# Source code for pde.pdes.swift_hohenberg

"""
The Swift-Hohenberg equation

.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""

from typing import Callable, Optional

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: Optional[BoundariesData] = 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
"""
assert isinstance(state, ScalarField), "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