# Source code for pde.pdes.kuramoto_sivashinsky

"""
The Kardar–Parisi–Zhang (KPZ) equation describing the evolution of an interface

.. 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 KuramotoSivashinskyPDE(PDEBase):
r"""The Kuramoto-Sivashinsky equation

The mathematical definition is

.. math::
\partial_t u = -\nu \nabla^4 u  - \nabla^2 u -
\frac{1}{2} \left(\nabla h\right)^2  + \eta(\boldsymbol r, t)

where :math:u is the height of the interface in Monge parameterization. The
dynamics are governed by the parameters :math:\nu , while :math:\eta is Gaussian
white noise, whose strength is controlled by the noise argument.
"""

explicit_time_dependence = False

@fill_in_docstring
def __init__(
self,
nu: float = 1,
*,
noise: float = 0,
bc: BoundariesData = "auto_periodic_neumann",
bc_lap: Optional[BoundariesData] = None,
):
r"""
Args:
nu (float):
Parameter :math:\nu for the strength of the fourth-order term
noise (float):
Variance of the (additive) noise term
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__(noise=noise)

self.nu = nu
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"""
expr = f"c + {expr_prod(self.nu, '∇²c')}"
return f"-∇²({expr}) - 0.5 * |∇c|²"

[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_lap = state.laplace(bc=self.bc, args={"t": t})
result = (
-self.nu * state_lap.laplace(bc=self.bc_lap, args={"t": t})
- state_lap
- 0.5 * state.gradient_squared(bc=self.bc, args={"t": t})
)
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)

nu_value = self.nu
laplace = state.grid.make_operator("laplace", bc=self.bc)
laplace2 = state.grid.make_operator("laplace", bc=self.bc_lap)