"""
The Swift-Hohenberg 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 SwiftHohenbergPDE(PDEBase):
r"""The Swift-Hohenberg equation
The mathematical definition is
.. math::
\partial_t c =
\left[\epsilon - \left(k_c^2 + \nabla^2\right)^2\right] c
+ \delta \, c^2 - c^3
where :math:`c` is a scalar field and :math:`\epsilon`, :math:`k_c^2`, and
:math:`\delta` are parameters of the equation.
"""
explicit_time_dependence = False
@fill_in_docstring
def __init__(
self,
rate: float = 0.1,
kc2: float = 1.0,
delta: float = 1.0,
*,
bc: BoundariesData = "auto_periodic_neumann",
bc_lap: BoundariesData | None = None,
):
r"""
Args:
rate (float):
The bifurcation parameter :math:`\epsilon`
kc2 (float):
Squared wave vector :math:`k_c^2` of the linear instability
delta (float):
Parameter :math:`\delta` of the non-linearity
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
bc_lap:
The boundary conditions applied to the second derivative of the scalar
field :math:`c`. If `None`, the same boundary condition as `bc` is
chosen. Otherwise, this supports the same options as `bc`.
"""
super().__init__()
self.rate = rate
self.kc2 = kc2
self.delta = delta
self.bc = bc
self.bc_lap = bc if bc_lap is None else bc_lap
@property
def expression(self) -> str:
"""str: the expression of the right hand side of this PDE"""
return (
f"{expr_prod(self.rate - self.kc2 ** 2, 'c')} - c³"
f" + {expr_prod(self.delta, 'c²')}"
f" - ∇²({expr_prod(2 * self.kc2, 'c')} + ∇²c)"
)
[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")
state_laplace = state.laplace(bc=self.bc, args={"t": t})
state_laplace2 = state_laplace.laplace(bc=self.bc_lap, args={"t": t})
result = (
(self.rate - self.kc2**2) * state
- 2 * self.kc2 * state_laplace
- state_laplace2
+ self.delta * state**2
- state**3
)
result.label = "evolution rate"
return result # 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)
rate = self.rate
kc2 = self.kc2
delta = self.delta
laplace = state.grid.make_operator("laplace", bc=self.bc)
laplace2 = state.grid.make_operator("laplace", bc=self.bc_lap)
@jit(signature)
def pde_rhs(state_data: np.ndarray, t: float):
"""compiled helper function evaluating right hand side"""
state_laplace = laplace(state_data, args={"t": t})
state_laplace2 = laplace2(state_laplace, args={"t": t})
return (
(rate - kc2**2) * state_data
- 2 * kc2 * state_laplace
- state_laplace2
+ delta * state_data**2
- state_data**3
)
return pde_rhs # type: ignore