r"""This module implements differential operators on polar grids.
.. autosummary::
:nosignatures:
make_laplace
make_gradient
make_gradient_squared
make_divergence
make_vector_gradient
make_tensor_divergence
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import TYPE_CHECKING, Literal
from .... import get_backend
from ....grids.spherical import PolarSymGrid
from ....tools.docstrings import fill_in_docstring
from ..backend import NumbaBackend
from ..utils import jit
if TYPE_CHECKING:
from ....tools.typing import NumericArray, OperatorImplType
[docs]
@NumbaBackend.register_operator(PolarSymGrid, "laplace", rank_in=0, rank_out=0)
@fill_in_docstring
def make_laplace(
grid: PolarSymGrid, *, backend: NumbaBackend | None = None
) -> OperatorImplType:
"""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
backend (:class:`~pde.backends.numba.backend.NumbaBackend`):
References to the backend to read configuration details
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, PolarSymGrid)
# calculate preliminary quantities
if backend is None:
backend = get_backend("numba") # type: ignore
dim_r = grid.shape[0]
dr = grid.discretization[0]
factor_r = 1 / (2 * grid.axes_coords[0] * dr)
dr_2 = 1 / dr**2
@jit(backend=backend)
def laplace(arr: NumericArray, out: NumericArray) -> 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
[docs]
@NumbaBackend.register_operator(PolarSymGrid, "gradient", rank_in=0, rank_out=1)
@fill_in_docstring
def make_gradient(
grid: PolarSymGrid,
*,
backend: NumbaBackend | None = None,
method: Literal["central", "forward", "backward"] = "central",
) -> OperatorImplType:
"""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
backend (:class:`~pde.backends.numba.backend.NumbaBackend`):
References to the backend to read configuration details
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
if backend is None:
backend = get_backend("numba") # type: ignore
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:
msg = f"Unknown derivative type `{method}`"
raise ValueError(msg)
@jit(backend=backend)
def gradient(arr: NumericArray, out: NumericArray) -> 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
[docs]
@NumbaBackend.register_operator(PolarSymGrid, "gradient_squared", rank_in=0, rank_out=0)
@fill_in_docstring
def make_gradient_squared(
grid: PolarSymGrid,
*,
backend: NumbaBackend | None = None,
central: bool = True,
) -> OperatorImplType:
"""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
backend (:class:`~pde.backends.numba.backend.NumbaBackend`):
References to the backend to read configuration details
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
if backend is None:
backend = get_backend("numba") # type: ignore
dim_r = grid.shape[0]
dr = grid.discretization[0]
if central:
# use central differences
scale = 0.25 / dr**2
@jit(backend=backend)
def gradient_squared(arr: NumericArray, out: NumericArray) -> 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(backend=backend)
def gradient_squared(arr: NumericArray, out: NumericArray) -> 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
[docs]
@NumbaBackend.register_operator(PolarSymGrid, "divergence", rank_in=1, rank_out=0)
@fill_in_docstring
def make_divergence(
grid: PolarSymGrid, *, backend: NumbaBackend | None = None
) -> OperatorImplType:
"""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
backend (:class:`~pde.backends.numba.backend.NumbaBackend`):
References to the backend to read configuration details
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, PolarSymGrid)
# calculate preliminary quantities
if backend is None:
backend = get_backend("numba") # type: ignore
dim_r = grid.shape[0]
dr = grid.discretization[0]
rs = grid.axes_coords[0]
scale_r = 1 / (2 * dr)
@jit(backend=backend)
def divergence(arr: NumericArray, out: NumericArray) -> 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
[docs]
@NumbaBackend.register_operator(PolarSymGrid, "vector_gradient", rank_in=1, rank_out=2)
@fill_in_docstring
def make_vector_gradient(
grid: PolarSymGrid, *, backend: NumbaBackend | None = None
) -> OperatorImplType:
"""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
backend (:class:`~pde.backends.numba.backend.NumbaBackend`):
References to the backend to read configuration details
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, PolarSymGrid)
# calculate preliminary quantities
if backend is None:
backend = get_backend("numba") # type: ignore
dim_r = grid.shape[0]
rs = grid.axes_coords[0]
dr = grid.discretization[0]
scale_r = 1 / (2 * dr)
@jit(backend=backend)
def vector_gradient(arr: NumericArray, out: NumericArray) -> 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
[docs]
@NumbaBackend.register_operator(
PolarSymGrid, "tensor_divergence", rank_in=2, rank_out=1
)
@fill_in_docstring
def make_tensor_divergence(
grid: PolarSymGrid, *, backend: NumbaBackend | None = None
) -> OperatorImplType:
"""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
backend (:class:`~pde.backends.numba.backend.NumbaBackend`):
References to the backend to read configuration details
Returns:
A function that can be applied to an array of values
"""
assert isinstance(grid, PolarSymGrid)
# calculate preliminary quantities
if backend is None:
backend = get_backend("numba") # type: ignore
dim_r = grid.shape[0]
rs = grid.axes_coords[0]
dr = grid.discretization[0]
scale_r = 1 / (2 * dr)
@jit(backend=backend)
def tensor_divergence(arr: NumericArray, out: NumericArray) -> 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