r"""
This module implements differential operators on spherical grids
.. autosummary::
:nosignatures:
make_laplace
make_gradient
make_divergence
make_vector_gradient
make_tensor_divergence
make_poisson_solver
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import Literal
import numpy as np
from ...tools.docstrings import fill_in_docstring
from ...tools.numba import jit
from ...tools.typing import OperatorType
from ..boundaries import Boundaries
from ..spherical import SphericalSymGrid
from .common import make_general_poisson_solver
[docs]
@SphericalSymGrid.register_operator("laplace", rank_in=0, rank_out=0)
@fill_in_docstring
def make_laplace(grid: SphericalSymGrid, conservative: bool = True) -> OperatorType:
"""make a discretized laplace operator for a spherical grid
{DESCR_SPHERICAL_GRID}
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
conservative (bool):
Flag indicating whether the laplace operator should be conservative (which
results in slightly slower computations). Conservative operators ensure mass
conservation.
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
dr = grid.discretization[0]
rs = grid.axes_coords[0]
r_min, r_max = grid.axes_bounds[0]
if conservative:
# create a conservative spherical laplace operator
rl = rs - dr / 2 # inner radii of spherical shells
rh = rs + dr / 2 # outer radii
assert np.isclose(rl[0], r_min) and np.isclose(rh[-1], r_max)
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
factor_l = rl**2 / (dr * volumes)
factor_h = rh**2 / (dr * volumes)
@jit
def laplace(arr: np.ndarray, out: np.ndarray) -> None:
"""apply laplace operator to array `arr`"""
for i in range(1, dim_r + 1): # iterate inner radial points
term_h = factor_h[i - 1] * (arr[i + 1] - arr[i])
term_l = factor_l[i - 1] * (arr[i] - arr[i - 1])
out[i - 1] = term_h - term_l
else: # create an operator that is not conservative
dr2 = 1 / dr**2
@jit
def laplace(arr: np.ndarray, out: np.ndarray) -> None:
"""apply laplace operator to array `arr`"""
for i in range(1, dim_r + 1): # iterate inner radial points
diff_2 = (arr[i + 1] - 2 * arr[i] + arr[i - 1]) * dr2
diff_1 = (arr[i + 1] - arr[i - 1]) / (rs[i - 1] * dr)
out[i - 1] = diff_2 + diff_1
return laplace # type: ignore
[docs]
@SphericalSymGrid.register_operator("gradient", rank_in=0, rank_out=1)
@fill_in_docstring
def make_gradient(grid: SphericalSymGrid) -> OperatorType:
"""make a discretized gradient operator for a spherical grid
{DESCR_SPHERICAL_GRID}
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
dr = grid.discretization[0]
scale_r = 1 / (2 * dr)
@jit
def gradient(arr: np.ndarray, out: np.ndarray) -> None:
"""apply gradient operator to array `arr`"""
for i in range(1, dim_r + 1): # iterate inner radial points
out[0, i - 1] = (arr[i + 1] - arr[i - 1]) * scale_r
out[1, i - 1] = out[2, i - 1] = 0 # no angular dependence by definition
return gradient # type: ignore
[docs]
@SphericalSymGrid.register_operator("gradient_squared", rank_in=0, rank_out=0)
@fill_in_docstring
def make_gradient_squared(grid: SphericalSymGrid, central: bool = True) -> OperatorType:
"""make a discretized gradient squared operator for a spherical grid
{DESCR_SPHERICAL_GRID}
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
central (bool):
Whether a central difference approximation is used for the gradient
operator. If this is False, the squared gradient is calculated as
the mean of the squared values of the forward and backward
derivatives.
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
dr = grid.discretization[0]
if central:
# use central differences
scale = 0.25 / dr**2
@jit
def gradient_squared(arr: np.ndarray, out: np.ndarray) -> None:
"""apply squared gradient operator to array `arr`"""
for i in range(1, dim_r + 1): # iterate inner radial points
out[i - 1] = (arr[i + 1] - arr[i - 1]) ** 2 * scale
else:
# use forward and backward differences
scale = 0.5 / dr**2
@jit
def gradient_squared(arr: np.ndarray, out: np.ndarray) -> None:
"""apply squared gradient operator to array `arr`"""
for i in range(1, dim_r + 1): # iterate inner radial points
term = (arr[i + 1] - arr[i]) ** 2 + (arr[i] - arr[i - 1]) ** 2
out[i - 1] = term * scale
return gradient_squared # type: ignore
[docs]
@SphericalSymGrid.register_operator("divergence", rank_in=1, rank_out=0)
@fill_in_docstring
def make_divergence(
grid: SphericalSymGrid, safe: bool = True, conservative: bool = True
) -> OperatorType:
"""make a discretized divergence operator for a spherical grid
{DESCR_SPHERICAL_GRID}
Warning:
This operator ignores the θ-component of the field when calculating the
divergence. This is because the resulting scalar field could not be expressed
on a :class:`~pde.grids.spherical_sym.SphericalSymGrid`.
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
safe (bool):
Add extra checks for the validity of the input
conservative (bool):
Flag indicating whether the operator should be conservative (which results
in slightly slower computations). Conservative operators ensure mass
conservation.
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
dr = grid.discretization[0]
rs = grid.axes_coords[0]
if conservative:
# implement conservative version of the divergence operator
rl = rs - dr / 2 # inner radii of spherical shells
rh = rs + dr / 2 # outer radii
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
factor_l = rl**2 / (2 * volumes)
factor_h = rh**2 / (2 * volumes)
@jit
def divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""apply divergence operator to array `arr`"""
if safe:
# the θ-component of the vector field are required to be zero. If this
# was not the case the scale field resulting from the divergence would
# contain components that cannot be expressed in spherically symmetric
# coordinates
assert np.all(arr[1, 1:-1] == 0)
arr_r = arr[0, :]
for i in range(1, dim_r + 1): # iterate radial points
term_h = factor_h[i - 1] * (arr_r[i] + arr_r[i + 1])
term_l = factor_l[i - 1] * (arr_r[i - 1] + arr_r[i])
out[i - 1] = term_h - term_l
else:
# implement naive divergence operator
scale_r = 1 / (2 * dr)
factors = 2 / rs # factors that need to be multiplied below
@jit
def divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""apply divergence operator to array `arr`"""
if safe:
# the θ-component of the vector field are required to be zero. If this
# was not the case the scale field resulting from the divergence would
# contain components that cannot be expressed in spherically symmetric
# coordinates
assert np.all(arr[1, 1:-1] == 0)
arr_r = arr[0, :]
for i in range(1, dim_r + 1): # iterate radial points
diff_r = (arr_r[i + 1] - arr_r[i - 1]) * scale_r
out[i - 1] = diff_r + factors[i - 1] * arr_r[i]
return divergence # type: ignore
[docs]
@SphericalSymGrid.register_operator("vector_gradient", rank_in=1, rank_out=2)
@fill_in_docstring
def make_vector_gradient(grid: SphericalSymGrid, safe: bool = True) -> OperatorType:
"""make a discretized vector gradient operator for a spherical grid
Warning:
This operator ignores the two angular components of the field when calculating
the gradient. This is because the resulting field could not be expressed on a
:class:`~pde.grids.spherical_sym.SphericalSymGrid`.
{DESCR_SPHERICAL_GRID}
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
safe (bool):
Add extra checks for the validity of the input
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
rs = grid.axes_coords[0]
dr = grid.discretization[0]
scale_r = 1 / (2 * dr)
@jit
def vector_gradient(arr: np.ndarray, out: np.ndarray) -> None:
"""apply vector gradient operator to array `arr`"""
if safe:
# the θ- and φ-components are required to be zero. If this was not the case
# the tensor field resulting from the gradient would contain components that
# cannot be expressed in spherically symmetric coordinates
assert np.all(arr[1:, 1:-1] == 0)
# assign aliases
arr_r = arr[0, :]
out_rr, out_rθ, out_rφ = out[0, 0, :], out[0, 1, :], out[0, 2, :]
out_θr, out_θθ, out_θφ = out[1, 0, :], out[1, 1, :], out[1, 2, :]
out_φr, out_φθ, out_φφ = out[2, 0, :], out[2, 1, :], out[2, 2, :]
# set all components to zero that are not affected
out_rθ[:] = 0
out_rφ[:] = 0
out_θr[:] = 0
out_θφ[:] = 0
out_φr[:] = 0
out_φθ[:] = 0
# inner radial boundary condition
for i in range(1, dim_r + 1): # iterate radial points
out_rr[i - 1] = (arr_r[i + 1] - arr_r[i - 1]) * scale_r
out_θθ[i - 1] = arr_r[i] / rs[i - 1]
out_φφ[i - 1] = arr_r[i] / rs[i - 1]
return vector_gradient # type: ignore
[docs]
@SphericalSymGrid.register_operator("tensor_divergence", rank_in=2, rank_out=1)
@fill_in_docstring
def make_tensor_divergence(
grid: SphericalSymGrid, safe: bool = True, conservative: bool = False
) -> OperatorType:
"""make a discretized tensor divergence operator for a spherical grid
{DESCR_SPHERICAL_GRID}
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
safe (bool):
Add extra checks for the validity of the input
conservative (bool):
Flag indicating whether the operator should be conservative (which results
in slightly slower computations). Conservative operators ensure mass
conservation.
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
rs = grid.axes_coords[0]
dr = grid.discretization[0]
if conservative:
# conservative implementation of the tensor divergence
rl = rs - dr / 2 # inner radii of spherical shells
rh = rs + dr / 2 # outer radii
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
factor_l = rl**2 / (2 * volumes)
factor_h = rh**2 / (2 * volumes)
area_factor = (rh**2 - rl**2) / volumes
@jit
def tensor_divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""apply tensor divergence operator to array `arr`"""
# assign aliases
arr_rr, arr_rθ, arr_rφ = arr[0, 0, :], arr[0, 1, :], arr[0, 2, :]
arr_θr, arr_θθ, arr_θφ = arr[1, 0, :], arr[1, 1, :], arr[1, 2, :]
arr_φr, arr_φθ, arr_φφ = arr[2, 0, :], arr[2, 1, :], arr[2, 2, :]
out_r, out_θ, out_φ = out[0, :], out[1, :], out[2, :]
# check inputs
if safe:
# the following conditions need to be met. Otherwise, the vector resulting
# from the divergence might contain components that cannot be expressed in
# spherically symmetric coordinates
assert np.all(arr_φr[1:-1] == 0)
assert np.all(arr_rφ[1:-1] == 0)
assert np.all(arr_rθ[1:-1] == 0)
assert np.all(arr_θr[1:-1] == 0)
assert np.all(arr_θθ[1:-1] == arr_φφ[1:-1])
assert np.all(arr_φθ[1:-1] == -arr_θφ[1:-1])
# iterate over inner points
for i in range(1, dim_r + 1):
term_r_h = factor_h[i - 1] * (arr_rr[i] + arr_rr[i + 1])
term_r_l = factor_l[i - 1] * (arr_rr[i - 1] + arr_rr[i])
out_r[i - 1] = term_r_h - term_r_l - area_factor[i - 1] * arr_φφ[i]
out_θ[i - 1] = 0
out_φ[i - 1] = 0
else:
# naive implementation of the tensor divergence
scale_r = 1 / (2 * dr)
@jit
def tensor_divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""apply tensor divergence operator to array `arr`"""
# assign aliases
arr_rr, arr_rθ, arr_rφ = arr[0, 0, :], arr[0, 1, :], arr[0, 2, :]
arr_θr, arr_θθ, arr_θφ = arr[1, 0, :], arr[1, 1, :], arr[1, 2, :]
arr_φr, arr_φθ, arr_φφ = arr[2, 0, :], arr[2, 1, :], arr[2, 2, :]
out_r, out_θ, out_φ = out[0, :], out[1, :], out[2, :]
# check inputs
if safe:
# the following conditions need to be met. Otherwise, the vector resulting
# from the divergence might contain components that cannot be expressed in
# spherically symmetric coordinates
assert np.all(arr_rθ[1:-1] == 0)
assert np.all(arr_θθ[1:-1] == arr_φφ[1:-1])
assert np.all(arr_φθ[1:-1] == -arr_θφ[1:-1])
# iterate over inner points
for i in range(1, dim_r + 1):
deriv_r = (arr_rr[i + 1] - arr_rr[i - 1]) * scale_r
out_r[i - 1] = deriv_r + 2 * (arr_rr[i] - arr_φφ[i]) / rs[i - 1]
deriv_r = (arr_θr[i + 1] - arr_θr[i - 1]) * scale_r
out_θ[i - 1] = deriv_r + 2 * arr_θr[i] / rs[i - 1]
deriv_r = (arr_φr[i + 1] - arr_φr[i - 1]) * scale_r
out_φ[i - 1] = deriv_r + (2 * arr_φr[i] + arr_rφ[i]) / rs[i - 1]
return tensor_divergence # type: ignore
[docs]
@SphericalSymGrid.register_operator("tensor_double_divergence", rank_in=2, rank_out=0)
@fill_in_docstring
def make_tensor_double_divergence(
grid: SphericalSymGrid, safe: bool = True, conservative: bool = True
) -> OperatorType:
"""make a discretized tensor double divergence operator for a spherical grid
{DESCR_SPHERICAL_GRID}
Args:
grid (:class:`~pde.grids.spherical.SphericalSymGrid`):
The polar grid for which this operator will be defined
safe (bool):
Add extra checks for the validity of the input
conservative (bool):
Flag indicating whether the operator should be conservative (which results
in slightly slower computations). Conservative operators ensure mass
conservation.
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, SphericalSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
rs = grid.axes_coords[0]
dr = grid.discretization[0]
if conservative:
# create a conservative double divergence laplace operator
rl = rs - dr / 2 # inner radii of spherical shells
rh = rs + dr / 2 # outer radii
r_min, r_max = grid.axes_bounds[0]
assert np.isclose(rl[0], r_min) and np.isclose(rh[-1], r_max)
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
factor_l = rl / volumes
factor_h = rh / volumes
factor2_l = rl**2 / (dr * volumes)
factor2_h = rh**2 / (dr * volumes)
@jit
def tensor_double_divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""apply double divergence operator to tensor array `arr`"""
# assign aliases
arr_rr, arr_rθ, ______ = arr[0, 0, :], arr[0, 1, :], arr[0, 2, :]
arr_θr, arr_θθ, ______ = arr[1, 0, :], arr[1, 1, :], arr[1, 2, :]
______, ______, arr_φφ = arr[2, 0, :], arr[2, 1, :], arr[2, 2, :]
# check inputs
if safe:
# the following conditions need to be met. Otherwise, the vector
# resulting from the divergence might contain components that cannot be
# expressed in spherically symmetric coordinates
assert np.all(arr_rθ[1:-1] == -arr_θr[1:-1])
assert np.all(arr_θθ[1:-1] == arr_φφ[1:-1])
# iterate over inner points
for i in range(1, dim_r + 1):
# radial part
arr_rr_h = arr_rr[i] + arr_rr[i + 1]
arr_rr_l = arr_rr[i - 1] + arr_rr[i]
arr_rr_dr_h = arr_rr[i + 1] - arr_rr[i]
arr_rr_dr_l = arr_rr[i] - arr_rr[i - 1]
div2_rr_h = factor_h[i - 1] * arr_rr_h + factor2_h[i - 1] * arr_rr_dr_h
div2_rr_l = factor_l[i - 1] * arr_rr_l + factor2_l[i - 1] * arr_rr_dr_l
div2_rr = div2_rr_h - div2_rr_l
# angular part
arr_φφ_h = arr_φφ[i] + arr_φφ[i + 1]
arr_φφ_l = arr_φφ[i - 1] + arr_φφ[i]
div2_φφ = factor_h[i - 1] * arr_φφ_h - factor_l[i - 1] * arr_φφ_l
out[i - 1] = div2_rr - div2_φφ
else:
# naive, non-conservative implementation of the double divergence
dr2 = 1 / dr**2
scale_r = 1 / (2 * dr)
@jit
def tensor_double_divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""apply double divergence operator to tensor array `arr`"""
# assign aliases
arr_rr, arr_rθ, ______ = arr[0, 0, :], arr[0, 1, :], arr[0, 2, :]
arr_θr, arr_θθ, ______ = arr[1, 0, :], arr[1, 1, :], arr[1, 2, :]
______, ______, arr_φφ = arr[2, 0, :], arr[2, 1, :], arr[2, 2, :]
# check inputs
if safe:
# the following conditions need to be met. Otherwise, the vector
# resulting from the divergence might contain components that cannot be
# expressed in spherically symmetric coordinates
assert np.all(arr_rθ[1:-1] == -arr_θr[1:-1])
assert np.all(arr_θθ[1:-1] == arr_φφ[1:-1])
# iterate over inner points
for i in range(1, dim_r + 1):
# first derivatives
arr_rr_dr = (arr_rr[i + 1] - arr_rr[i - 1]) * scale_r
arr_φφ_dr = (arr_φφ[i + 1] - arr_φφ[i - 1]) * scale_r
# second derivative
term1 = (arr_rr[i + 1] - arr_rr[i - 1]) / (rs[i - 1] * dr)
term2 = (arr_rr[i + 1] - 2 * arr_rr[i] + arr_rr[i - 1]) * dr2
lap_rr = term1 + term2
enum = (arr_rr[i] - arr_φφ[i]) / rs[i - 1] + arr_rr_dr - arr_φφ_dr
out[i - 1] = lap_rr + 2 * enum / rs[i - 1]
return tensor_double_divergence # type: ignore
@fill_in_docstring
def _get_laplace_matrix(bcs: Boundaries) -> tuple[np.ndarray, np.ndarray]:
"""get sparse matrix for laplace operator on a polar grid
Args:
bcs (:class:`~pde.grids.boundaries.axes.Boundaries`):
{ARG_BOUNDARIES_INSTANCE}
Returns:
tuple: A sparse matrix and a sparse vector that can be used to evaluate
the discretized laplacian
"""
from scipy import sparse
assert isinstance(bcs.grid, SphericalSymGrid)
bcs.check_value_rank(0)
# calculate preliminary quantities
dim_r = bcs.grid.shape[0]
dr = bcs.grid.discretization[0]
rs = bcs.grid.axes_coords[0]
r_min, r_max = bcs.grid.axes_bounds[0]
# create a conservative spherical laplace operator
rl = r_min + dr * np.arange(dim_r) # inner radii of spherical shells
rh = rl + dr # outer radii
assert np.isclose(rh[-1], r_max)
volumes = (rh**3 - rl**3) / 3 # volume of the spherical shells
factor_l = (rs - 0.5 * dr) ** 2 / (dr * volumes)
factor_h = (rs + 0.5 * dr) ** 2 / (dr * volumes)
matrix = sparse.dok_matrix((dim_r, dim_r))
vector = sparse.dok_matrix((dim_r, 1))
for i in range(dim_r):
matrix[i, i] += -factor_l[i] - factor_h[i]
if i == 0:
if r_min == 0:
matrix[i, i + 1] = factor_l[i]
else:
const, entries = bcs[0].get_sparse_matrix_data((-1,))
vector[i] += const * factor_l[i]
for k, v in entries.items():
matrix[i, k] += v * factor_l[i]
else:
matrix[i, i - 1] = factor_l[i]
if i == dim_r - 1:
const, entries = bcs[0].get_sparse_matrix_data((dim_r,))
vector[i] += const * factor_h[i]
for k, v in entries.items():
matrix[i, k] += v * factor_h[i]
else:
matrix[i, i + 1] = factor_h[i]
return matrix, vector
[docs]
@SphericalSymGrid.register_operator("poisson_solver", rank_in=0, rank_out=0)
@fill_in_docstring
def make_poisson_solver(
bcs: Boundaries, method: Literal["auto", "scipy"] = "auto"
) -> OperatorType:
"""make a operator that solves Poisson's equation
{DESCR_POLAR_GRID}
Args:
bcs (:class:`~pde.grids.boundaries.axes.Boundaries`):
{ARG_BOUNDARIES_INSTANCE}
method (str):
The chosen method for implementing the operator
Returns:
A function that can be applied to an array of values
"""
matrix, vector = _get_laplace_matrix(bcs)
return make_general_poisson_solver(matrix, vector, method)