def test_diffusion_polar():
# test H2 and H7 for spatially-dependent H2. Diffusion in polar coordinates
dt = 1e6
L = 1 # size of domain
N = 500 # number of spatial grid points
dr = L / (N - 0.5) # spatial grid size
r = np.linspace(dr/2, L, N) # location corresponding to grid points j=0, ..., N-1
nL = 0.03
# initial conditions
n_initial = 0.3*np.sin(np.pi * r) + nL * r
# diffusion problem coefficients
D = 1.3
H1 = r
H2 = D*r
H7 = r**2
n_final = HToMatrixFD.H_to_matrix_and_solve(dt, dr, nL, n_initial, H1=H1, H2=H2, H7=H7 )
# compare with analytic and plots
n_ss_analytic = nL + 1/(9*D) * (1-r**3)
tol = 1e-8 # for N=500
mean_square_error = np.linalg.norm(n_final - n_ss_analytic) / N
obs = mean_square_error
exp = 0
assert abs(obs - exp) < tol
def test_diffusion_cartesian():
# test H2 and H7
dt = 1e5 # test only steady state
L = 1 # size of domain
N = 500 # number of spatial grid points
dx = L / (N-1) # spatial grid size
x = np.arange(N)*dx # location corresponding to grid points j=0, ..., N-1
nL = 0.2
# initial conditions
n_initial = np.sin(np.pi * x) + nL * x
# diffusion problem coefficients
D = 0.8
H1 = np.ones_like(x)
H2 = D * np.ones_like(x)
H7 = 1 - x**2
# solve it for steady state: only one iteration required for linear problem
n_final = HToMatrixFD.H_to_matrix_and_solve(dt, dx, nL, n_initial, H1=H1, H2=H2, H7=H7)
# compare with analytic and plots
n_ss_analytic = nL + 1/(12*D) * (1-x**2) * (5 - x**2)
tol = 1e-4
mean_square_error = np.linalg.norm(n_final - n_ss_analytic) / N
obs = mean_square_error
exp = 0
assert abs(obs - exp) < tol
def test_H4():
# test H4 in a problem with an analytic solution
dt = 1e9 # test only steady state
L = 1 # size of domain
N = 500 # number of spatial grid points
dx = L / (N - 1) # spatial grid size
x = np.arange(
N) * dx # location corresponding to grid points j=0, ..., N-1
UL = 2.2
U_initial = np.sin(np.pi * x) + UL * x
H1 = np.ones_like(x)
H2 = np.ones_like(x)
H4 = -np.sin(x)
H7 = 1 - x**2
U_final = HToMatrixFD.H_to_matrix_and_solve(dt,
dx,
UL,
U_initial,
H1=H1,
H2=H2,
H4=H4,
H7=H7)
U_ss_analytic = UL - (np.cos(x) -
np.cos(L)) - (x**2 - L**2) / 2 + (x**4 - L**4) / 12
assert np.allclose(U_final, U_ss_analytic, rtol=1e-3, atol=1e-3)
def test_diffusion_convection_time_polar():
# test H1, H2, H3, H7 -- diffusion-convection problem in polar coordinates, including time dependence
# use a manufactured solution to generate a problem with analytic solution
L = 1 # size of domain
N = 300 # number of spatial grid points
dr = L / (N - 0.5) # spatial grid size
r = np.linspace(dr / 2, L,
N) # location corresponding to grid points j=0, ..., N-1
nL = 0.5
# initial conditions
n_initial = nL * np.ones_like(r)
n = n_initial
# specify diffusion problem coefficients
D = 1.3
c0 = 0.7
gamma = 0.9
H1 = r
H2 = D * r
H3 = -c0 * r**2
fH7 = lambda r, t: r/D * gamma * np.exp(-gamma*t) * (1-r**5) + \
25 * r**4 * (1 - np.exp(-gamma*t)) + \
c0 * (2*r*nL + 1/D * (1 - np.exp(-gamma*t)) * (2*r - 7*r**6))
# set up time loop
dt = 1e-2
tf = 0.1
tvec = np.arange(0, tf + 1e-10, dt)
for m in range(1, len(tvec)):
t = tvec[m]
H7 = fH7(r, t)
n = HToMatrixFD.H_to_matrix_and_solve(dt,
dr,
nL,
n,
H1=H1,
H2=H2,
H3=H3,
H7=H7)
n_final = n
# compare with analytic and plots
#n_ss_analytic = nL + 1/D * (1-r**5)
n_timedep_analytic = nL + 1 / D * (1 - np.exp(-gamma * t)) * (1 - r**5)
tol = 2e-5 # for N=300
mean_square_timedep_error = np.linalg.norm(n_final -
n_timedep_analytic) / N
obs = mean_square_timedep_error
exp = 0
assert abs(obs - exp) < tol