11.3 Multigrid Methods for Flow Calculation
349
systematic refinement or coarsening - the grids may be arbitrary; it is only
important that the solution domain and boundary conditions be the same on
all levels, and that one grid is substantially coarser than the other (otherwise
computational efficiency is not improved). The restriction and prolongation
operators are then based on generic interpolation methods; such multigrid
methods are called algebraic multigrid methods (see e.g. Raw, 1995, and Weiss
et al., 1999).
For unsteady flows with implicit methods and small time steps, the outer
iterations usually converge very rapidly (residual reduction by an order of
magnitude per outer iteration), so multigrid acceleration is not necessary.
The biggest savings are achieved for fully elliptic (diffusion-dominated) problems, and the smallest for convection-dominated problems (Euler equations).
Typical acceleration factors range from 10 to 100 when five grid levels are
used. An example is given below.
When computing turbulent flows with the k-E turbulence model, interpolation may produce negative values of k and/or E in the early cycles of a
multigrid method; the corrections then have t o be limited to maintain positivity. In problems with variable properties, the properties may vary by several
orders of magnitude within the solution domain. This strong non-linear coupling of the equations may cause the multigrid method to become unstable.
It may be better to update some quantities (e.g. the turbulent viscosity in
the k - E turbulence model) only on the finest grid and keep them constant
within a multigrid cycle.
40
Fig. 11.7. Number of outer iterations on the finest grid in a
0
multigrid method a s a function of
the under-relaxation factor a , for
0.5
0.6
0.7
0.8
0.9 1.0 the lid-driven cavity flow at Re =
a,
1000
Under-relaxation factors are relatively unimportant in multigrid methods
for laminar flows; the methods are less sensitive to these parameters than the
single grid method. In Fig. 11.7 we show the dependence of the number of
required outer iterations to solve the lid-driven cavity problem a t Re = 1000
on the under-relaxation factor for velocity (using optimum under-relaxation
of pressure correction, see Sect. 7.8) for two grids. The number of iterations
varies by about 30% in the range of a , between 0.5 and 0.9, while for the
349
systematic refinement or coarsening - the grids may be arbitrary; it is only
important that the solution domain and boundary conditions be the same on
all levels, and that one grid is substantially coarser than the other (otherwise
computational efficiency is not improved). The restriction and prolongation
operators are then based on generic interpolation methods; such multigrid
methods are called algebraic multigrid methods (see e.g. Raw, 1995, and Weiss
et al., 1999).
For unsteady flows with implicit methods and small time steps, the outer
iterations usually converge very rapidly (residual reduction by an order of
magnitude per outer iteration), so multigrid acceleration is not necessary.
The biggest savings are achieved for fully elliptic (diffusion-dominated) problems, and the smallest for convection-dominated problems (Euler equations).
Typical acceleration factors range from 10 to 100 when five grid levels are
used. An example is given below.
When computing turbulent flows with the k-E turbulence model, interpolation may produce negative values of k and/or E in the early cycles of a
multigrid method; the corrections then have t o be limited to maintain positivity. In problems with variable properties, the properties may vary by several
orders of magnitude within the solution domain. This strong non-linear coupling of the equations may cause the multigrid method to become unstable.
It may be better to update some quantities (e.g. the turbulent viscosity in
the k - E turbulence model) only on the finest grid and keep them constant
within a multigrid cycle.
40
Fig. 11.7. Number of outer iterations on the finest grid in a
0
multigrid method a s a function of
the under-relaxation factor a , for
0.5
0.6
0.7
0.8
0.9 1.0 the lid-driven cavity flow at Re =
a,
1000
Under-relaxation factors are relatively unimportant in multigrid methods
for laminar flows; the methods are less sensitive to these parameters than the
single grid method. In Fig. 11.7 we show the dependence of the number of
required outer iterations to solve the lid-driven cavity problem a t Re = 1000
on the under-relaxation factor for velocity (using optimum under-relaxation
of pressure correction, see Sect. 7.8) for two grids. The number of iterations
varies by about 30% in the range of a , between 0.5 and 0.9, while for the