def volume_int_convergence():
'''
convergence test for volume int flux
'''
N_LGL = np.arange(15).astype(float) + 3
L1_norm_option_3 = np.zeros([15])
L1_norm_option_1 = np.zeros([15])
for i in range(0, 15):
test_waveEqn.change_parameters(i + 3, 16, i + 4)
vol_int_analytical = np.zeros([params.N_Elements, params.N_LGL])
for j in range (params.N_Elements):
for k in range (params.N_LGL):
vol_int_analytical[j][k] = (analytical_volume_integral\
(af.transpose(params.element_array[j]), k))
vol_int_analytical = af.transpose(af.np_to_af_array\
(vol_int_analytical))
L1_norm_option_3[i] = af.mean(af.abs(vol_int_analytical\
- wave_equation.volume_integral_flux(params.u_init, 0)))
for i in range(0, 15):
test_waveEqn.change_parameters(i + 3, 16, i + 3)
vol_int_analytical = np.zeros([params.N_Elements, params.N_LGL])
for j in range (params.N_Elements):
for k in range (params.N_LGL):
vol_int_analytical[j][k] = analytical_volume_integral(\
af.transpose(params.element_array[j]), k)
vol_int_analytical = af.transpose(af.np_to_af_array(vol_int_analytical))
L1_norm_option_1[i] = af.mean(af.abs(vol_int_analytical\
- wave_equation.volume_integral_flux(params.u_init, 0)))
normalization = 0.0023187 / (3 ** (-3))
print(L1_norm_option_1, L1_norm_option_3)
plt.loglog(N_LGL, L1_norm_option_1, marker='o', label='L1 norm option 1')
plt.loglog(N_LGL, L1_norm_option_3, marker='o', label='L1 norm option 3')
plt.loglog(N_LGL, normalization * N_LGL **(-N_LGL), color='black', linestyle='--', label='$N_{LGL}^{-N_{LGL}}$')
plt.title('L1 norm of volume integral term')
plt.xlabel('LGL points')
plt.ylabel('L1 norm')
plt.legend(loc='best')
plt.show()
def test_volume_integral_flux():
'''
A test function to check the volume_integral_flux function in wave_equation
module by analytically calculated Gauss-Lobatto quadrature.
Reference
---------
The link to the sage worksheet where the calculations were caried out is
given below.
`https://goo.gl/5Mub8M`
'''
threshold = 4e-9
params.c = 1
change_parameters(8, 10, 11, 'gaussian')
referenceFluxIntegral = af.transpose(af.interop.np_to_af_array(np.array
([
[-0.002016634876668093, -0.000588597708116113, -0.0013016773719126333,\
-0.002368387579324652, -0.003620502047659841, -0.004320197094090966,
-0.003445512010153811, 0.0176615086879261],\
[-0.018969769374, -0.00431252844519,-0.00882630935977,-0.0144355176966,\
-0.019612124119, -0.0209837936827, -0.0154359890788, 0.102576031756], \
[-0.108222418798, -0.0179274222595, -0.0337807018822, -0.0492589052599,\
-0.0588472807471, -0.0557970236273, -0.0374764132459, 0.361310165819],\
[-0.374448714304, -0.0399576371245, -0.0683852285846, -0.0869229749357,\
-0.0884322503841, -0.0714664112839, -0.0422339853622, 0.771847201979], \
[-0.785754362849, -0.0396035640187, -0.0579313769517, -0.0569022801117,\
-0.0392041960688, -0.0172295769141, -0.00337464521455, 1.00000000213],\
[-1.00000000213, 0.00337464521455, 0.0172295769141, 0.0392041960688,\
0.0569022801117, 0.0579313769517, 0.0396035640187, 0.785754362849],\
[-0.771847201979, 0.0422339853622, 0.0714664112839, 0.0884322503841, \
0.0869229749357, 0.0683852285846, 0.0399576371245, 0.374448714304],\
[-0.361310165819, 0.0374764132459, 0.0557970236273, 0.0588472807471,\
0.0492589052599, 0.0337807018822, 0.0179274222595, 0.108222418798], \
[-0.102576031756, 0.0154359890788, 0.0209837936827, 0.019612124119, \
0.0144355176966, 0.00882630935977, 0.00431252844519, 0.018969769374],\
[-0.0176615086879, 0.00344551201015 ,0.00432019709409, 0.00362050204766,\
0.00236838757932, 0.00130167737191, 0.000588597708116, 0.00201663487667]\
])))
numerical_flux = wave_equation.volume_integral_flux(params.u[:, :, 0])
assert (af.mean(af.abs(numerical_flux - referenceFluxIntegral)) <
threshold)
def test_b_vector():
'''
A test function to check the b vector obtained analytically and compare it
with the one returned by b_vector function in wave_equation module.
'''
threshold = 1e-13
params.c = 1
change_parameters(8, 10, 8, 'gaussian')
u_n_A_matrix = af.blas.matmul(wave_equation.A_matrix(),\
params.u[:, :, 0])
volume_integral_flux = wave_equation.volume_integral_flux(params.u[:, :,
0])
surface_term = test_surface_term()
b_vector_analytical = u_n_A_matrix + (volume_integral_flux -\
(surface_term)) * params.delta_t
b_vector_array = wave_equation.b_vector(params.u[:, :, 0])
assert (b_vector_analytical - b_vector_array) < threshold