r"""This module implements differential operators on polar 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 PolarSymGrid
from .common import make_general_poisson_solver
[docs]
@PolarSymGrid.register_operator("laplace", rank_in=0, rank_out=0)
@fill_in_docstring
def make_laplace(grid: PolarSymGrid) -> OperatorType:
"""Make a discretized laplace operator for a polar grid.
{DESCR_POLAR_GRID}
Args:
grid (:class:`~pde.grids.spherical.PolarSymGrid`):
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, PolarSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
dr = grid.discretization[0]
factor_r = 1 / (2 * grid.axes_coords[0] * dr)
dr_2 = 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
out[i - 1] = (arr[i + 1] - 2 * arr[i] + arr[i - 1]) * dr_2
out[i - 1] += (arr[i + 1] - arr[i - 1]) * factor_r[i - 1]
return laplace # type: ignore
[docs]
@PolarSymGrid.register_operator("gradient", rank_in=0, rank_out=1)
@fill_in_docstring
def make_gradient(
grid: PolarSymGrid, *, method: Literal["central", "forward", "backward"] = "central"
) -> OperatorType:
"""Make a discretized gradient operator for a polar grid.
{DESCR_POLAR_GRID}
Args:
grid (:class:`~pde.grids.spherical.PolarSymGrid`):
The polar grid for which this operator will be defined
method (str):
The method for calculating the derivative. Possible values are 'central',
'forward', and 'backward'.
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, PolarSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
if method == "central":
scale_r = 0.5 / grid.discretization[0]
elif method in {"forward", "backward"}:
scale_r = 1 / grid.discretization[0]
else:
raise ValueError(f"Unknown derivative type `{method}`")
@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
if method == "central":
out[0, i - 1] = (arr[i + 1] - arr[i - 1]) * scale_r
elif method == "forward":
out[0, i - 1] = (arr[i + 1] - arr[i]) * scale_r
elif method == "backward":
out[0, i - 1] = (arr[i] - arr[i - 1]) * scale_r
out[1, i - 1] = 0 # no angular dependence by definition
return gradient # type: ignore
[docs]
@PolarSymGrid.register_operator("gradient_squared", rank_in=0, rank_out=0)
@fill_in_docstring
def make_gradient_squared(grid: PolarSymGrid, *, central: bool = True) -> OperatorType:
"""Make a discretized gradient squared operator for a polar grid.
{DESCR_POLAR_GRID}
Args:
grid (:class:`~pde.grids.spherical.PolarSymGrid`):
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, PolarSymGrid)
# 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]
@PolarSymGrid.register_operator("divergence", rank_in=1, rank_out=0)
@fill_in_docstring
def make_divergence(grid: PolarSymGrid) -> OperatorType:
"""Make a discretized divergence operator for a polar grid.
{DESCR_POLAR_GRID}
Args:
grid (:class:`~pde.grids.spherical.PolarSymGrid`):
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, PolarSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
dr = grid.discretization[0]
rs = grid.axes_coords[0]
scale_r = 1 / (2 * dr)
@jit
def divergence(arr: np.ndarray, out: np.ndarray) -> None:
"""Apply divergence operator to array `arr`"""
# inner radial boundary condition
for i in range(1, dim_r + 1): # iterate radial points
out[i - 1] = (arr[0, i + 1] - arr[0, i - 1]) * scale_r
out[i - 1] += arr[0, i] / rs[i - 1]
return divergence # type: ignore
[docs]
@PolarSymGrid.register_operator("vector_gradient", rank_in=1, rank_out=2)
@fill_in_docstring
def make_vector_gradient(grid: PolarSymGrid) -> OperatorType:
"""Make a discretized vector gradient operator for a polar grid.
{DESCR_POLAR_GRID}
Args:
grid (:class:`~pde.grids.spherical.PolarSymGrid`):
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, PolarSymGrid)
# 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`"""
# assign aliases
arr_r, arr_φ = arr
out_rr, out_rφ = out[0, 0, :], out[0, 1, :]
out_φr, out_φφ = out[1, 0, :], out[1, 1, :]
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_rφ[i - 1] = -arr_φ[i] / rs[i - 1]
out_φr[i - 1] = (arr_φ[i + 1] - arr_φ[i - 1]) * scale_r
out_φφ[i - 1] = arr_r[i] / rs[i - 1]
return vector_gradient # type: ignore
[docs]
@PolarSymGrid.register_operator("tensor_divergence", rank_in=2, rank_out=1)
@fill_in_docstring
def make_tensor_divergence(grid: PolarSymGrid) -> OperatorType:
"""Make a discretized tensor divergence operator for a polar grid.
{DESCR_POLAR_GRID}
Args:
grid (:class:`~pde.grids.spherical.PolarSymGrid`):
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, PolarSymGrid)
# calculate preliminary quantities
dim_r = grid.shape[0]
rs = grid.axes_coords[0]
dr = grid.discretization[0]
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_rφ = arr[0, 0, :], arr[0, 1, :]
arr_φr, arr_φφ = arr[1, 0, :], arr[1, 1, :]
out_r, out_φ = out[0, :], out[1, :]
# iterate over inner points
for i in range(1, dim_r + 1):
term = (arr_rr[i] - arr_φφ[i]) / rs[i - 1]
out_r[i - 1] = (arr_rr[i + 1] - arr_rr[i - 1]) * scale_r + term
term = (arr_rφ[i] + arr_φr[i]) / rs[i - 1]
out_φ[i - 1] = (arr_φr[i + 1] - arr_φr[i - 1]) * scale_r + term
return tensor_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, PolarSymGrid)
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, _ = bcs.grid.axes_bounds[0]
scale = 1 / dr**2
matrix = sparse.dok_matrix((dim_r, dim_r))
vector = sparse.dok_matrix((dim_r, 1))
for i in range(dim_r):
matrix[i, i] += -2 * scale
scale_i = 1 / (2 * rs[i] * dr)
if i == 0:
if r_min == 0:
matrix[i, i + 1] = 2 * scale
continue # the special case of the inner boundary is handled
const, entries = bcs[0].get_sparse_matrix_data((-1,))
factor = scale - scale_i
vector[i] += const * factor
for k, v in entries.items():
matrix[i, k] += v * factor
else:
matrix[i, i - 1] = scale - scale_i
if i == dim_r - 1:
const, entries = bcs[0].get_sparse_matrix_data((dim_r,))
factor = scale + scale_i
vector[i] += const * factor
for k, v in entries.items():
matrix[i, k] += v * factor
else:
matrix[i, i + 1] = scale + scale_i
return matrix, vector
[docs]
@PolarSymGrid.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)