def time_evolution(u=None):
'''
Solves the wave equation
.. math:: u^{t_n + 1} = b(t_n) \\times A
iterated over time.shape[0] time steps t_n
Second order time stepping is used.
It increases the accuracy of the wave evolution.
The second order time-stepping would be
`U^{n + 1/2} = U^n + dt / 2 (A^{-1} B(U^n))`
`U^{n + 1} = U^n + dt (A^{-1} B(U^{n + 1/2}))`
Returns
-------
None
'''
# Creating a folder to store hdf5 files. If it doesn't exist.
results_directory = 'results/hdf5_%02d' % (int(params.N_LGL))
if not os.path.exists(results_directory):
os.makedirs(results_directory)
element_LGL = params.element_LGL
delta_t = params.delta_t
shape_u_n = utils.shape(u)
time = params.time
A_inverse = af.tile(af.inverse(A_matrix()), d0=1, d1=1, d2=shape_u_n[2])
element_boundaries = af.np_to_af_array(params.np_element_array)
for t_n in trange(0, time.shape[0]):
if (t_n % 20) == 0:
h5file = h5py.File(
'results/hdf5_%02d/dump_timestep_%06d' %
(int(params.N_LGL), int(t_n)) + '.hdf5', 'w')
dset = h5file.create_dataset('u_i', data=u, dtype='d')
dset[:, :] = u[:, :]
# Code for solving 1D Maxwell's Equations
# Storing u at timesteps which are multiples of 100.
#if (t_n % 5) == 0:
#h5file = h5py.File('results/hdf5_%02d/dump_timestep_%06d' \
#%(int(params.N_LGL), int(t_n)) + '.hdf5', 'w')
#Ez_dset = h5file.create_dataset('E_z', data = u[:, :, 0],
#dtype = 'd')
#By_dset = h5file.create_dataset('B_y', data = u[:, :, 1],
#dtype = 'd')
#Ez_dset[:, :] = u[:, :, 0]
#By_dset[:, :] = u[:, :, 1]
u += RK4_timestepping(A_inverse, u, delta_t)
def simple_lapack(verbose=False):
display_func = _util.display_func(verbose)
print_func = _util.print_func(verbose)
a = af.randu(5, 5)
l, u, p = af.lu(a)
display_func(l)
display_func(u)
display_func(p)
p = af.lu_inplace(a, "full")
display_func(a)
display_func(p)
a = af.randu(5, 3)
q, r, t = af.qr(a)
display_func(q)
display_func(r)
display_func(t)
af.qr_inplace(a)
display_func(a)
a = af.randu(5, 5)
a = af.matmulTN(a, a.copy()) + 10 * af.identity(5, 5)
R, info = af.cholesky(a)
display_func(R)
print_func(info)
af.cholesky_inplace(a)
display_func(a)
a = af.randu(5, 5)
ai = af.inverse(a)
display_func(a)
display_func(ai)
x0 = af.randu(5, 3)
b = af.matmul(a, x0)
x1 = af.solve(a, b)
display_func(x0)
display_func(x1)
p = af.lu_inplace(a)
x2 = af.solve_lu(a, p, b)
display_func(x2)
print_func(af.rank(a))
print_func(af.det(a))
print_func(af.norm(a, af.NORM.EUCLID))
print_func(af.norm(a, af.NORM.MATRIX_1))
print_func(af.norm(a, af.NORM.MATRIX_INF))
print_func(af.norm(a, af.NORM.MATRIX_L_PQ, 1, 1))
a = af.randu(10, 10)
display_func(a)
u, s, vt = af.svd(a)
display_func(af.matmul(af.matmul(u, af.diag(s, 0, False)), vt))
u, s, vt = af.svd_inplace(a)
display_func(af.matmul(af.matmul(u, af.diag(s, 0, False)), vt))
def simple_lapack(verbose=False):
display_func = _util.display_func(verbose)
print_func = _util.print_func(verbose)
a = af.randu(5,5)
l,u,p = af.lu(a)
display_func(l)
display_func(u)
display_func(p)
p = af.lu_inplace(a, "full")
display_func(a)
display_func(p)
a = af.randu(5,3)
q,r,t = af.qr(a)
display_func(q)
display_func(r)
display_func(t)
af.qr_inplace(a)
display_func(a)
a = af.randu(5, 5)
a = af.matmulTN(a, a) + 10 * af.identity(5,5)
R,info = af.cholesky(a)
display_func(R)
print_func(info)
af.cholesky_inplace(a)
display_func(a)
a = af.randu(5,5)
ai = af.inverse(a)
display_func(a)
display_func(ai)
x0 = af.randu(5, 3)
b = af.matmul(a, x0)
x1 = af.solve(a, b)
display_func(x0)
display_func(x1)
p = af.lu_inplace(a)
x2 = af.solve_lu(a, p, b)
display_func(x2)
print_func(af.rank(a))
print_func(af.det(a))
print_func(af.norm(a, af.NORM.EUCLID))
print_func(af.norm(a, af.NORM.MATRIX_1))
print_func(af.norm(a, af.NORM.MATRIX_INF))
print_func(af.norm(a, af.NORM.MATRIX_L_PQ, 1, 1))
af.qr_inplace(a) af.display(a) a = af.randu(5, 5) a = af.matmulTN(a, a) + 10 * af.identity(5, 5) R, info = af.cholesky(a) af.display(R) print(info) af.cholesky_inplace(a) af.display(a) a = af.randu(5, 5) ai = af.inverse(a) af.display(a) af.display(ai) x0 = af.randu(5, 3) b = af.matmul(a, x0) x1 = af.solve(a, b) af.display(x0) af.display(x1) p = af.lu_inplace(a) x2 = af.solve_lu(a, p, b)