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 secondorder
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 L_{2} and L_{∞} norms of the error in
the xcomponent of the velocity is computed. Table 3
gives the errors and order of convergence obtained.
Close to secondorder convergence is obtained (asymptotically in
L) for the L_{2} and L_{∞} norms. The values
obtained are comparable to that in [29, 1].
Table 3: Errors and convergence orders in the xcomponent of the velocity
for a simple periodic problem. The reference solution values are given in blue. 
 L_{2} 
 L=5  O_{2}  L=6  O_{2}  L=7 
Uniform  8.27e03  2.84  1.15e03  2.48  2.07e04 
 8.27e03  2.84  1.15e03  2.48  2.07e04 
r=1  8.35e03  2.29  1.70e03  2.11  3.96e04 
 8.35e03  2.29  1.70e03  2.11  3.96e04 
r=2  1.06e02  2.08  2.50e03  2.00  6.24e04 
 1.06e02  2.08  2.50e03  2.00  6.24e04 
 L_{∞} 
 L=5  O_{∞}  L=6  O_{∞}  L=7 
Uniform  1.97e02  2.66  3.10e03  2.61  5.08e04 
 1.97e02  2.66  3.10e03  2.61  5.08e04 
r=1  2.20e02  2.21  4.77e03  2.06  1.14e03 
 2.20e02  2.21  4.77e03  2.06  1.14e03 
r=2  2.82e02  2.11  6.54e03  1.99  1.65e03 
 2.82e02  2.11  6.54e03  1.99  1.65e03 