r"""
Initialize a smooth incompressible convergence test. Here, the
velocities are initialized as
.. math::
u(x,y) = 1 - 2 \cos(2 \pi x) \sin(2 \pi y)
v(x,y) = 1 + 2 \sin(2 \pi x) \cos(2 \pi y)
and the exact solution at some later time t is then
.. math::
u(x,y,t) = 1 - 2 \cos(2 \pi (x - t)) \sin(2 \pi (y - t))
v(x,y,t) = 1 + 2 \sin(2 \pi (x - t)) \cos(2 \pi (y - t))
p(x,y,t) = -\cos(4 \pi (x - t)) - \cos(4 \pi (y - t))
The numerical solution can be compared to the exact solution to
measure the convergence rate of the algorithm.
"""
import math
import numpy as np
from pyro.mesh import patch
from pyro.util import msg
[docs]
def init_data(my_data, rp):
""" initialize the incompressible converge problem """
del rp # this problem doesn't use runtime params
msg.bold("initializing the incompressible converge problem...")
# make sure that we are passed a valid patch object
if not isinstance(my_data, patch.CellCenterData2d):
print(my_data.__class__)
msg.fail("ERROR: patch invalid in converge.py")
# get the velocities
u = my_data.get_var("x-velocity")
v = my_data.get_var("y-velocity")
myg = my_data.grid
if (myg.xmin != 0 or myg.xmax != 1 or
myg.ymin != 0 or myg.ymax != 1):
msg.fail("ERROR: domain should be a unit square")
u[:, :] = 1.0 - 2.0*np.cos(2.0*math.pi*myg.x2d)*np.sin(2.0*math.pi*myg.y2d)
v[:, :] = 1.0 + 2.0*np.sin(2.0*math.pi*myg.x2d)*np.cos(2.0*math.pi*myg.y2d)
[docs]
def finalize():
""" print out any information to the user at the end of the run """
ostr = """
Comparisons to the analytic solution can be done using
analysis/incomp_converge_error.py
"""
print(ostr)