"""
A Allen-Cahn equation
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from typing import Callable # @UnusedImport
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 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
interface_width: float
@fill_in_docstring
def __init__(
self, interface_width: float = 1, bc: BoundariesData = "auto_periodic_neumann"
):
"""
Args:
interface_width (float):
The diffusivity of the described species
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
"""
super().__init__()
self.interface_width = interface_width
self.bc = bc
@property
def expression(self) -> str:
"""str: the expression of the right hand side of this PDE"""
return f"{expr_prod(self.interface_width, '∇²c')} - c³ + 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.interface_width * laplace - state**3 + state # 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
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 ( # type: ignore
interface_width * laplace(state_data, args={"t": t})
- state_data**3
+ state_data
)
return pde_rhs # type: ignore