5.8 Examples
133
Since the SIP solver is used in CFD on structured grids, we show the
dependence of the number of iterations required to reach convergence on the
parameter a in Fig. 5.6. For a = 0 the SIP reduces to the standard ILU
solver. With the optimum value of a , SIP is about six times as fast as ILU.
The problem with SIP is that the optimum value of a lies at the end of
the range of usable values: for a slightly larger than the optimum value, the
method does not converge. The optimum value usually lies between 0.92 and
0.96. It is safe to use a = 0.92, which is usually not optimum, but it provides
about five times the speed of the standard ILU method.
Some solvers are affected by cell aspect ratio, because the magnitudes of
coefficients becomes non-uniform. Using a grid with Ax = 10 Ay makes AN
and As 100 times larger than Aw and AE (see example section of the previous
chapter). To investigate this effect we solved the Laplace equation problem
described above on a rectangular region X x Y = 10 x 1 using the same
number of grid nodes in each direction. Table 5.4 shows numbers of iterations
required to reduce the normalized L1 residual norm below
for various
solvers. The GS solver is not affected, but it is no longer a suitable smoother
for the MG method. LGS-AD1 and SIP solvers become substantially faster
compared to the square grid problem. ICCG also performs slightly better.
MG-SIP is not affected but MG-GS deteriorates considerably.
Table 5.4. Numbers of iterations required by various solvers to reduce the normalized L1 residual norm below 10-"or
the 2D Laplace equation with Dirichlet
boundary conditions on a rectangular domain X x Y = 10 x 1 with uniform grid
in both directions
Grid
GS LGS-AD1 SIP ICCG MG-GS MG-SIP
This behavior is typical and is also found in convection/diffusion problems
(although the effect is less pronounced) and on non-uniform grids which have
both small and large cell aspect ratios. A mathematical explanation for the
worsening of GS and improvement of ILU performance with increasing aspect
ratio is given by Brandt (1984).
Finally we present some results for the solution of a Poisson equation with
Neumann boundary conditions in 3D. The pressure and pressure-correction
equations in CFD are of this type. The equation solved is:
133
Since the SIP solver is used in CFD on structured grids, we show the
dependence of the number of iterations required to reach convergence on the
parameter a in Fig. 5.6. For a = 0 the SIP reduces to the standard ILU
solver. With the optimum value of a , SIP is about six times as fast as ILU.
The problem with SIP is that the optimum value of a lies at the end of
the range of usable values: for a slightly larger than the optimum value, the
method does not converge. The optimum value usually lies between 0.92 and
0.96. It is safe to use a = 0.92, which is usually not optimum, but it provides
about five times the speed of the standard ILU method.
Some solvers are affected by cell aspect ratio, because the magnitudes of
coefficients becomes non-uniform. Using a grid with Ax = 10 Ay makes AN
and As 100 times larger than Aw and AE (see example section of the previous
chapter). To investigate this effect we solved the Laplace equation problem
described above on a rectangular region X x Y = 10 x 1 using the same
number of grid nodes in each direction. Table 5.4 shows numbers of iterations
required to reduce the normalized L1 residual norm below
for various
solvers. The GS solver is not affected, but it is no longer a suitable smoother
for the MG method. LGS-AD1 and SIP solvers become substantially faster
compared to the square grid problem. ICCG also performs slightly better.
MG-SIP is not affected but MG-GS deteriorates considerably.
Table 5.4. Numbers of iterations required by various solvers to reduce the normalized L1 residual norm below 10-"or
the 2D Laplace equation with Dirichlet
boundary conditions on a rectangular domain X x Y = 10 x 1 with uniform grid
in both directions
Grid
GS LGS-AD1 SIP ICCG MG-GS MG-SIP
This behavior is typical and is also found in convection/diffusion problems
(although the effect is less pronounced) and on non-uniform grids which have
both small and large cell aspect ratios. A mathematical explanation for the
worsening of GS and improvement of ILU performance with increasing aspect
ratio is given by Brandt (1984).
Finally we present some results for the solution of a Poisson equation with
Neumann boundary conditions in 3D. The pressure and pressure-correction
equations in CFD are of this type. The equation solved is: