Source code for pde.pdes.diffusion

"""
A simple diffusion equation

.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""

from typing import Callable

import numba as nb
import numpy as np

from ..fields import ScalarField
from ..grids.boundaries.axes import BoundariesData
from ..tools.docstrings import fill_in_docstring
from ..tools.numba import jit
from .base import PDEBase, expr_prod

[docs]class DiffusionPDE(PDEBase):
r"""A simple diffusion equation

The mathematical definition is

.. math::
\partial_t c = D \nabla^2 c

where :math:c is a scalar field and :math:D denotes the diffusivity.
"""

explicit_time_dependence = False

@fill_in_docstring
def __init__(
self,
diffusivity: float = 1,
noise: float = 0,
bc: BoundariesData = "auto_periodic_neumann",
):
"""
Args:
diffusivity (float):
The diffusivity of the described species
noise (float):
Variance of the (additive) noise term
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
"""
super().__init__(noise=noise)

self.diffusivity = diffusivity
self.bc = bc

@property
def expression(self) -> str:
"""str: the expression of the right hand side of this PDE"""
return expr_prod(self.diffusivity, "∇²(c)")

[docs]    def evolution_rate(  # type: ignore
self,
state: ScalarField,
t: float = 0,
) -> ScalarField:
"""evaluate the right hand side of the PDE

Args:
state (:class:~pde.fields.ScalarField):
The scalar field describing the concentration distribution
t (float): The current time point

Returns:
:class:~pde.fields.ScalarField:
Scalar field describing the evolution rate of the PDE
"""
assert isinstance(state, ScalarField), "state must be ScalarField"
laplace = state.laplace(bc=self.bc, label="evolution rate", args={"t": t})
return self.diffusivity * laplace  # type: ignore

def _make_pde_rhs_numba(  # type: ignore
self, state: ScalarField
) -> Callable[[np.ndarray, float], np.ndarray]:
"""create a compiled function evaluating the right hand side of the PDE

Args:
state (:class:~pde.fields.ScalarField):
An example for the state defining the grid and data types

Returns:
A function with signature (state_data, t), which can be called
with an instance of :class:~numpy.ndarray of the state data and
the time to obtained an instance of :class:~numpy.ndarray giving
the evolution rate.
"""
arr_type = nb.typeof(state.data)
signature = arr_type(arr_type, nb.double)

diffusivity_value = self.diffusivity
laplace = state.grid.make_operator("laplace", bc=self.bc)

@jit(signature)
def pde_rhs(state_data: np.ndarray, t: float):
"""compiled helper function evaluating right hand side"""
return diffusivity_value * laplace(state_data, args={"t": t})

return pde_rhs  # type: ignore