Source code for pde.pdes.diffusion

A simple diffusion equation

.. codeauthor:: David Zwicker <> 

from typing import Callable, Optional

import numba as nb
import numpy as np

from ..fields import ScalarField
from ..grids.boundaries.axes import BoundariesData
from import fill_in_docstring
from 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, *, bc: BoundariesData = "auto_periodic_neumann", noise: float = 0, rng: Optional[np.random.Generator] = None, ): """ Args: diffusivity (float): The diffusivity of the described species bc: The boundary conditions applied to the field. {ARG_BOUNDARIES} noise (float): Variance of the (additive) noise term rng (:class:`~numpy.random.Generator`): Random number generator (default: :func:`~numpy.random.default_rng()`) used for stochastic simulations. Note that this random number generator is only used for numpy function, while compiled numba code uses the random number generator of numba. Moreover, in simulations using multiprocessing, setting the same generator in all processes might yield unintended correlations in the simulation results. """ super().__init__(noise=noise, rng=rng) 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( 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