Source code for pyro.multigrid.examples.mg_test_vc_constant

#!/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)

"""


import matplotlib.pyplot as plt
import numpy as np

import pyro.mesh.boundary as bnd
import pyro.multigrid.variable_coeff_MG as MG
from pyro.mesh import patch


# 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') img1 = 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(f"nx = {nx}") plt.colorbar(img1) plt.savefig("mg_alpha.png") # 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="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" % (e.norm(), 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) img2 = 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(f"nx = {nx}") plt.colorbar(img2) plt.subplot(122) img3 = 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(img3) 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)