from FIAT import ufc_simplex, create_quadrature
from FIAT.quadrature import RadauQuadratureLineRule, GaussLobattoLegendreQuadratureLineRule
ufc_line = ufc_simplex(1)
[docs]
class GalerkinScheme:
"""
Base class for describing Galerkin-in-time methods in lieu of
a Butcher tableau.
:arg order: An integer indicating the order of the method
:kwarg basis_type: A string (or tuple of strings) indicating the finite
element family (either ``'Lagrange'`` or ``'Bernstein'``) or the
Lagrange variant (either ``'equispaced'``, ``'spectral'``, ``'chebyshev'``,
or ``'integral'``) for the test/trial spaces.
Defaults to equispaced Lagrange elements.
:kwarg quadrature_degree: An integer or string indicating the degree of the
quadrature to be used in time. Defaults to the sum
of the degrees of the trial and test spaces. The string value ``'auto'``
enables automatic degree estimation for each term in the form.
:kwarg quadrature_scheme: A string indicating the quadrature scheme
to be used in time. Defaults to Gauss-Legendre.
:kwarg max_quadrature_degree: An integer indicating the maximum quadrature
degree allowed in the automatic degree estimation.
If ``None``, then the estimated quadrature degree will always be used.
"""
def __init__(self, order,
basis_type=None,
quadrature_degree=None,
quadrature_scheme=None,
max_quadrature_degree=None):
self.order = order
self.basis_type = basis_type
self.quadrature_degree = quadrature_degree
self.quadrature_scheme = quadrature_scheme
self.max_quadrature_degree = max_quadrature_degree
[docs]
class DiscontinuousGalerkinScheme(GalerkinScheme):
"""Class for describing DG-in-time methods
:arg order: An integer indicating the order of the method
:kwarg basis_type: A string (or tuple of strings) indicating the finite
element family (either ``'Lagrange'`` or ``'Bernstein'``) or the
Lagrange variant (either ``'equispaced'``, ``'spectral'``, ``'chebyshev'``,
or ``'integral'``) for the test/trial spaces.
Defaults to equispaced Lagrange elements.
:kwarg quadrature_degree: An integer or string indicating the degree of the
quadrature to be used in time. Defaults to the sum
of the degrees of the trial and test spaces. The string value ``'auto'``
enables automatic degree estimation for each term in the form.
:kwarg quadrature_scheme: A string indicating the quadrature scheme
to be used in time. Defaults to Gauss-Legendre.
:kwarg max_quadrature_degree: An integer indicating the maximum quadrature
degree allowed in the automatic degree estimation.
If ``None``, then the estimated quadrature degree will always be used.
:kwarg deriv_type: A string indicating how to integrate terms with time derivatives.
Valid values are:
- `'weak'`: Time derivatives act on the test function (integrating by parts once).
- `'strong'`: Time derivatives act on the unknown (integrating by parts twice).
"""
def __init__(self, order,
basis_type=None,
quadrature_degree=None,
quadrature_scheme=None,
max_quadrature_degree=None,
deriv_type="strong"):
if order < 0:
raise ValueError(f"{type(self).__name__} must have order >= 0")
if deriv_type not in {'weak', 'strong'}:
raise ValueError("deriv_type must be either 'weak' or 'strong'.")
self.deriv_type = deriv_type
super().__init__(order, basis_type=basis_type,
quadrature_degree=quadrature_degree,
quadrature_scheme=quadrature_scheme,
max_quadrature_degree=max_quadrature_degree)
[docs]
class ContinuousPetrovGalerkinScheme(GalerkinScheme):
"""Class for describing cPG-in-time methods"""
def __init__(self, order,
basis_type=None,
quadrature_degree=None,
quadrature_scheme=None,
max_quadrature_degree=None):
if order < 1:
raise ValueError(f"{type(self).__name__} must have order >= 1")
super().__init__(order, basis_type=basis_type,
quadrature_degree=quadrature_degree,
quadrature_scheme=quadrature_scheme,
max_quadrature_degree=max_quadrature_degree)
[docs]
class GalerkinCollocationScheme(ContinuousPetrovGalerkinScheme):
"""Class for describing collocation cPG-in-time methods"""
def __init__(self, order,
stage_type="deriv",
quadrature_degree=None,
quadrature_scheme=None,
max_quadrature_degree=None):
if order < 1:
raise ValueError(f"{type(self).__name__} must have order >= 1")
if quadrature_scheme is None:
test_type = "spectral"
elif quadrature_scheme in {"radau", "lobatto"}:
test_type = quadrature_scheme
if quadrature_degree is None:
# Default to under-integration!
if quadrature_scheme == "radau":
quadrature_degree = 2*order-2
elif quadrature_scheme == "lobatto":
quadrature_degree = 2*order-3
else:
raise ValueError(f"Unsupported quadrature scheme {quadrature_scheme}.")
if stage_type not in {"deriv", "value"}:
raise ValueError(f"Unsupported stage type {stage_type}.")
trial_type = stage_type
basis_type = (trial_type, test_type)
super().__init__(order, basis_type=basis_type,
quadrature_degree=quadrature_degree,
quadrature_scheme=quadrature_scheme,
max_quadrature_degree=max_quadrature_degree)
[docs]
def create_time_quadrature(degree, scheme=None):
"""Return a :class:`FIAT.QuadratureRule` on the unit interval.
:arg degree: The degree of polynomial that the rule should integrate exactly.
:kwarg scheme: The quadrature scheme. Can be either `'default'` for Gauss-Legendre,
`'Lobatto'` for Gauss-Lobatto-Legendre, or `'Radau'` for Gauss-Radau.
"""
if scheme is not None and scheme.lower() == "radau":
num_points = (degree + 1) // 2 + 1
return RadauQuadratureLineRule(ufc_line, num_points)
elif scheme is not None and scheme.lower() == "lobatto":
num_points = degree // 2 + 2
return GaussLobattoLegendreQuadratureLineRule(ufc_line, num_points)
else:
return create_quadrature(ufc_line, degree, scheme=scheme or "default")
