Source code for pde.pdes.allen_cahn

"""
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.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, mobility: float = 1, bc: BoundariesData = "auto_periodic_neumann", ): """ 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 = 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