def test_legendre_spacetime_massmatrix(order):
'''
This test compares the analytical solution of the space-time MM to
the computed one.
'''
basis = basis_defs.LegendreQuad(order)
mesh = mesh_common.mesh_1D(num_elems=1, xmin=-1., xmax=1.)
# Set quadrature
basis.set_elem_quadrature_type("GaussLegendre")
basis.set_face_quadrature_type("GaussLegendre")
MM = basis_st_tools.get_elem_mass_matrix_ader(mesh, basis, order, -1)
'''
Reference:
Dumbser, M., Enaux, C., and Toro, E. JCP 2008
Vol. 227, Issue 8, Pgs: 3971 - 4001 Appendix B
'''
# Multiply by 4.0 caused by reference element shift [0, 1] -> [-1, 1]
if order == 1:
expected = 4.0 * np.array([[1., 0., 0., 0.], [0., 1. / 3., 0., 0.],
[0., 0., 1. / 3., 0], [0., 0., 0., 1. / 9.]
])
elif order == 2:
expected = 4.0 * np.array([[1., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 1. / 3., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 1. / 5., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 1. / 3., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1. / 9., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 1. / 15., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 1. / 5, 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 1. / 15., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 1. / 25.]])
# Assert
np.testing.assert_allclose(MM, expected, rtol, atol)
def set_basis(order, basis_name):
'''
Sets the basis class given the basis_name string argument
Inputs:
-------
order: solution order
basis_name: name of the basis function we wish to instantiate
as a class
Outputs:
--------
basis: instantiated basis class
Raise:
------
If the basis class is not defined, returns a NotImplementedError
'''
if BasisType[basis_name] == BasisType.LagrangeSeg:
basis = basis_defs.LagrangeSeg(order)
elif BasisType[basis_name] == BasisType.LegendreSeg:
basis = basis_defs.LegendreSeg(order)
elif BasisType[basis_name] == BasisType.LagrangeQuad:
basis = basis_defs.LagrangeQuad(order)
elif BasisType[basis_name] == BasisType.LegendreQuad:
basis = basis_defs.LegendreQuad(order)
elif BasisType[basis_name] == BasisType.LagrangeTri:
basis = basis_defs.LagrangeTri(order)
elif BasisType[basis_name] == BasisType.HierarchicH1Tri:
basis = basis_defs.HierarchicH1Tri(order)
else:
raise NotImplementedError
return basis
def set_basis_spacetime(mesh, order, basis_name):
'''
Sets the space-time basis class given the basis_name string argument
Inputs:
-------
order: solution order
basis_name: name of the spatial basis function used to determine
the space-time basis function we wish to instantiate
as a class
Outputs:
--------
basis_st: instantiated space-time basis class
Raise:
------
If the basis class is not defined returns a NotImplementedError
'''
if BasisType[basis_name] == BasisType.LagrangeSeg:
basis_st = basis_defs.LagrangeQuad(order)
elif BasisType[basis_name] == BasisType.LegendreSeg:
basis_st = basis_defs.LegendreQuad(order)
elif BasisType[basis_name] == BasisType.LagrangeQuad:
basis_st = basis_defs.LagrangeHex(order)
elif BasisType[basis_name] == BasisType.LagrangeTri:
basis_st = basis_defs.LagrangePrism(order)
else:
raise NotImplementedError
return basis_st
def test_legendre_spacetime_stiffnessmatrix_spacedir_phys(dt, order):
'''
This test compares the analytical solution of the space-time
stiffness matrix in space to the computed one.
'''
basis = basis_defs.LegendreSeg(order)
basis_st = basis_defs.LegendreQuad(order)
mesh = mesh_common.mesh_1D(num_elems=1, xmin=0., xmax=1.)
# Set quadrature
basis.set_elem_quadrature_type("GaussLegendre")
basis.set_face_quadrature_type("GaussLegendre")
basis_st.set_elem_quadrature_type("GaussLegendre")
basis_st.set_face_quadrature_type("GaussLegendre")
mesh.gbasis.set_elem_quadrature_type("GaussLegendre")
mesh.gbasis.set_face_quadrature_type("GaussLegendre")
# Get stiffness matrix in space
SMS = basis_st_tools.get_stiffness_matrix_ader(mesh,
basis,
basis_st,
order,
dt,
elem_ID=0,
grad_dir=0,
physical_space=True)
'''
Reference:
Dumbser, M., Enaux, C., and Toro, E. JCP 2008
Vol. 227, Issue 8, Pgs: 3971 - 4001 Appendix B
Note: We multiply by 4 here because (unlike Dumbser's paper), we
multiply by the inverse jacobian to account for non-ideal meshes
See the Quail documentation (Section for ADER-DG: space-time
predictor step where the K matrices are defined)
'''
if order == 1:
expected = 4.0 * np.array([[0., 2., 0., 0.], [0., 0., 0., 0.],
[0., 0., 0., 2. / 3.], [0., 0., 0., 0.]])
elif order == 2:
expected = 4.0 * np.array([[0., 2., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 2., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 2. / 3., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 2. / 3., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 2. / 5., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 2. / 5.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.]])
# Assert
np.testing.assert_allclose(SMS.transpose(), expected, rtol, atol)
def test_legendre_spacetime_stiffnessmatrix_timedir(dt, order):
'''
This test compares the analytical solution of the space-time
stiffness matrix in time to the computed one.
'''
basis = basis_defs.LegendreSeg(order)
basis_st = basis_defs.LegendreQuad(order)
mesh = mesh_common.mesh_1D(num_elems=1, xmin=-1., xmax=1.)
# Set quadrature
basis.set_elem_quadrature_type("GaussLegendre")
basis.set_face_quadrature_type("GaussLegendre")
basis_st.set_elem_quadrature_type("GaussLegendre")
basis_st.set_face_quadrature_type("GaussLegendre")
mesh.gbasis.set_elem_quadrature_type("GaussLegendre")
mesh.gbasis.set_face_quadrature_type("GaussLegendre")
# Get stiffness matrix in time
SMT = basis_st_tools.get_stiffness_matrix_ader(mesh,
basis,
basis_st,
order,
dt,
elem_ID=0,
grad_dir=-1,
physical_space=False)
'''
Reference:
Dumbser, M., Enaux, C., and Toro, E. JCP 2008
Vol. 227, Issue 8, Pgs: 3971 - 4001 Appendix B
'''
# Multiply by 2.0 caused by reference element shift [0, 1] -> [-1, 1]
if order == 1:
expected = 2.0 * np.array([[0., 0., 0., 0.], [0., 0., 0., 0.],
[2., 0., 0., 0.], [0., 2. / 3., 0., 0.]])
elif order == 2:
expected = 2.0 * np.array([[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0.],
[2., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 2. / 3., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 2. / 5., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 2., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 2. / 3., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 2. / 5., 0., 0., 0.]])
# Assert
np.testing.assert_allclose(SMT, expected, rtol, atol)
def test_legendre_spacetime_massmatrix(order):
'''
This test makes sure that the already tested mass matrix yields
the corrected inverse mass matrix with the diagonal inverted.
'''
basis = basis_defs.LegendreQuad(order)
mesh = mesh_common.mesh_1D(num_elems=1, xmin=-1., xmax=1.)
# Set quadrature
basis.set_elem_quadrature_type("GaussLegendre")
basis.set_face_quadrature_type("GaussLegendre")
MM = basis_st_tools.get_elem_mass_matrix_ader(mesh, basis, order, -1)
iMM = basis_st_tools.get_elem_inv_mass_matrix_ader(mesh, basis, order, -1)
np.testing.assert_allclose(np.diagonal(iMM), 1. / np.diagonal(MM), rtol,
atol)
def test_temporal_flux_ader_tnp1(order):
'''
This test compares the analytical solution of the space-time temporal
flux evaluated at t^{n+1} to the computed one.
'''
basis_st = basis_defs.LegendreQuad(order)
mesh = mesh_common.mesh_1D(num_elems=1, xmin=-1., xmax=1.)
# Set quadrature
basis_st.set_elem_quadrature_type("GaussLegendre")
basis_st.set_face_quadrature_type("GaussLegendre")
mesh.gbasis.set_elem_quadrature_type("GaussLegendre")
mesh.gbasis.set_face_quadrature_type("GaussLegendre")
# Get flux matrices in time
FTL = basis_st_tools.get_temporal_flux_ader(mesh,
basis_st,
basis_st,
order,
physical_space=False)
'''
Reference:
Dumbser, M., Enaux, C., and Toro, E. JCP 2008
Vol. 227, Issue 8, Pgs: 3971 - 4001 Appendix B
'''
# Multiply by 2.0 caused by reference element shift [0, 1] -> [-1, 1]
if order == 1:
expected = 2.0 * np.array([[1., 0., 1., 0.], [
0., 1. / 3., 0., 1. / 3.
], [1., 0., 1., 0], [0., 1. / 3., 0., 1. / 3.]])
elif order == 2:
expected = 2.0 * np.array(
[[1., 0., 0., 1., 0., 0., 1., 0., 0.],
[0., 1. / 3., 0., 0., 1. / 3., 0., 0., 1. / 3., 0.],
[0., 0., 1. / 5., 0., 0., 1. / 5., 0., 0., 1. / 5.],
[1., 0., 0., 1., 0., 0., 1., 0., 0.],
[0., 1. / 3., 0., 0., 1. / 3., 0., 0., 1. / 3., 0.],
[0., 0., 1. / 5., 0., 0., 1. / 5., 0., 0., 1. / 5.],
[1., 0., 0., 1., 0., 0., 1., 0., 0.],
[0., 1. / 3., 0., 0., 1. / 3., 0., 0., 1. / 3., 0.],
[0., 0., 1. / 5., 0., 0., 1. / 5., 0., 0., 1. / 5.]])
# Assert
np.testing.assert_allclose(FTL, expected, rtol, atol)
def test_legendre_spacetime_massmatrix_phys(order):
'''
This test compares the analytical solution of the space-time MM to
the computed one, but uses a physics element that spans [0.0, 0.5]
'''
basis = basis_defs.LegendreQuad(order)
mesh = mesh_common.mesh_1D(num_elems=1, xmin=0., xmax=0.5)
# Set quadrature
basis.set_elem_quadrature_type("GaussLegendre")
basis.set_face_quadrature_type("GaussLegendre")
MM = basis_st_tools.get_elem_mass_matrix_ader(mesh,
basis,
order,
0,
physical_space=True)
'''
Reference:
Dumbser, M., Enaux, C., and Toro, E. JCP 2008
Vol. 227, Issue 8, Pgs: 3971 - 4001 Appendix B
'''
if order == 1:
expected = np.array([[1., 0., 0., 0.], [0., 1. / 3., 0., 0.],
[0., 0., 1. / 3., 0], [0., 0., 0., 1. / 9.]])
elif order == 2:
expected = np.array([[1., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 1. / 3., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 1. / 5., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 1. / 3., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1. / 9., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 1. / 15., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 1. / 5, 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 1. / 15., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 1. / 25.]])
# Assert
np.testing.assert_allclose(MM, expected, rtol, atol)