1 Introduction
2 Problem formulation
2.1 Unsteady case
2.1.1 Governing equations
2.1.2 Discretization
2.2 Steady case
2.2.1 Governing equations
2.2.2 Discretization
3 Numerical method
3.1 Nonlinear multigrid method
3.2 Multigrid components
Grid | Smoother | Conv. | Time (s) |
\(\frac{\parallel p_{h}-p\parallel _{\infty }}{\parallel p\parallel _{\infty }}\)
|
\(\frac{\parallel u_{h}-u\parallel _{\infty }}{\parallel u\parallel _{\infty }}\)
|
---|---|---|---|---|---|
\(16\times 16\times 2\)
| PGS | 0.20 | 0.06 |
\(6.89\times 10^{-3}\)
|
\(1.96\times 10^{-3}\)
|
Vanka | 0.19 | 0.08 | |||
\(32\times 32\times 4\)
| PGS | 0.20 | 0.24 |
\(9.83\times 10^{-4}\)
|
\(5.69\times 10^{-4}\)
|
Vanka | 0.18 | 0.46 | |||
\(64\times 64\times 8\)
| PGS | 0.20 | 1.56 |
\(1.81\times 10^{-4}\)
|
\(1.39\times 10^{-4}\)
|
Vanka | 0.17 | 3.30 | |||
\(128\times 128\times 16\)
| PGS | 0.20 | 11.48 |
\(5.69\times 10^{-5}\)
|
\(3.50\times 10^{-5}\)
|
Vanka | 0.17 | 25.58 |
4 Numerical results
4.1 Unsteady case
Grid | Smoother | Conv. | Time (s) |
---|---|---|---|
\(16\times 16\times 2\)
| PGS | 0.52 | 0.09 |
Vanka | 0.33 | 0.12 | |
\(32\times 32\times 4\)
| PGS | 0.67 | 0.90 |
Vanka | 0.44 | 0.89 | |
\(64\times 64\times 8\)
| PGS | 0.79 | 12.60 |
Vanka | 0.52 | 8.83 | |
\(128\times 128\times 16\)
| PGS | 0.88 | 161.67 |
Vanka | 0.60 | 89.44 | |
\(256\times 256\times 32\)
| PGS |
\(\diagup \)
|
\(\diagup \)
|
Vanka | 0.68 | 1001.82 |
Grid | Smoother | Conv. | Time (s) |
---|---|---|---|
\(16\times 16\times 2\)
| PGS | 0.42 | 0.07 |
Vanka | 0.35 | 0.11 | |
\(32\times 32\times 4\)
| PGS | 0.45 | 0.39 |
Vanka | 0.34 | 0.61 | |
\(64\times 64\times 8\)
| PGS | 0.45 | 2.88 |
Vanka | 0.32 | 4.45 | |
\(128\times 128\times 16\)
| PGS | 0.45 | 22.91 |
Vanka | 0.30 | 34.74 |
Grid | Smoother | Conv. | Time (s) |
---|---|---|---|
\(16\times 16\times 2\)
| PGS |
\(\diagup \)
|
\(\diagup \)
|
Vanka | 0.81 | 0.47 | |
\(32\times 32\times 4\)
| PGS |
\(\diagup \)
|
\(\diagup \)
|
Vanka | 0.76 | 2.82 | |
\(64\times 64\times 8\)
| PGS |
\(\diagup \)
|
\(\diagup \)
|
Vanka | 0.79 | 25.42 | |
\(128\times 128\times 16\)
| PGS |
\(\diagup \)
|
\(\diagup \)
|
Vanka | 0.80 | 215.44 | |
\(256\times 256\times 32\)
| PGS |
\(\diagup \)
|
\(\diagup \)
|
Vanka | 0.78 | 1736.54 |
4.2 Steady case
4.2.1 Homogeneous test
Smoother | F-cycle |
\(32\times 32\)
|
\(64\times 64\)
|
\(128\times 128\)
|
---|---|---|---|---|
PGS | F (2, 2) | 0.22 (0.13) | 0.21 (0.44) | 0.21 (1.64) |
F (3, 3) | 0.19 (0.15) | 0.19 (0.49) | 0.18 (2.00) | |
Vanka | F (2, 2) | 0.23 (0.25) | 0.23 (0.88) | 0.23 (3.58) |
F (3, 3) | 0.18 (0.28) | 0.18 (1.05) | 0.18 (3.98) |
4.2.2 Heterogeneous test
Smoother | F-cycle |
\(32\times 32\)
|
\(64\times 64\)
|
\(128\times 128\)
|
---|---|---|---|---|
PGS | F (2, 2) | 0.43 (0.22) | 0.44 (0.82) | 0.54 (4.04) |
F (3, 3) | 0.38 (0.25) | 0.40 (0.97) | 0.48 (4.60) | |
Vanka | F (2, 2) | 0.47 (0.50) | 0.44 (1.78) | 0.65 (13.47) |
F (3, 3) | 0.38 (0.54) | 0.40 (2.45) | 0.56 (14.17) |
Smoother | F-cycle |
\(32\times 32\)
|
\(64\times 64\)
|
\(128\times 128\)
|
---|---|---|---|---|
PGS | F (2, 2) | 0.22 (0.17) | 0.22 (0.49) | 0.21 (1.82) |
F (3, 3) | 0.18 (0.19) | 0.19 (0.55) | 0.18 (2.30) | |
Vanka | F (2, 2) | 0.27 (0.30) | 0.24 (1.00) | 0.26 (4.35) |
F (3, 3) | 0.17 (0.45) | 0.18 (1.18) | 0.17 (4.60) |