Convergence of upwind schemeΒΆ
First, we test the upwind-advection scheme with pointwise multiplication
| solution | vorticity | enstrophy | energy | ||||||
|---|---|---|---|---|---|---|---|---|---|
| error | order | error | order | error | order | error | order | ||
| n | N | ||||||||
| 1 | 72 | 3.11e-01 | n/a | -4.62e-06 | n/a | -6.51e-03 | n/a | 7.32e-03 | n/a |
| 144 | 1.74e-01 | 0.84 | -5.87e-07 | 2.98 | -2.67e-02 | -2.03 | -2.38e-02 | -1.70 | |
| 288 | 9.15e-02 | 0.93 | -7.70e-08 | 2.93 | -2.12e-02 | 0.33 | -2.03e-02 | 0.23 | |
| 576 | 4.68e-02 | 0.97 | -7.49e-09 | 3.36 | -1.28e-02 | 0.72 | -1.25e-02 | 0.70 | |
| solution | vorticity | enstrophy | energy | ||||||
|---|---|---|---|---|---|---|---|---|---|
| error | order | error | order | error | order | error | order | ||
| n | N | ||||||||
| 3 | 24 | 8.96e-03 | n/a | 1.72e-04 | n/a | -2.52e-04 | n/a | -1.83e-04 | n/a |
| 48 | 1.63e-03 | 2.46 | 4.06e-06 | 5.41 | -9.49e-06 | 4.73 | -7.12e-06 | 4.68 | |
| 96 | 2.14e-04 | 2.93 | 8.39e-08 | 5.59 | -4.13e-07 | 4.52 | -2.59e-07 | 4.78 | |
| 192 | 2.53e-05 | 3.08 | 1.65e-09 | 5.67 | -7.37e-08 | 2.49 | 5.76e-09 | 5.49 | |
| solution | vorticity | enstrophy | energy | ||||||
|---|---|---|---|---|---|---|---|---|---|
| error | order | error | order | error | order | error | order | ||
| n | N | ||||||||
| 5 | 12 | 1.12e-03 | n/a | 3.09e-07 | n/a | 3.52e-06 | n/a | 3.23e-06 | n/a |
| 24 | 6.86e-05 | 4.03 | 8.42e-09 | 5.20 | -1.69e-07 | 4.38 | -3.52e-08 | 6.52 | |
| 48 | 2.70e-06 | 4.66 | 1.19e-11 | 9.46 | -7.55e-08 | 1.16 | 9.19e-09 | 1.94 | |
| 96 | 1.02e-07 | 4.73 | 1.39e-14 | 9.74 | -5.73e-08 | 0.40 | 1.10e-08 | -0.26 | |
In the last table we might see the effects of the accuracy of the third order time-stepper