350
11. Efficiency and Accuracy Improvement
single grid method, the variation was by a factor of five to seven (higher for
finer grids; see Fig. 7.15). However, for turbulent flows, heat transfer, etc.,
under-relaxation may affect the multigrid method substantially.
The full multigrid method described above provides solutions on all grids.
The cost of solving on all of the coarse grids is about 30% of the cost of the
finest grid solution. If the solution process were started on the finest grid with
zero fields, more total effort is needed than if the FMG approach is used. The
savings resulting from more accurate initial fields usually outweigh the cost
needed to obtain them. In addition, having solutions on a series of grids allows
evaluation of discretization errors, as discussed in Sect. 3.9 and provides a
basis for grid refinement. The grid refinement process may be stopped when
the desired accuracy is achieved. Also, Richardson extrapolation may be used.
The multigrid approach to accelerating outer iterations described above
can be applied to any solution method for the Navier-Stokes equations.
Vanka (1986) applied it to the point-coupled solution method; Hutchinson
and Raithby (1986) and Hutchinson et al. (1988) use it with a line-coupled
solution technique. Methods of the SIMPLE type and fractional-step methods are also well suited for multigrid acceleration; see e.g. Hortmann et al.
(1990) and Thompson and Ferziger (1989). The role of the smoother is now
taken by the basic algorithm (e.g. SIMPLE); the linear equation solver plays
a minor role.
In Table 11.1 we compare the numbers of outer iterations needed to solve
the 2D lid-driven cavity flow problem at Reynolds numbers Re = 100 and
Re = 1000 using different solution strategies. SG denotes the single-grid
method with a zero initial field. PG denotes the prolongation scheme, in
which the solution from the next coarser grid is used to provide the initial
field. MG denotes the multigrid method using V-cycles, with the finest grid
having zero initial fields. Finally, FMG denotes the multigrid method described above, which can be considered a combination of the PG and MG
schemes.
The results show that, for the Re = 100 case, MG and FMG need about
the same number of outer iterations on the finest grid level - a quality initial
guess does not save much. For the single-grid scheme, the savings are substantial: on the 128 x 128 CV grid, the number of iterations is reduced by a
factor of 3.5. The multigrid method reduces the number of iterations on that
grid by a factor of 15; this factor increases as the grid is refined. The number
of iterations remains constant in the MG and FMG methods from the third
grid onwards, while it increases by a factor of four in SG and by a factor of
2.5 in the PG scheme.
For high Reynolds number flows, the situation changes a bit. SG needs
fewer iterations than at Re = 100, except on coarse grids, on which the
use of CDS slows convergence. PG reduces the number of iterations by less
than a factor of two. MG needs about twice as many iterations as it did for
11. Efficiency and Accuracy Improvement
single grid method, the variation was by a factor of five to seven (higher for
finer grids; see Fig. 7.15). However, for turbulent flows, heat transfer, etc.,
under-relaxation may affect the multigrid method substantially.
The full multigrid method described above provides solutions on all grids.
The cost of solving on all of the coarse grids is about 30% of the cost of the
finest grid solution. If the solution process were started on the finest grid with
zero fields, more total effort is needed than if the FMG approach is used. The
savings resulting from more accurate initial fields usually outweigh the cost
needed to obtain them. In addition, having solutions on a series of grids allows
evaluation of discretization errors, as discussed in Sect. 3.9 and provides a
basis for grid refinement. The grid refinement process may be stopped when
the desired accuracy is achieved. Also, Richardson extrapolation may be used.
The multigrid approach to accelerating outer iterations described above
can be applied to any solution method for the Navier-Stokes equations.
Vanka (1986) applied it to the point-coupled solution method; Hutchinson
and Raithby (1986) and Hutchinson et al. (1988) use it with a line-coupled
solution technique. Methods of the SIMPLE type and fractional-step methods are also well suited for multigrid acceleration; see e.g. Hortmann et al.
(1990) and Thompson and Ferziger (1989). The role of the smoother is now
taken by the basic algorithm (e.g. SIMPLE); the linear equation solver plays
a minor role.
In Table 11.1 we compare the numbers of outer iterations needed to solve
the 2D lid-driven cavity flow problem at Reynolds numbers Re = 100 and
Re = 1000 using different solution strategies. SG denotes the single-grid
method with a zero initial field. PG denotes the prolongation scheme, in
which the solution from the next coarser grid is used to provide the initial
field. MG denotes the multigrid method using V-cycles, with the finest grid
having zero initial fields. Finally, FMG denotes the multigrid method described above, which can be considered a combination of the PG and MG
schemes.
The results show that, for the Re = 100 case, MG and FMG need about
the same number of outer iterations on the finest grid level - a quality initial
guess does not save much. For the single-grid scheme, the savings are substantial: on the 128 x 128 CV grid, the number of iterations is reduced by a
factor of 3.5. The multigrid method reduces the number of iterations on that
grid by a factor of 15; this factor increases as the grid is refined. The number
of iterations remains constant in the MG and FMG methods from the third
grid onwards, while it increases by a factor of four in SG and by a factor of
2.5 in the PG scheme.
For high Reynolds number flows, the situation changes a bit. SG needs
fewer iterations than at Re = 100, except on coarse grids, on which the
use of CDS slows convergence. PG reduces the number of iterations by less
than a factor of two. MG needs about twice as many iterations as it did for