def test_heat_2d_step_fe():
"""
Test step()
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
heat_2d = Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
method='FE',
rhs=lambda x, y, t: 2 * x * y,
t_start=0,
t_stop=1,
nt=11)
heat_2d_res = heat_2d.step(u_start=heat_2d.vector_t_start,
t_start=0,
t_stop=0.1)
np.testing.assert_almost_equal(
heat_2d_res.get_values(),
np.array([[0., 0., 0., 0., 0.], [0., 0.1625, 0.175, 0.1875, 0.],
[0., 0.325, 0.35, 0.375, 0.],
[0., 0.4875, 0.525, 0.5625, 0.], [0., 0., 0., 0., 0.]]))
def test_heat_2d_step_be():
"""
Test step()
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
heat_2d = Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
method='BE',
rhs=lambda x, y, t: 2 * x * y,
t_start=0,
t_stop=1,
nt=11)
heat_2d_res = heat_2d.step(u_start=heat_2d.vector_t_start,
t_start=0,
t_stop=0.1)
np.testing.assert_almost_equal(
heat_2d_res.get_values(),
np.array([[0., 0., 0., 0., 0.],
[0., 0.06659024, 0.08719337, 0.07227713, 0.],
[0., 0.11922399, 0.15502696, 0.12990086, 0.],
[0., 0.12666875, 0.16193148, 0.1391124, 0.],
[0., 0., 0., 0., 0.]]))
def test_heat_2d_step_cn():
"""
Test step()
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
heat_2d = Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
method='CN',
rhs=lambda x, y, t: 2 * x * y,
t_start=0,
t_stop=1,
nt=11)
heat_2d_res = heat_2d.step(u_start=heat_2d.vector_t_start,
t_start=0,
t_stop=0.1)
np.testing.assert_almost_equal(
heat_2d_res.get_values(),
np.array([[0., 0., 0., 0., 0.],
[0., 0.09547237, 0.12246323, 0.10480116, 0.],
[0., 0.17564171, 0.22390841, 0.19336787, 0.],
[0., 0.1964882, 0.24654636, 0.21772176, 0.],
[0., 0., 0., 0., 0.]]))
def test_heat_2d_constructor_boundary():
"""
Test constructor
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
bc_top=lambda y: 2,
bc_left=lambda y: 2,
bc_right=lambda y: 2,
bc_bottom=lambda y: 2,
rhs=lambda x, y, t: 2 * x * y,
t_start=0,
t_stop=1,
nt=11)
def test_heat_2d_constructor_cn():
"""
Test constructor
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
heat_2d = Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
method='CN',
rhs=lambda x, y, t: 2 * x * y,
t_start=0,
t_stop=1,
nt=11)
np.testing.assert_almost_equal(
heat_2d.compute_rhs(u_start=4 * np.ones((5, 5)),
t_start=0.2,
t_stop=0.3),
np.array([
0., 0., 0., 0., 0., 0., 4.1625, 4.175, 4.1875, 0., 0., 4.325, 4.35,
4.375, 0., 0., 4.4875, 4.525, 4.5625, 0., 0., 0., 0., 0., 0.
]))
def test_heat_2d_constructor_exception_boundary_top():
"""
Test constructor
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
with pytest.raises(Exception):
Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
bc_top='2',
rhs=lambda x, y, t: 2 * x * y,
t_start=0,
t_stop=1,
nt=11)
def test_heat_2d_constructor():
"""
Test constructor
"""
x_start = 0
x_end = 1
y_start = 3
y_end = 4
nx = 5
ny = 5
a = 1
heat_2d = Heat2D(a=a,
x_start=x_start,
x_end=x_end,
y_start=y_start,
y_end=y_end,
nx=nx,
ny=ny,
t_start=0,
t_stop=1,
nt=11)
np.testing.assert_equal(heat_2d.x_start, x_start)
np.testing.assert_equal(heat_2d.x_end, x_end)
np.testing.assert_equal(heat_2d.y_start, y_start)
np.testing.assert_equal(heat_2d.y_end, y_end)
np.testing.assert_equal(heat_2d.nx, nx)
np.testing.assert_equal(heat_2d.ny, ny)
np.testing.assert_almost_equal(heat_2d.dx, 0.25)
np.testing.assert_almost_equal(heat_2d.dy, 0.25)
np.testing.assert_equal(heat_2d.x, np.linspace(x_start, x_end, nx))
np.testing.assert_equal(heat_2d.y, np.linspace(y_start, y_end, ny))
np.testing.assert_equal(heat_2d.x_2d,
np.linspace(x_start, x_end, nx)[:, np.newaxis])
np.testing.assert_equal(heat_2d.y_2d,
np.linspace(y_start, y_end, ny)[np.newaxis, :])
np.testing.assert_equal(heat_2d.a, a)
np.testing.assert_equal(True,
isinstance(heat_2d.vector_template, VectorHeat2D))
np.testing.assert_equal(True,
isinstance(heat_2d.vector_t_start, VectorHeat2D))
np.testing.assert_equal(heat_2d.vector_t_start.get_values(),
np.zeros((5, 5)))
matrix = [[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., -16., 0., 0., 0., -16., 64., -16., 0., 0., 0., -16., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., -16., 0., 0., 0., -16., 64., -16., 0., 0., 0., -16.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., -16., 0., 0., 0., -16., 64., -16., 0., 0., 0.,
-16., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., -16., 0., 0., 0., -16., 64., -16.,
0., 0., 0., -16., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., -16., 0., 0., 0., -16., 64.,
-16., 0., 0., 0., -16., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., -16., 0., 0., 0., -16., 64.,
-16., 0., 0., 0., -16., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -16., 0., 0., 0.,
-16., 64., -16., 0., 0., 0., -16., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -16., 0., 0.,
0., -16., 64., -16., 0., 0., 0., -16., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., -16., 0.,
0., 0., -16., 64., -16., 0., 0., 0., -16., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
],
[
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0., 0., 0., 0., 0., 0., 0., 0.
]]
np.testing.assert_equal(heat_2d.space_disc.toarray(), matrix)
def main():
def output_fcn(self):
# Set path to solution
path = 'results/heat_equation_2d'
# Create path if not existing
pathlib.Path(path).mkdir(parents=True, exist_ok=True)
# Save solution with corresponding time point to file
np.save(path + '/heat_equation_2d-rank' + str(self.comm_time_rank),
[[[self.t[0][i], self.u[0][i]] for i in self.index_local[0]]])
# example problem parameters
x_end = 0.75
y_end = 1.5
a = 3.5
def rhs(x, y, t):
"""
Right-hand side of 2D heat equation example problem at a given space-time point (x,y,t),
5x(x_end - x)y(y_end - y) + 10at(y(y_end-y)) + x(x_end - x), a = 3.5
:param x: x-coordinate of spatial grid point
:param y: y_coordinate of spatial grid point
:param t: time point
:return: right-hand side of 2D heat equation example problem at point (x,y,t)
"""
return 5 * x * (x_end - x) * y * (y_end -
y) + 10 * a * t * (y *
(y_end - y) + x *
(x_end - x))
def exact_sol(x, y, t):
"""
Exact solution of 2D heat equation example problem at a given space-time point (x,y,t)
Note: Can be used for computing the error of the MGRIT approximation
:param x: x_coordinate of spatial grid point
:param y: y_coordinate of spatial grid point
:param t: time point
:return: exact solution of 2D heat equation example problem at point (x,y,t)
"""
return 5 * t * x * (x_end - x) * y * (y_end - y)
heat0 = Heat2D(x_start=0,
x_end=x_end,
y_start=0,
y_end=y_end,
nx=55,
ny=125,
a=a,
rhs=rhs,
t_start=0,
t_stop=1,
nt=33)
heat1 = Heat2D(x_start=0,
x_end=x_end,
y_start=0,
y_end=y_end,
nx=55,
ny=125,
a=a,
rhs=rhs,
t_interval=heat0.t[::2])
# Setup two-level MGRIT solver and solve the problem
mgrit = Mgrit(problem=[heat0, heat1],
cycle_type='V',
output_fcn=output_fcn)
info = mgrit.solve()
if MPI.COMM_WORLD.Get_rank() == 0:
time.sleep(2)
sol = []
path = 'results/heat_equation_2d/'
for filename in os.listdir(path):
data = np.load(path + filename, allow_pickle=True).tolist()[0]
sol += data
sol.sort(key=lambda tup: tup[0])
diff = 0
for i in range(len(sol)):
u_e = exact_sol(x=heat0.x_2d, y=heat0.y_2d, t=heat0.t[i])
diff += abs(sol[i][1].get_values() - u_e).max()
print("Difference between MGRIT solution and exact solution:", diff)