#!/usr/bin/env python3
"""
Test the variable-coefficient MG solver with periodic data.
Here we solve::
div . ( alpha grad phi ) = f
with::
alpha = 2.0 + cos(2.0*pi*x)*cos(2.0*pi*y)
f = -16.0*pi**2*(cos(2*pi*x)*cos(2*pi*y) + 1)*sin(2*pi*x)*sin(2*pi*y)
This has the exact solution::
phi = sin(2.0*pi*x)*sin(2.0*pi*y)
on [0,1] x [0,1]
We use Dirichlet BCs on phi. For alpha, we do not have to impose the
same BCs, since that may represent a different physical quantity.
Here we take alpha to have Neumann BCs. (Dirichlet BCs for alpha will
force it to 0 on the boundary, which is not correct here)
"""
import matplotlib.pyplot as plt
import numpy as np
import pyro.mesh.boundary as bnd
import pyro.multigrid.variable_coeff_MG as MG
import pyro.util.io_pyro as io
from pyro.mesh import patch
from pyro.util import compare, msg
# the analytic solution
[docs]
def true(x, y):
return np.sin(2.0*np.pi*x)*np.sin(2.0*np.pi*y)
# the coefficients
[docs]
def alpha(x, y):
return 2.0 + np.cos(2.0*np.pi*x)*np.cos(2.0*np.pi*y)
# the righthand side
[docs]
def f(x, y):
return -16.0*np.pi**2*(np.cos(2*np.pi*x)*np.cos(2*np.pi*y) + 1) * \
np.sin(2*np.pi*x)*np.sin(2*np.pi*y)
[docs]
def test_vc_poisson_periodic(N, store_bench=False, comp_bench=False, bench_dir="tests/",
make_plot=False, verbose=1, rtol=1.e-12):
"""
test the variable-coefficient MG solver. The return value
here is the error compared to the exact solution, UNLESS
comp_bench=True, in which case the return value is the
error compared to the stored benchmark
"""
# test the multigrid solver
nx = N
ny = nx
# create the coefficient variable
g = patch.Grid2d(nx, ny, ng=1)
d = patch.CellCenterData2d(g)
bc_c = bnd.BC(xlb="periodic", xrb="periodic",
ylb="periodic", yrb="periodic")
d.register_var("c", bc_c)
d.create()
c = d.get_var("c")
c[:, :] = alpha(g.x2d, g.y2d)
# check whether the RHS sums to zero (necessary for periodic data)
rhs = f(g.x2d, g.y2d)
print(f"rhs sum: {np.sum(rhs[g.ilo:g.ihi+1, g.jlo:g.jhi+1])}")
# create the multigrid object
a = MG.VarCoeffCCMG2d(nx, ny,
xl_BC_type="periodic", yl_BC_type="periodic",
xr_BC_type="periodic", yr_BC_type="periodic",
coeffs=c, coeffs_bc=bc_c,
verbose=verbose, vis=0, true_function=true)
# initialize the solution to 0
a.init_zeros()
# initialize the RHS using the function f
rhs = f(a.x2d, a.y2d)
a.init_RHS(rhs)
# solve to a relative tolerance of 1.e-11
a.solve(rtol=1.e-11)
# alternately, we can just use smoothing by uncommenting the following
# a.smooth(a.nlevels-1,10000)
# get the solution
v = a.get_solution()
# get the true solution
b = true(a.x2d, a.y2d)
# compute the error from the analytic solution -- note that with
# periodic BCs all around, there is nothing to normalize the
# solution. We subtract off the average of phi from the MG
# solution (we do the same for the true solution to put them on
# the same footing)
e = v - np.sum(v.v())/(nx*ny) - (b - np.sum(b[a.ilo:a.ihi+1, a.jlo:a.jhi+1])/(nx*ny))
enorm = e.norm()
print(" L2 error from true solution = %g\n rel. err from previous cycle = %g\n num. cycles = %d" %
(enorm, a.relative_error, a.num_cycles))
# plot the solution
if make_plot:
plt.clf()
plt.figure(figsize=(10.0, 4.0), dpi=100, facecolor='w')
plt.subplot(121)
img1 = plt.imshow(np.transpose(v.v()),
interpolation="nearest", origin="lower",
extent=[a.xmin, a.xmax, a.ymin, a.ymax])
plt.xlabel("x")
plt.ylabel("y")
plt.title(f"nx = {nx}")
plt.colorbar(img1)
plt.subplot(122)
img2 = plt.imshow(np.transpose(e.v()),
interpolation="nearest", origin="lower",
extent=[a.xmin, a.xmax, a.ymin, a.ymax])
plt.xlabel("x")
plt.ylabel("y")
plt.title("error")
plt.colorbar(img2)
plt.tight_layout()
plt.savefig("mg_vc_periodic_test.png")
# store the output for later comparison
bench = "mg_vc_poisson_periodic"
my_data = a.get_solution_object()
if store_bench:
my_data.write(f"{bench_dir}/{bench}")
# do we do a comparison?
if comp_bench:
compare_file = f"{bench_dir}/{bench}"
msg.warning(f"comparing to {compare_file}")
bench = io.read(compare_file)
result = compare.compare(my_data, bench, rtol)
if result == 0:
msg.success(f"results match benchmark to within relative tolerance of {rtol}\n")
else:
msg.warning(f"ERROR: {compare.errors[result]}\n")
return result
# normal return -- error wrt true solution
return enorm
[docs]
def main():
N = [16, 32, 64, 128, 256, 512]
err = []
plot = False
store = False
do_compare = False
for nx in N:
if nx == max(N):
plot = True
enorm = test_vc_poisson_periodic(nx, make_plot=plot,
store_bench=store, comp_bench=do_compare)
err.append(enorm)
# plot the convergence
N = np.array(N, dtype=np.float64)
err = np.array(err)
plt.clf()
plt.loglog(N, err, "x", color="r")
plt.loglog(N, err[0]*(N[0]/N)**2, "--", color="k")
plt.xlabel("N")
plt.ylabel("error")
fig = plt.gcf()
fig.set_size_inches(7.0, 6.0)
plt.tight_layout()
plt.savefig("mg_vc_periodic_converge.png")
if __name__ == "__main__":
main()
