#!/usr/bin/env python3
"""
Test the variable coefficient MG solver with a CONSTANT coefficient
problem -- the same one from the multigrid class test. This ensures
we didn't screw up the base functionality here.
We solve::
u_xx + u_yy = -2[(1-6x**2)y**2(1-y**2) + (1-6y**2)x**2(1-x**2)]
u = 0 on the boundary
this is the example from page 64 of the book `A Multigrid Tutorial, 2nd Ed.`
The analytic solution is u(x,y) = (x**2 - x**4)(y**4 - y**2)
"""
from __future__ import print_function
import numpy as np
import mesh.boundary as bnd
import mesh.patch as patch
import multigrid.variable_coeff_MG as MG
import matplotlib.pyplot as plt
# the analytic solution
[docs]def true(x, y):
return (x**2 - x**4)*(y**4 - y**2)
# the coefficients
[docs]def alpha(x, y):
return np.ones_like(x)
# the righthand side
[docs]def f(x, y):
return -2.0*((1.0-6.0*x**2)*y**2*(1.0-y**2) + (1.0-6.0*y**2)*x**2*(1.0-x**2))
[docs]def test_vc_constant(N):
# test the multigrid solver
nx = N
ny = nx
# create the coefficient variable -- note we don't want Dirichlet here,
# because that will try to make alpha = 0 on the interface. alpha can
# have different BCs than phi
g = patch.Grid2d(nx, ny, ng=1)
d = patch.CellCenterData2d(g)
bc_c = bnd.BC(xlb="neumann", xrb="neumann",
ylb="neumann", yrb="neumann")
d.register_var("c", bc_c)
d.create()
c = d.get_var("c")
c[:, :] = alpha(g.x2d, g.y2d)
plt.clf()
plt.figure(num=1, figsize=(5.0, 5.0), dpi=100, facecolor='w')
plt.imshow(np.transpose(c[g.ilo:g.ihi+1, g.jlo:g.jhi+1]),
interpolation="nearest", origin="lower",
extent=[g.xmin, g.xmax, g.ymin, g.ymax])
plt.xlabel("x")
plt.ylabel("y")
plt.title("nx = {}".format(nx))
plt.colorbar()
plt.savefig("mg_alpha.png")
# check whether the RHS sums to zero (necessary for periodic data)
rhs = f(g.x2d, g.y2d)
print("rhs sum: {}".format(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="dirichlet", yl_BC_type="dirichlet",
xr_BC_type="dirichlet", yr_BC_type="dirichlet",
coeffs=c, coeffs_bc=bc_c,
verbose=1)
# 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-1,50000)
# get the solution
v = a.get_solution()
# compute the error from the analytic solution
b = true(a.x2d, a.y2d)
e = v - b
print(" L2 error from true solution = %g\n rel. err from previous cycle = %g\n num. cycles = %d" %
(a.soln_grid.norm(e), a.relative_error, a.num_cycles))
# plot it
plt.clf()
plt.figure(num=1, figsize=(10.0, 5.0), dpi=100, facecolor='w')
plt.subplot(121)
plt.imshow(np.transpose(v[a.ilo:a.ihi+1, a.jlo:a.jhi+1]),
interpolation="nearest", origin="lower",
extent=[a.xmin, a.xmax, a.ymin, a.ymax])
plt.xlabel("x")
plt.ylabel("y")
plt.title("nx = {}".format(nx))
plt.colorbar()
plt.subplot(122)
plt.imshow(np.transpose(e[a.ilo:a.ihi+1, a.jlo:a.jhi+1]),
interpolation="nearest", origin="lower",
extent=[a.xmin, a.xmax, a.ymin, a.ymax])
plt.xlabel("x")
plt.ylabel("y")
plt.title("error")
plt.colorbar()
plt.tight_layout()
plt.savefig("mg_test.png")
# store the output for later comparison
my_data = a.get_solution_object()
my_data.write("mg_test")
if __name__ == "__main__":
test_vc_constant(256)