4.2 PASS:
Convergence for a simple periodic problem
-
Author
- Stéphane Popinet
- Command
- sh periodic.sh periodic.gfs
- Version
- 0.6.4
- Required files
- periodic.gfs (view) (download)
periodic.sh r0.ref r1.ref r2.ref
- Running time
- 47 minutes 45 seconds
This is one of the test cases presented in Popinet [31].
Following Minion [29] and Almgren et al. [1],
this convergence test illustrates the second-order
accuracy of Gerris for flows without solid boundaries. This
problem uses a square unit domain with periodic boundary conditions in
both directions. The initial conditions are taken as
| u(x,y) | = | 1−2cos(2π x)sin(2π y), |
| v(x,y) | = | 1+2sin(2π x)cos(2π y). |
|
The exact solution of the Euler equations for these initial conditions
is
| u(x,y,t) | = | 1−2cos(2π(x−t))sin(2π(y−t)), |
| v(x,y,t) | = | 1+2sin(2π(x−t))cos(2π(y−t)), |
| p(x,y,t) | = | −cos(4π(x−t))−cos(4π(y−t)). |
|
As in [1] nine runs are performed on grids with L=5,6
and 7 levels of refinement (labelled “uniform”) and with one
(labelled r=1) or two (labelled r=2) additional levels added only
within the square defined by the points (−0.25,−0.25) and
(0.25,0.25). The length of the run for each case is 0.5, the CFL number is
0.75. For each run both the L2 and L∞ norms of the error in
the x-component of the velocity is computed. Table 3
gives the errors and order of convergence obtained.
Close to second-order convergence is obtained (asymptotically in
L) for the L2 and L∞ norms. The values
obtained are comparable to that in [29, 1].
| Table 3: Errors and convergence orders in the x-component of the velocity
for a simple periodic problem. The reference solution values are given in blue. |
| | L2 |
| | L=5 | O2 | L=6 | O2 | L=7 |
| Uniform | 8.27e-03 | 2.84 | 1.15e-03 | 2.48 | 2.07e-04 |
| | 8.27e-03 | 2.84 | 1.15e-03 | 2.48 | 2.07e-04 |
| r=1 | 8.35e-03 | 2.29 | 1.70e-03 | 2.11 | 3.96e-04 |
| | 8.35e-03 | 2.29 | 1.70e-03 | 2.11 | 3.96e-04 |
| r=2 | 1.06e-02 | 2.08 | 2.50e-03 | 2.00 | 6.24e-04 |
| | 1.06e-02 | 2.08 | 2.50e-03 | 2.00 | 6.24e-04 |
| | L∞ |
| | L=5 | O∞ | L=6 | O∞ | L=7 |
| Uniform | 1.97e-02 | 2.66 | 3.10e-03 | 2.61 | 5.08e-04 |
| | 1.97e-02 | 2.66 | 3.10e-03 | 2.61 | 5.08e-04 |
| r=1 | 2.20e-02 | 2.21 | 4.77e-03 | 2.06 | 1.14e-03 |
| | 2.20e-02 | 2.21 | 4.77e-03 | 2.06 | 1.14e-03 |
| r=2 | 2.82e-02 | 2.11 | 6.54e-03 | 1.99 | 1.65e-03 |
| | 2.82e-02 | 2.11 | 6.54e-03 | 1.99 | 1.65e-03 |