"""A Allen-Cahn equation.
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import Callable
import numba as nb
import numpy as np
from ..fields import ScalarField
from ..grids.boundaries import set_default_bc
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 AllenCahnPDE(PDEBase):
r"""A simple Allen-Cahn equation.
The mathematical definition is
.. math::
\partial_t c = \gamma \nabla^2 c - c^3 + c
where :math:`c` is a scalar field and :math:`\gamma` sets the (squared) interfacial
width.
"""
explicit_time_dependence = False
default_bc = "auto_periodic_neumann"
"""Default boundary condition used when no specific conditions are chosen."""
interface_width: float
@fill_in_docstring
def __init__(
self,
interface_width: float = 1,
mobility: float = 1,
*,
bc: BoundariesData | None = None,
):
"""
Args:
interface_width (float):
The diffusivity of the described species
mobility (float):
The rate at which the structures evolve
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
"""
super().__init__()
self.interface_width = interface_width
self.mobility = mobility
self.bc = set_default_bc(bc, self.default_bc)
@property
def expression(self) -> str:
"""str: the expression of the right hand side of this PDE"""
expr = f"{expr_prod(self.interface_width, '∇²c')} - c³ + c"
if np.isclose(self.mobility, 1):
return expr
else:
return expr_prod(self.mobility, f"({expr})")
[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
"""
if not isinstance(state, ScalarField):
raise ValueError("`state` must be ScalarField")
laplace = state.laplace(bc=self.bc, label="evolution rate", args={"t": t})
rhs = self.interface_width * laplace - state**3 + state
return self.mobility * rhs # 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)
interface_width = self.interface_width
mobility = self.mobility
laplace = state.grid.make_operator("laplace", bc=self.bc)
@jit(signature)
def pde_rhs(state_data: np.ndarray, t: float) -> np.ndarray:
"""Compiled helper function evaluating right hand side."""
return mobility * ( # type: ignore
interface_width * laplace(state_data, args={"t": t})
- state_data**3
+ state_data
)
return pde_rhs # type: ignore