#!/usr/bin/env python3
"""
an example of using the multigrid class to solve Laplace's equation. 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 multigrid.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 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 doit(nx, ny):
# test the multigrid solver
# create the multigrid object
a = MG.CellCenterMG2d(nx, ny,
xl_BC_type="dirichlet", yl_BC_type="dirichlet",
xr_BC_type="dirichlet", yr_BC_type="dirichlet",
verbose=0,
nsmooth=5, nsmooth_bottom=10,
vis=1, true_function=true,
vis_title=r"$u_{xx} + u_{yy} = -2[(1-6x^2)y^2(1-y^2) + (1-6y^2)x^2(1-x^2)]$")
plt.ion()
plt.figure(num=1, figsize=(12.8, 7.2), dpi=100, facecolor='w')
# initialize the solution to 0
init = a.soln_grid.scratch_array()
a.init_solution(init)
# 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.figure(num=1, figsize=(2.10,2.10), dpi=100, facecolor='w')
plt.figure(num=1, figsize=(5.0, 5.0), dpi=100, facecolor='w')
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.axis("off")
# plt.subplots_adjust(bottom=0.0, top=1.0, left=0.0, right=1.0)
plt.xlabel("x")
plt.ylabel("y")
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__":
doit(64, 64)