the finest grid
by roughly 1.5 to account for the cost of iterations on coarse grids. MG-GS
is therefore computationally the most efficient solver in this case.
Since the rate of convergence is different for each solver, the relative cost
depends on how accurately we want to solve the equations. To analyze this
issue we have plotted the variation of the sum of absolute residuals, and the
variation of iteration error with iterations in Fig. 5.5. Two observations can
be made:
0 The fall of the residual sum is irregular initially, but after a certain number
of iterations, the convergence rate becomes constant. An exception is the
ICCG solver, which becomes faster as iterations go on. When very accurate
solution is needed, MG solvers and ICCG are the best choice. If - as is the