r"""
Module for defining function spaces in the Chebyshev family.
A function is approximated in the Chebyshev basis as
.. math::
u(x) = \sum_{i=0}^{N-1} \hat{u}_i T_i(x)
where :math:`T_i(x)` is the i'th Chebyshev polynomial of the first kind.
The Chebyshev polynomials are orthogonal with weight :math:`\omega=1/\sqrt{1-x^2}`
.. math::
\int_{-1}^1 T_i T_k \omega dx = \frac{c_k \pi}{2} \delta_{ki},
where :math:`c_0=2` and :math:`c_i=1` for :math:`i>0`.
All other bases defined in this module are combinations of :math:`T_i`'s.
For example, a Dirichlet basis is
.. math::
\phi_i = T_i - T_{i+2}
The basis is implemented using a stencil matrix :math:`K \in \mathbb{R}^{N-2 \times N}`,
such that
.. math::
\boldsymbol{\phi} = K \boldsymbol{T},
where :math:`\boldsymbol{\phi}=(\phi_0, \phi_1, \ldots, \phi_{N-3})` and
:math:`\boldsymbol{T}=(T_0, T_1, \ldots, T_{N-1})`. For the Dirichlet basis
:math:`K = (\delta_{i, j} - \delta_{i+2, j})_{i,j=0}^{N-2, N}`.
All composite bases make use of the fast transforms that exists for
:math:`\boldsymbol{T}` through fast cosine transforms. The stencil matrix
is used to transfer any composite basis back and forth to the orthogonal basis.
"""
from __future__ import division
import functools
import numpy as np
from numpy.polynomial import chebyshev as n_cheb
import sympy as sp
from scipy.special import eval_chebyt
from mpi4py_fft import fftw
from shenfun.spectralbase import SpectralBase, Transform, FuncWrap, \
islicedict, slicedict, getCompositeBase, getBCGeneric, BoundaryConditions
from shenfun.matrixbase import SparseMatrix
from shenfun.optimization import optimizer
from shenfun.config import config
from shenfun.jacobi.recursions import half, cn
from shenfun.jacobi import JacobiBase
from shenfun.utilities import n
bases = ['Orthogonal',
'ShenDirichlet',
'Heinrichs',
'ShenNeumann',
'CombinedShenNeumann',
'MikNeumann',
'ShenBiharmonic',
'UpperDirichlet',
'LowerDirichlet',
'UpperDirichletNeumann',
'LowerDirichletNeumann',
'ShenBiPolar',
'PolarDirichlet',
'DirichletNeumann',
'NeumannDirichlet',
'Compact3',
'Compact4',
'Generic']
bcbases = ['BCGeneric']
testbases = ['Phi1', 'Phi2', 'Phi3', 'Phi4', 'Phi6']
__all__ = bases + bcbases + testbases
#pylint: disable=abstract-method, not-callable, method-hidden, no-self-use, cyclic-import
chebval = optimizer(n_cheb.chebval)
xp = sp.Symbol('x', real=True)
class DCTWrap(FuncWrap):
"""DCT for complex input"""
@property
def dct(self):
return object.__getattribute__(self, '_func')
def __call__(self, **kw):
dct_obj = self.dct
dct_obj.input_array[...] = self.input_array.real
dct_obj(None, None, **kw)
self.output_array.real[...] = dct_obj.output_array
dct_obj.input_array[...] = self.input_array.imag
dct_obj(None, None, **kw)
self.output_array.imag[...] = dct_obj.output_array
return self.output_array
[docs]
class Orthogonal(JacobiBase):
r"""Function space for regular Chebyshev series
The orthogonal basis is
.. math::
T_k, \quad k = 0, 1, \ldots, N-1,
where :math:`T_k` is the :math:`k`'th Chebyshev polynomial of the first
kind.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad='GC', domain=(-1, 1), dtype=float, padding_factor=1,
dealias_direct=False, coordinates=None, **kw):
JacobiBase.__init__(self, N, quad=quad, alpha=-half, beta=-half, domain=domain, dtype=dtype,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
assert quad in ('GC', 'GL', 'GU')
self.gn = cn
if quad == 'GC':
self._xfftn_fwd = functools.partial(fftw.dctn, type=2)
self._xfftn_bck = functools.partial(fftw.dctn, type=3)
self._xfftn_fwd.opts = self._xfftn_bck.opts = config['fftw']['dct']
elif quad in ('GL', 'GU'):
self._xfftn_fwd = functools.partial(fftw.dctn, type=1)
self._xfftn_bck = functools.partial(fftw.dctn, type=1)
self._xfftn_fwd.opts = self._xfftn_bck.opts = config['fftw']['dct']
self.plan((int(padding_factor*N),), 0, dtype, {})
# Comment due to curvilinear issues
#def apply_inverse_mass(self, array):
# coors = self.tensorproductspace.coors if self.dimensions > 1 else self.coors
# if not coors.hi.prod() == 1:
# return JacobiBase.apply_inverse_mass(self, array)
# array *= (2/np.pi*self.domain_factor())
# array[self.si[0]] /= 2
# if self.quad == 'GL':
# array[self.si[-1]] /= 2
# return array
[docs]
@staticmethod
def family():
return 'chebyshev'
[docs]
def points_and_weights(self, N=None, map_true_domain=False, weighted=True, **kw):
if N is None:
N = self.shape(False)
if weighted:
if self.quad == "GL":
points = -(n_cheb.chebpts2(N)).astype(float)
weights = np.full(N, np.pi/(N-1))
weights[0] /= 2
weights[-1] /= 2
elif self.quad == "GC":
points, weights = n_cheb.chebgauss(N)
points = points.astype(float)
weights = weights.astype(float)
elif self.quad == "GU":
points = np.cos((np.arange(N)+1)*np.pi/(N+1))
weights = np.full(N, np.pi/(N+1))
else:
if self.quad == "GL":
points = np.cos(np.arange(N)*np.pi/(N-1))
d = fftw.aligned(N, fill=0)
k = 2*(1 + np.arange((N-1)//2))
d[::2] = (2./(N-1))/np.hstack((1., 1.-k*k))
w = fftw.aligned_like(d)
dct = fftw.dctn(w, axes=(0,), type=1)
weights = dct(d, w)
weights[0] *= 0.5
weights[-1] *= 0.5
elif self.quad == "GC":
points = n_cheb.chebgauss(N)[0]
d = fftw.aligned(N, fill=0)
k = 2*(1 + np.arange((N-1)//2))
d[::2] = (2./N)/np.hstack((1., 1.-k*k))
w = fftw.aligned_like(d)
dct = fftw.dctn(w, axes=(0,), type=3)
weights = dct(d, w)
elif self.quad == "GU":
theta = (np.arange(N)+1)*np.pi/(N+1)
points = np.cos(theta)
d = fftw.aligned(N, fill=0)
k = np.arange(N)
d[::2] = 2/(k[::2]+1)
w = fftw.aligned_like(d)
dst = fftw.dstn(w, axes=(0,), type=1)
weights = dst(d, w)
weights *= (np.sin(theta))/(N+1)
if map_true_domain is True:
points = self.map_true_domain(points)
return points, weights
[docs]
def vandermonde(self, x):
return n_cheb.chebvander(x, self.shape(False)-1)
[docs]
def weight(self, x=xp):
return 1/sp.sqrt(1-x**2)
[docs]
def orthogonal_basis_function(self, i=0, x=xp):
return sp.chebyshevt(i, x)
[docs]
def L2_norm_sq(self, i):
return (1+int(i==0))*sp.pi/2
[docs]
def l2_norm_sq(self, i=None):
if i is None:
f = np.full(self.N, np.pi/2)
f[0] *= 2
if self.quad == 'GL':
f[-1] *= 2
return f
elif i == 0 or i == self.N-1 and self.quad == 'GL':
return np.pi
return np.pi/2
[docs]
def evaluate_basis(self, x, i=0, output_array=None):
x = np.atleast_1d(x)
if output_array is None:
output_array = np.zeros(x.shape)
#output_array[:] = np.cos(i*np.arccos(x))
output_array[:] = eval_chebyt(i, x)
return output_array
[docs]
def evaluate_basis_derivative(self, x=None, i=0, k=0, output_array=None):
if x is None:
x = self.mesh(False, False)
if output_array is None:
output_array = np.zeros(x.shape)
x = np.atleast_1d(x)
basis = np.zeros(self.shape(True))
basis[i] = 1
basis = n_cheb.Chebyshev(basis)
if k > 0:
basis = basis.deriv(k)
output_array[:] = basis(x)
return output_array
[docs]
def evaluate_basis_derivative_all(self, x=None, k=0, argument=0):
if x is None:
x = self.mesh(False, False)
V = self.vandermonde(x)
M = V.shape[1]
if k > 0:
D = np.zeros((M, M))
D[:-k] = n_cheb.chebder(np.eye(M, M), k)
V = np.dot(V, D)
return V
def _evaluate_expansion_all(self, input_array, output_array, x=None, kind='fast'):
if kind != 'fast':
SpectralBase._evaluate_expansion_all(self, input_array, output_array, x, kind=kind)
return
assert input_array is self.backward.tmp_array
assert output_array is self.backward.output_array
output_array = self.backward.xfftn()
if self.quad == "GC":
s0 = self.sl[slice(0, 1)]
output_array *= 0.5
output_array += input_array[s0]/2
elif self.quad == "GL":
output_array *= 0.5
output_array += input_array[self.sl[slice(0, 1)]]/2
s0 = self.sl[slice(-1, None)]
s2 = self.sl[slice(0, None, 2)]
output_array[s2] += input_array[s0]/2
s2 = self.sl[slice(1, None, 2)]
output_array[s2] -= input_array[s0]/2
def _evaluate_scalar_product(self, kind='fast'):
if kind != 'fast':
SpectralBase._evaluate_scalar_product(self, kind=kind)
return
if self.quad == "GC":
out = self.scalar_product.xfftn()
out *= (np.pi/(2*self.domain_factor()*self.N*self.padding_factor))
elif self.quad == "GL":
out = self.scalar_product.xfftn()
out *= (np.pi/(2*self.domain_factor()*(self.N*self.padding_factor-1)))
#@profile
[docs]
def eval(self, x, u, output_array=None):
x = np.atleast_1d(x)
x = self.map_reference_domain(x)
#oa = chebval(x, u)
oa = n_cheb.chebval(x, u, False)
if output_array is not None:
output_array[:] = oa
return output_array
return oa
@property
def is_orthogonal(self):
return True
[docs]
@staticmethod
def short_name():
return 'T'
[docs]
def stencil_matrix(self, N=None):
N = self.N if N is None else N
return SparseMatrix({0: 1}, (N, N))
[docs]
def sympy_stencil(self, i=sp.Symbol('i', integer=True), j=sp.Symbol('j', integer=True)):
return sp.KroneckerDelta(i, j)
[docs]
def to_ortho(self, input_array, output_array=None):
assert input_array.__class__.__name__ == 'Orthogonal'
if output_array:
output_array[:] = input_array
return output_array
return input_array
[docs]
def get_orthogonal(self, **kwargs):
d = dict(quad=self.quad,
domain=self.domain,
dtype=self.dtype,
padding_factor=self.padding_factor,
dealias_direct=self.dealias_direct,
coordinates=self.coors.coordinates)
d.update(kwargs)
return Orthogonal(self.N, **d)
[docs]
def get_bc_space(self):
if self._bc_space:
return self._bc_space
self._bc_space = BCGeneric(self.N, bc=self.bcs, domain=self.domain)
return self._bc_space
[docs]
def plan(self, shape, axis, dtype, options):
if shape in (0, (0,)):
return
if isinstance(axis, tuple):
assert len(axis) == 1
axis = axis[-1]
if isinstance(self.forward, Transform):
if self.forward.input_array.shape == shape and self.axis == axis:
# Already planned
return
plan_fwd = self._xfftn_fwd
plan_bck = self._xfftn_bck
opts = plan_fwd.opts
opts['overwrite_input'] = 'FFTW_DESTROY_INPUT'
opts.update(options)
flags = (fftw.flag_dict[opts['planner_effort']],
fftw.flag_dict[opts['overwrite_input']])
threads = opts['threads']
U = fftw.aligned(shape, dtype=float)
xfftn_fwd = plan_fwd(U, axes=(axis,), threads=threads, flags=flags)
V = xfftn_fwd.output_array
xfftn_bck = plan_bck(V, axes=(axis,), threads=threads, flags=flags, output_array=U)
V.fill(0)
U.fill(0)
if np.dtype(dtype) is np.dtype('complex'):
# dct only works on real data, so need to wrap it
U = fftw.aligned(shape, dtype=complex)
V = fftw.aligned(shape, dtype=complex)
U.fill(0)
V.fill(0)
xfftn_fwd = DCTWrap(xfftn_fwd, U, V)
xfftn_bck = DCTWrap(xfftn_bck, V, U)
self.axis = axis
if self.padding_factor != 1:
trunc_array = self._get_truncarray(shape, V.dtype)
self.scalar_product = Transform(self.scalar_product, xfftn_fwd, U, V, trunc_array)
self.forward = Transform(self.forward, xfftn_fwd, U, V, trunc_array)
self.backward = Transform(self.backward, xfftn_bck, trunc_array, V, U)
else:
self.scalar_product = Transform(self.scalar_product, xfftn_fwd, U, V, V)
self.forward = Transform(self.forward, xfftn_fwd, U, V, V)
self.backward = Transform(self.backward, xfftn_bck, V, V, U)
self.si = islicedict(axis=self.axis, dimensions=U.ndim)
self.sl = slicedict(axis=self.axis, dimensions=U.ndim)
# Note that all composite spaces rely on the fast transforms of
# the orthogonal space. For this reason we have an intermediate
# class CompositeBase for all composite spaces, where common code
# is implemented and reused by all.
CompositeBase = getCompositeBase(Orthogonal)
BCGeneric = getBCGeneric(CompositeBase)
[docs]
class ShenDirichlet(CompositeBase):
r"""Function space for Dirichlet boundary conditions.
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_k - T_{k+2}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\
\phi_{N-1} &= \frac{1}{2}(T_0+T_1),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a \text{ and } u(1) = b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of floats, optional
Boundary conditions at, respectively, x=(-1, 1).
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {0: 1, 2: -1}
[docs]
@staticmethod
def boundary_condition():
return 'Dirichlet'
[docs]
@staticmethod
def short_name():
return 'SD'
#def _evaluate_scalar_product(self, kind='fast'):
# if kind != 'fast':
# SpectralBase._evaluate_scalar_product(self, kind=kind)
# self.scalar_product.tmp_array[self.si[-2]] = 0
# self.scalar_product.tmp_array[self.si[-1]] = 0
# return
# Orthogonal._evaluate_scalar_product(self, kind=kind)
# output = self.scalar_product.tmp_array
# s0 = self.sl[slice(0, self.N-2)]
# s1 = self.sl[slice(2, self.N)]
# output[s0] -= output[s1]
# output[self.si[-2]] = 0
# output[self.si[-1]] = 0
#def to_ortho(self, input_array, output_array=None):
# if output_array is None:
# output_array = np.zeros_like(input_array)
# else:
# output_array.fill(0)
# s0 = self.sl[slice(0, self.N-2)]
# s1 = self.sl[slice(2, self.N)]
# output_array[s0] = input_array[s0]
# output_array[s1] -= input_array[s0]
# self.bc._add_to_orthogonal(output_array, input_array)
# return output_array
[docs]
class Phi1(CompositeBase):
r"""Function space for Dirichlet boundary conditions.
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= \frac{T_k - T_{k+2}}{\pi (k+1)} = \frac{2(1-x^2)}{\pi k(k+1)} T'_{k+1}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\
\phi_{N-1} &= \frac{1}{2}(T_0+T_1),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a \text{ and } u(1) = b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of floats, optional
Boundary conditions at, respectively, x=(-1, 1).
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1., 1.), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: sp.simplify(b(-half, -half, n+1, n, cn) / (h(-half, -half, n, 0, cn))),
# 2: sp.simplify(b(-half, -half, n+1, n+2, cn) / (h(-half, -half, n+2, 0, cn)))}
self._stencil = {0: 1/sp.pi/(n+1), 2: -1/sp.pi/(n+1)}
[docs]
@staticmethod
def boundary_condition():
return 'Dirichlet'
[docs]
@staticmethod
def short_name():
return 'P1'
[docs]
class Heinrichs(CompositeBase):
r"""Function space for Dirichlet boundary conditions
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= \alpha_k(1-x^2) T_{k}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\
\phi_{N-1} &= \frac{1}{2}(T_0+T_1),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a \text{ and } u(1) = b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`. If the parameter `scaled=True`, then
:math:`\alpha_k=1/(k+1)/(k+2)`, otherwise, :math:`\alpha_k=1`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of floats, optional
Boundary conditions at, respectively, x=(-1, 1).
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optiona
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
scaled : bool, optional
Whether or not to use scaled basis
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float, scaled=False,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc, scaled=scaled,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
[docs]
@staticmethod
def boundary_condition():
return 'Dirichlet'
[docs]
@staticmethod
def short_name():
return 'HH'
[docs]
def stencil_matrix(self, N=None):
N = self.N if N is None else N
d = 0.5*np.ones(N, dtype=int)
d[-2:] = 0
d[1] = 0.25
dm2 = -0.25*np.ones(N-2, dtype=int)
dm2[-2:] = 0
dp2 = -0.25*np.ones(N-2, dtype=int)
dp2[0] = -0.5
if self.is_scaled():
k = np.arange(N)
d /= ((k+1)*(k+2))
dm2 /= ((k[:-2]+3)*(k[:-2]+4))
dp2 /= ((k[:-2]+1)*(k[:-2]+2))
return SparseMatrix({-2: dm2, 0: d, 2: dp2}, (N, N))
[docs]
def sympy_stencil(self, i=sp.Symbol('i', integer=True), j=sp.Symbol('j', integer=True)):
return RuntimeError, "Not possible for current basis"
[docs]
class ShenNeumann(CompositeBase):
r"""Function space for Neumann boundary conditions
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} - \left(\frac{k}{k+2}\right)^2 T_{k+2}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= \frac{1}{8}(4T_1-T_2), \\
\phi_{N-1} &= \frac{1}{8}(4T_1+T_2),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u'(-1) &= a \text{ and } u'(1) = b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of floats, optional
Boundary condition values at, respectively, x=(-1, 1).
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float, padding_factor=1,
dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'left': {'N': bc[0]}, 'right': {'N': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {0: 1, 2: -(n/(n+2))**2}
[docs]
@staticmethod
def boundary_condition():
return 'Neumann'
[docs]
@staticmethod
def short_name():
return 'SN'
[docs]
class CombinedShenNeumann(CompositeBase):
r"""Function space for Neumann boundary conditions
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k = \begin{cases}
T_0, \quad &k=0, \\
T_1 - T_3/9, \quad &k=1, \\
T_2/4 - T_4/16, \quad &k=2, \\
-\frac{T_{k-2}}{(k-2)^2} +2\frac{T_k}{k^2} - \frac{T_{k+2}}{(k+2)^2}, &k=3, 4, \ldots, N-3, \\
\frac{1}{8}(4T_1-T_2), \quad &k=N-2 \\
\frac{1}{8}(4T_1+T_2), \quad &k=N-1
\end{cases}
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u'(-1) &= a \text{ and } u'(1) = b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of floats, optional
Boundary condition values at, respectively, x=(-1, 1).
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float, padding_factor=1,
dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'left': {'N': bc[0]}, 'right': {'N': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
[docs]
@staticmethod
def boundary_condition():
return 'Neumann'
[docs]
@staticmethod
def short_name():
return 'CN'
[docs]
def stencil_matrix(self, N=None):
N = self.N if N is None else N
k = np.arange(N)
k[0] = 1
d = 2/k**2
d[-2:] = 0
d[0] = 1
d[1] = 1
d[2] = 0.25
dm2 = -1/k[:-2]**2
dm2[0] = 0
dm2[-2:] = 0
dp2 = -1/k[2:]**2
dp2[0] = 0
return SparseMatrix({-2: dm2, 0: d, 2: dp2}, (N, N))
[docs]
def sympy_stencil(self, i=sp.Symbol('i', integer=True), j=sp.Symbol('j', integer=True)):
return RuntimeError, "Not possible for current basis"
[docs]
class MikNeumann(CompositeBase):
r"""Function space for Neumann boundary conditions
The basis functions :math:`\phi_k` for :math:`k=0,1, \ldots, N-3` are
.. math::
\phi_k &= \frac{2}{k+1}\int (T_{k-1}-T_{k+1}),
which (with also boundary functions) leads to the basis
.. math::
\phi_k = \frac{1}{k+1} \begin{cases}
T_0, &k=0, \\
3T_1-T_3/3, &k=1, \\
T_2-T_4/4, &k=2, \\
-\frac{T_{k-2}}{k-2} + 2\frac{T_k}{k} - \frac{T_{k+2}}{k+2} , &k=3, 4, \ldots, N-3, \\
\frac{1}{8}(4T_1-T_2), \quad &k=N-2 \\
\frac{1}{8}(4T_1+T_2), \quad &k=N-1
\end{cases}
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u'(-1) &= a \text{ and } u'(1) = b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of floats, optional
Boundary condition values at, respectively, x=(-1, 1).
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float, padding_factor=1,
dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'left': {'N': bc[0]}, 'right': {'N': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
[docs]
@staticmethod
def boundary_condition():
return 'Neumann'
[docs]
@staticmethod
def short_name():
return 'MN'
[docs]
def stencil_matrix(self, N=None):
N = self.N if N is None else N
k = np.arange(N)
k[0] = 1
d = 2/k/(k+1)
d[-2:] = 0
d[0] = 1
d[1] = 3/2
d[2] = 1/3
dm2 = -1/k[:-2]/(k[2:]+1)
dm2[0] = 0
dm2[-2:] = 0
dp2 = -1/k[2:]/(k[2:]-1)
dp2[0] = 0
#dp2[1] = -1/6
#dp2[2] = -1/12
return SparseMatrix({-2: dm2, 0: d, 2: dp2}, (N, N))
[docs]
def sympy_stencil(self, i=sp.Symbol('i', integer=True), j=sp.Symbol('j', integer=True)):
return RuntimeError, "Not possible for current basis"
[docs]
class ShenBiharmonic(CompositeBase):
r"""Function space for biharmonic equation.
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_n - \frac{2(k+2)}{k+3}T_{k+2}+\frac{k+1}{k+3}T_{k+4}, \, k=0, 1, \ldots, N-5, \\
\phi_{N-4} &= \frac{1}{16}(8T_0-9T_1+T_3), \\
\phi_{N-3} &= \frac{1}{16}(2T_0-T_1-2T_2+T_3), \\
\phi_{N-2} &= \frac{1}{16}(8T_0+9T_1-T_3), \\
\phi_{N-1} &= \frac{1}{16}(2T_0-T_1+2T_2+T_3),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1)&=a, u'(-1) = b, u(1)=c, u'(1) = d.
The last four bases are for boundary conditions and only used if a, b, c or d are
different from 0. In one dimension :math:`\hat{u}_{N-4}=a`, :math:`\hat{u}_{N-3}=b`,
:math:`\hat{u}_{N-2}=c` and :math:`\hat{u}_{N-1}=d`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 4-tuple of numbers
The values of the 4 boundary conditions at x=(-1, 1).
The two conditions on x=-1 first, and then x=1.
With (a, b, c, d) corresponding to
bc = {'left': [('D', a), ('N', b)], 'right': [('D', c), ('N', d)]}
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0, 0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {0: 1, 2: -(2*n + 4)/(n + 3), 4: (n + 1)/(n + 3)}
[docs]
@staticmethod
def boundary_condition():
return 'Biharmonic'
[docs]
@staticmethod
def short_name():
return 'SB'
[docs]
class PolarDirichlet(CompositeBase):
r"""Function space for polar coordinates.
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_k - T_{k+2}, \, k=0, 1
\phi_k &= T_{k-2} - T_{k+2}, \, k=2, 3, \ldots, N-3, \\
\phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\
\phi_{N-1} &= \frac{1}{2}(T_0+T_1),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1)&=a, u(1)=b
The last four bases are for boundary conditions and only used if a, b, c or d are
different from 0. In one dimension :math:`\hat{u}_{N-4}=a`, :math:`\hat{u}_{N-3}=b`,
:math:`\hat{u}_{N-2}=c` and :math:`\hat{u}_{N-1}=d`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 4-tuple of numbers
The values of the 4 boundary conditions at x=(-1, 1).
The two conditions on x=-1 first, and then x=1.
With (a, b, c, d) corresponding to
bc = {'left': [('D', a), ('N', b)], 'right': [('D', c), ('N', d)]}
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
[docs]
@staticmethod
def boundary_condition():
return 'Dirichlet'
[docs]
def stencil_matrix(self, N=None):
N = self.N if N is None else N
d = np.ones(N)
d[2:] = 0
dm2 = np.ones(N)
dm2[-2:] = 0
return SparseMatrix({-2: dm2, 0: d, 2: -1}, (N, N))
[docs]
@staticmethod
def short_name():
return 'PD'
[docs]
def sympy_stencil(self, i=sp.Symbol('i', integer=True), j=sp.Symbol('j', integer=True)):
return RuntimeError, "Not possible for current basis"
[docs]
class Phi2(CompositeBase):
r"""Function space for biharmonic equation.
The basis functions :math:`\phi_k` for :math:`k=0, 1, \ldots, N-5` are
.. math::
\phi_k = \frac{2(1-x^2)^2 T''_{k+2}}{\pi (k+1)(k+2)^2(k+3)} ,
which (along with boundary functions) gives the basis
.. math::
\phi_k &= \frac{1}{2 \pi (k+1)(k+2)}(T_k - \frac{2(k+2)}{k+3}T_{k+2} + \frac{k+1}{k+3}T_{k+4}), \, k=0, 1, \ldots, N-5, \\
\phi_{N-4} &= \frac{1}{16}(8T_0-9T_1+T_3), \\
\phi_{N-3} &= \frac{1}{16}(2T_0-T_1-2T_2+T_3), \\
\phi_{N-2} &= \frac{1}{16}(8T_0+9T_1-T_3), \\
\phi_{N-1} &= \frac{1}{16}(2T_0-T_1+2T_2+T_3),
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1)&=a, u'(-1) = b, u(1)=c, u'(1) = d.
The last four bases are for boundary conditions and only used if a, b, c or d are
different from 0. In one dimension :math:`\hat{u}_{N-4}=a`, :math:`\hat{u}_{N-3}=b`,
:math:`\hat{u}_{N-2}=c` and :math:`\hat{u}_{N-1}=d`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 4-tuple of numbers
The values of the 4 boundary conditions at x=(-1, 1).
The two conditions at x=-1 first and then x=1.
With (a, b, c, d) corresponding to
`bc = {'left': {'D': a, 'N': b}, 'right': {'D': c, 'N': d}}`
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0, 0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: sp.simplify(matpow(b, 2, -half, -half, n+2, n, cn) / h(-half, -half, n, 0, cn)),
# 2: sp.simplify(matpow(b, 2, -half, -half, n+2, n+2, cn) / h(-half, -half, n+2, 0, cn)),
# 4: sp.simplify(matpow(b, 2, -half, -half, n+2, n+4, cn) / h(-half, -half, n+4, 0, cn))}
self._stencil = {
0: 1/(2*sp.pi*(n + 1)*(n + 2)),
2: -1/(sp.pi*(n**2 + 4*n + 3)),
4: 1/(2*sp.pi*(n + 2)*(n + 3))
}
[docs]
@staticmethod
def boundary_condition():
return 'Biharmonic'
[docs]
@staticmethod
def short_name():
return 'P2'
[docs]
class Phi3(CompositeBase):
r"""Function space for 6'th order equation
The basis functions :math:`\phi_k` for :math:`k=0, 1, \ldots, N-7` are
.. math::
\phi_k &= \frac{(1-x^2)^3}{h^{(3)}_{k+3}} T^{(3)}_{k+3} \\
h^{(3)}_{k+3} &= \frac{\pi (k+3) \Gamma (k+6)}{2k!} = \int_{-1}^1 T^{(3)}_k T^{(3)}_k (1-x^2)^{2.5} dx.
where :math:`T^{(3)}_k` is the 3rd derivative of :math:`T_k`. The boundary
basis for inhomogeneous boundary conditions is too messy to print, but can
be obtained using :func:`~shenfun.utilities.findbasis.get_bc_basis`. We have
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a, u'(-1)=b, u''(-1)=c, u(1)=d u'(1)=e, u''(1)=f.
The last 6 basis functions are for boundary conditions and only used if there
are nonzero boundary conditions.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 6-tuple of numbers
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0,)*6, domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: sp.simplify(matpow(b, 3, -half, -half, n+3, n, cn) / h(-half, -half, n, 0, cn)),
# 2: sp.simplify(matpow(b, 3, -half, -half, n+3, n+2, cn) / h(-half, -half, n+2, 0, cn)),
# 4: sp.simplify(matpow(b, 3, -half, -half, n+3, n+4, cn) / h(-half, -half, n+4, 0, cn)),
# 6: sp.simplify(matpow(b, 3, -half, -half, n+3, n+6, cn) / h(-half, -half, n+6, 0, cn))}
# Below is the same but faster since already simplified
self._stencil = {
0: 1/(4*sp.pi*(n + 1)*(n + 2)*(n + 3)),
2: -3/(4*sp.pi*(n + 1)*(n + 3)*(n + 4)),
4: 3/(4*sp.pi*(n + 2)*(n + 3)*(n + 5)),
6: -1/(4*sp.pi*(n + 3)*(n + 4)*(n + 5))
}
[docs]
@staticmethod
def boundary_condition():
return '6th order'
[docs]
@staticmethod
def short_name():
return 'P3'
[docs]
class Phi4(CompositeBase):
r"""Function space with 4 Dirichlet and 4 Neumann boundary conditions
The basis functions :math:`\phi_k` for :math:`k=0, 1, \ldots, N-9` are
.. math::
\phi_k &= \frac{(1-x^2)^4}{h^{(4)}_{k+4}} T^{(4)}_{k+4} \\
h^{(4)}_k &= \frac{\pi k \Gamma (k+4)}{2(k-4)!} = \int_{-1}^1 T^{(4)}_k T^{(4)}_k (1-x^2)^{3.5} dx,
where :math:`T^{(4)}_k` is the 4th derivative of :math:`T_k`. The boundary
basis for inhomogeneous boundary conditions is too messy to print, but can
be obtained using :func:`~shenfun.utilities.findbasis.get_bc_basis`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 8-tuple of numbers
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0,)*8, domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: sp.simplify(matpow(b, 4, -half, -half, n+4, n, cn) / h(-half, -half, n, 0, cn)),
# 2: sp.simplify(matpow(b, 4, -half, -half, n+4, n+2, cn) / h(-half, -half, n+2, 0, cn)),
# 4: sp.simplify(matpow(b, 4, -half, -half, n+4, n+4, cn) / h(-half, -half, n+4, 0, cn)),
# 6: sp.simplify(matpow(b, 4, -half, -half, n+4, n+6, cn) / h(-half, -half, n+6, 0, cn)),
# 8: sp.simplify(matpow(b, 4, -half, -half, n+4, n+8, cn) / h(-half, -half, n+8, 0, cn))}
# Below is the same but faster since already simplified
self._stencil = {
0: 1/(8*sp.pi*(n + 1)*(n + 2)*(n + 3)*(n + 4)),
2: -1/(2*sp.pi*(n + 1)*(n + 3)*(n + 4)*(n + 5)),
4: 3/(4*sp.pi*(n + 2)*(n + 3)*(n + 5)*(n + 6)),
6: -1/(2*sp.pi*(n + 3)*(n + 4)*(n + 5)*(n + 7)),
8: 1/(8*sp.pi*(n + 4)*(n + 5)*(n + 6)*(n + 7))
}
[docs]
@staticmethod
def boundary_condition():
return 'Biharmonic*2'
[docs]
@staticmethod
def short_name():
return 'P4'
[docs]
class Phi6(CompositeBase):
r"""Function space for 12th order equation
The basis functions :math:`\phi_k` for :math:`k=0, 1, \ldots, N-13` are
.. math::
\phi_k &= \frac{(1-x^2)^6}{h^{(6)}_{k+6}} T^{(6)}_{k+6} \\
h^{(6)}_k &= \frac{\pi k (k+5)!}{2(k-6)!} = \int_{-1}^1 T^{(6)}_k T^{(6)}_k (1-x^2)^{5.5} dx,
where :math:`T^{(6)}_k` is the 6th derivative of :math:`T_k`. The boundary
basis for inhomogeneous boundary conditions is too messy to print, but can
be obtained using :func:`~shenfun.utilities.findbasis.get_bc_basis`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 12-tuple of numbers
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0,)*12, domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: sp.simplify(matpow(b, 6, -half, -half, n+6, n, cn) / h(-half, -half, n, 0, cn)),
# 2: sp.simplify(matpow(b, 6, -half, -half, n+6, n+2, cn) / h(-half, -half, n+2, 0, cn)),
# 4: sp.simplify(matpow(b, 6, -half, -half, n+6, n+4, cn) / h(-half, -half, n+4, 0, cn)),
# 6: sp.simplify(matpow(b, 6, -half, -half, n+6, n+6, cn) / h(-half, -half, n+6, 0, cn)),
# 8: sp.simplify(matpow(b, 6, -half, -half, n+6, n+8, cn) / h(-half, -half, n+8, 0, cn)),
# 10: sp.simplify(matpow(b, 6, -half, -half, n+6, n+10, cn) / h(-half, -half, n+10, 0, cn)),
# 12: sp.simplify(matpow(b, 6, -half, -half, n+6, n+12, cn) / h(-half, -half, n+12, 0, cn))}
# Below is the same but faster since already simplified
self._stencil = {
0: 1/(32*sp.pi*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)),
2: -3/(16*sp.pi*(n + 1)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)),
4: 15/(32*sp.pi*(n + 2)*(n + 3)*(n + 5)*(n + 6)*(n + 7)*(n + 8)),
6: -5/(8*sp.pi*(n + 3)*(n + 4)*(n + 5)*(n + 7)*(n + 8)*(n + 9)),
8: 15/(32*sp.pi*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(n + 9)*(n + 10)),
10: -3/(16*sp.pi*(n + 5)*(n + 6)*(n + 7)*(n + 8)*(n + 9)*(n + 11)),
12: 1/(32*sp.pi*(n + 6)*(n + 7)*(n + 8)*(n + 9)*(n + 10)*(n + 11))
}
[docs]
@staticmethod
def boundary_condition():
return '12th order'
[docs]
@staticmethod
def short_name():
return 'P6'
[docs]
class UpperDirichlet(CompositeBase):
r"""Function space with single Dirichlet on upper edge
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} - T_{k+1}, \, k=0, 1, \ldots, N-2, \\
\phi_{N-1} &= T_0,
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(1) &= a.
The last basis function is for boundary condition and only used if a is
different from 0. In one dimension :math:`\hat{u}_{N-1}=a`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of (None, number), optional
The number is the boundary condition value
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(None, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {0: 1, 1: -1}
[docs]
@staticmethod
def boundary_condition():
return 'UpperDirichlet'
[docs]
@staticmethod
def short_name():
return 'UD'
[docs]
class LowerDirichlet(CompositeBase):
r"""Function space with single Dirichlet boundary condition
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} + T_{k+1}, \, k=0, 1, \ldots, N-2, \\
\phi_{N-1} &= T_0,
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a.
The last basis funciton is for boundary condition and only used if a is
different from 0. In one dimension :math:`\hat{u}_{N-1}=a`.
Parameters
----------
N : int
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : tuple of (number, None)
Boundary conditions at edges of domain.
domain : 2-tuple of numbers, optional
The computational domain
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, None), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {0: 1, 1: 1}
[docs]
@staticmethod
def boundary_condition():
return 'LowerDirichlet'
[docs]
@staticmethod
def short_name():
return 'LD'
[docs]
class ShenBiPolar(CompositeBase):
"""Function space for the Biharmonic equation in polar coordinates
u(-1)=a, u(1)=c, u'(-1)=b and u'(1)=d
Parameters
----------
N : int
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 4-tuple of numbers
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0, 0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
[docs]
@staticmethod
def boundary_condition():
return 'Biharmonic'
[docs]
@staticmethod
def short_name():
return 'SP'
[docs]
def stencil_matrix(self, N=None):
N = self.N if N is None else N
k = np.arange(N)
d = (k+1)*3/8
d[0] = 0.5
d[1] = 0.5
d[-4:] = 0
dm2 = -(k[2:]+1)/8
dm2[-4:] = 0
dp2 = -(k[:-2]+1)*3/8
dp2[-2:] = 0
dp4 = (k[:-4]+1)/8
return SparseMatrix({-2: dm2, 0: d, 2: dp2, 4: dp4}, (N, N))
[docs]
def sympy_stencil(self, i=sp.Symbol('i', integer=True), j=sp.Symbol('j', integer=True)):
return RuntimeError, "Not possible for current basis"
[docs]
class DirichletNeumann(CompositeBase):
r"""Function space for mixed Dirichlet/Neumann boundary conditions
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} + \frac{4(k+1)}{2k^2+6k+5}T_{k+1} - \frac{2k^2+2k+1}{2k^2+6k+5}T_{k+2}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= T_0, \\
\phi_{N-1} &= T_0+T_1,
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a, u'(1)=b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of numbers
Boundary condition values at x=-1 and x=1
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'left': {'D': bc[0]}, 'right': {'N': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {
0: 1,
1: 4*(n + 1)/(2*n**2 + 6*n + 5),
2: -(2*n**2 + 2*n + 1)/(2*n**2 + 6*n + 5)
}
[docs]
@staticmethod
def boundary_condition():
return 'DirichletNeumann'
[docs]
@staticmethod
def short_name():
return 'DN'
[docs]
class NeumannDirichlet(CompositeBase):
r"""Function space for mixed Neumann/Dirichlet boundary conditions
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} - \frac{4(k+1)}{2k^2+6k+5}T_{k+1} - \frac{2k^2+2k+1}{2k^2+6k+5}T_{k+2}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= -T_0+T_1, \\
\phi_{N-1} &= T_0,
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u'(-1) &= a, u(1)=b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of numbers
Boundary condition values at x=-1 and x=1
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'left': {'N': bc[0]}, 'right': {'D': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {
0: 1,
1: -(4*n + 4)/(2*n**2 + 6*n + 5),
2: -(2*n**2 + 2*n + 1)/(2*n**2 + 6*n + 5)
}
[docs]
@staticmethod
def boundary_condition():
return 'NeumannDirichlet'
[docs]
@staticmethod
def short_name():
return 'ND'
[docs]
class UpperDirichletNeumann(CompositeBase):
r"""Function space for both Dirichlet and Neumann boundary conditions
on the right hand side.
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} - \frac{4(k+1)}{2k+3}T_{k+1} + \frac{2k+1}{2k+3}T_{k+2}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= T_0, \\
\phi_{N-1} &= -T_0+T_1,
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(1) &= a, u'(1)=b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of numbers
Boundary condition values at the right edge of domain
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'right': {'D': bc[0], 'N': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {
0: 1,
1: -(4*n + 4)/(2*n + 3),
2: (2*n + 1)/(2*n + 3)
}
[docs]
@staticmethod
def boundary_condition():
return 'UpperDirichletNeumann'
[docs]
@staticmethod
def short_name():
return 'US'
[docs]
class LowerDirichletNeumann(CompositeBase):
r"""Function space for both Dirichlet and Neumann boundary conditions
on the left hand side
The basis :math:`\{\phi_k\}_{k=0}^{N-1}` is
.. math::
\phi_k &= T_{k} + \frac{4(k+1)}{2k+3}T_{k+1} + \frac{2k+1}{2k+3}T_{k+2}, \, k=0, 1, \ldots, N-3, \\
\phi_{N-2} &= T_0, \\
\phi_{N-1} &= T_0+T_1,
such that
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a, u'(-1)=b.
The last two bases are for boundary conditions and only used if a or b are
different from 0. In one dimension :math:`\hat{u}_{N-2}=a` and
:math:`\hat{u}_{N-1}=b`.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 2-tuple of numbers
Boundary condition values at the left edge of domain
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0, 0), domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
if isinstance(bc, (tuple, list)):
bc = BoundaryConditions({'left': {'D': bc[0], 'N': bc[1]}}, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
self._stencil = {
0: 1,
1: 4*(n + 1)/(2*n + 3),
2: (2*n + 1)/(2*n + 3)
}
[docs]
@staticmethod
def boundary_condition():
return 'LowerDirichletNeumann'
[docs]
@staticmethod
def short_name():
return 'LS'
[docs]
class Compact3(CompositeBase):
r"""Function space for 6'th order equation
The basis functions :math:`\phi_k` for :math:`k=0, 1, \ldots, N-7` are
.. math::
\phi_k &= \frac{h_k}{b^{(3)}_{k+3,k}}\frac{(1-x^2)^3}{h^{(3)}_{k+3}} T^{(3)}_{k+3} \\
h^{(3)}_{k+3} &= \frac{\pi (k+3) \Gamma (k+6)}{2k!} = \int_{-1}^1 T^{(3)}_k T^{(3)}_k (1-x^2)^{2.5} dx.
where :math:`T^{(3)}_k` is the 3rd derivative of :math:`T_k`.
This is :class:`.Phi3` scaled such that the main diagonal of the stencil
matrix is unity.
The boundary basis for inhomogeneous boundary conditions is too messy to
print, but can be obtained using :func:`~shenfun.utilities.findbasis.get_bc_basis`.
We have
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a, u'(-1)=b, u''(-1)=c, u(1)=d u'(1)=e, u''(1)=f.
The last 6 basis functions are for boundary conditions and only used if there
are nonzero boundary conditions.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 6-tuple of numbers
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0,)*6, domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: 1,
# 2: sp.simplify(matpow(b, 3, -half, -half, n+3, n+2, cn) / matpow(b, 3, -half, -half, n+3, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+2, 0, cn)),
# 4: sp.simplify(matpow(b, 3, -half, -half, n+3, n+4, cn) / matpow(b, 3, -half, -half, n+3, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+4, 0, cn)),
# 6: sp.simplify(matpow(b, 3, -half, -half, n+3, n+6, cn) / matpow(b, 3, -half, -half, n+3, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+6, 0, cn))}
# Below is the same but faster since already simplified
# Can also use findbasis.get_stencil_matrix
self._stencil = {
0: 1,
2: -(3*n + 6)/(n + 4),
4: 3*(n + 1)/(n + 5),
6: -(n + 1)*(n + 2)/((n + 4)*(n + 5))
}
[docs]
@staticmethod
def boundary_condition():
return '6th order'
[docs]
@staticmethod
def short_name():
return 'C3'
[docs]
class Compact4(CompositeBase):
r"""Function space for 8'th order equation
The basis functions :math:`\phi_k` for :math:`k=0, 1, \ldots, N-9` are
.. math::
\phi_k &= \frac{h_k}{b^{(4)}_{k+4,k}}\frac{(1-x^2)^4}{h^{(4)}_{k+4}} T^{(4)}_{k+4} \\
h^{(4)}_{k+4} &= \int_{-1}^1 T^{(4)}_k T^{(4)}_k (1-x^2)^{3.5} dx.
where :math:`T^{(4)}_k` is the 4rd derivative of :math:`T_k`.
This is :class:`.Phi4` scaled such that the main diagonal of the stencil
matrix is unity.
The boundary basis for inhomogeneous boundary conditions is too messy to
print, but can be obtained using :func:`~shenfun.utilities.findbasis.get_bc_basis`.
We have
.. math::
u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\
u(-1) &= a, u'(-1)=b, u''(-1)=c, u'''(-1)=d, u(1)=e u'(1)=f, u''(1)=g, u'''(1)=h.
The last 8 basis functions are for boundary conditions and only used if there
are nonzero boundary conditions.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : 8-tuple of numbers
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
"""
def __init__(self, N, quad="GC", bc=(0,)*8, domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
#self._stencil = {
# 0: 1,
# 2: sp.simplify(matpow(b, 4, -half, -half, n+4, n+2, cn) / matpow(b, 4, -half, -half, n+4, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+2, 0, cn)),
# 4: sp.simplify(matpow(b, 4, -half, -half, n+4, n+4, cn) / matpow(b, 4, -half, -half, n+4, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+4, 0, cn)),
# 6: sp.simplify(matpow(b, 4, -half, -half, n+4, n+6, cn) / matpow(b, 4, -half, -half, n+4, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+6, 0, cn)),
# 8: sp.simplify(matpow(b, 4, -half, -half, n+4, n+8, cn) / matpow(b, 4, -half, -half, n+4, n, cn) * h(-half, -half, n, 0, cn) / h(-half, -half, n+8, 0, cn))}
# Below is the same but faster since already simplified
# Can also use findbasis.get_stencil_matrix
self._stencil = {
0: 1,
2: -(4*n + 8)/(n + 5),
4: 6*(n + 1)*(n + 4)/((n + 5)*(n + 6)),
6: -4*(n + 1)*(n + 2)/((n + 5)*(n + 7)),
8: (n + 1)*(n + 2)*(n + 3)/((n + 5)*(n + 6)*(n + 7))
}
#self._stencil = {
# 0: 1/(8*sp.pi*(n + 1)*(n + 2)),
# 2: -1/(2*sp.pi*(n + 1)*(n + 5)),
# 4: 3*(n + 4)/(4*sp.pi*(n + 2)*(n + 5)*(n + 6)),
# 6: -1/(2*sp.pi*(n + 5)*(n + 7)),
# 8: (n + 3)/(8*sp.pi*(n + 5)*(n + 6)*(n + 7))
#}
#self._stencil = {
# 0: 1/(8*sp.pi*(n + 1)*(n + 2)*(n + 3)*(n + 4)),
# 2: -1/(2*sp.pi*(n + 1)*(n + 3)*(n + 4)*(n + 5)),
# 4: 3/(4*sp.pi*(n + 2)*(n + 3)*(n + 5)*(n + 6)),
# 6: -1/(2*sp.pi*(n + 3)*(n + 4)*(n + 5)*(n + 7)),
# 8: 1/(8*sp.pi*(n + 4)*(n + 5)*(n + 6)*(n + 7))
#}
[docs]
@staticmethod
def boundary_condition():
return '8th order'
[docs]
@staticmethod
def short_name():
return 'C4'
[docs]
class Generic(CompositeBase):
r"""Function space for space with any boundary conditions
Any combination of Dirichlet and Neumann is possible.
Parameters
----------
N : int, optional
Number of quadrature points
quad : str, optional
Type of quadrature
- GL - Chebyshev-Gauss-Lobatto
- GC - Chebyshev-Gauss
bc : dict, optional
The dictionary must have keys 'left' and 'right', to describe boundary
conditions on the left and right boundaries, and a list of 2-tuples to
specify the condition. Specify Dirichlet on both ends with
{'left': {'D': a}, 'right': {'D': b}}
for some values `a` and `b`, that will be neglected in the current
function. Specify mixed Neumann and Dirichlet as
{'left': {'N': a}, 'right': {'N': b}}
For both conditions on the right do
{'right': {'N': a, 'D': b}}
Any combination should be possible, and it should also be possible to
use second derivatives `N2`. See :class:`~shenfun.spectralbase.BoundaryConditions`.
domain : 2-tuple of numbers, optional
The computational domain
dtype : data-type, optional
Type of input data in real physical space. Will be overloaded when
basis is part of a :class:`.TensorProductSpace`.
padding_factor : float, optional
Factor for padding backward transforms.
dealias_direct : bool, optional
Set upper 1/3 of coefficients to zero before backward transform
coordinates: 2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional
Map for curvilinear coordinatesystem, and parameters to :class:`~shenfun.coordinates.Coordinates`
Note
----
A test function is always using homogeneous boundary conditions.
"""
def __init__(self, N, quad="GC", bc={}, domain=(-1, 1), dtype=float,
padding_factor=1, dealias_direct=False, coordinates=None, **kw):
from shenfun.utilities.findbasis import get_stencil_matrix
self._stencil = get_stencil_matrix(bc, 'chebyshev', -half, -half, cn)
if not isinstance(bc, BoundaryConditions):
bc = BoundaryConditions(bc, domain=domain)
CompositeBase.__init__(self, N, quad=quad, domain=domain, dtype=dtype, bc=bc,
padding_factor=padding_factor, dealias_direct=dealias_direct,
coordinates=coordinates)
[docs]
@staticmethod
def boundary_condition():
return 'Generic'
[docs]
@staticmethod
def short_name():
return 'GT'