"""
The Kardar–Parisi–Zhang (KPZ) equation describing the evolution of an interface
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
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 KPZInterfacePDE(PDEBase):
r"""The Kardar–Parisi–Zhang (KPZ) equation
The mathematical definition is
.. math::
\partial_t h = \nu \nabla^2 h +
\frac{\lambda}{2} \left(\nabla h\right)^2 + \eta(\boldsymbol r, t)
where :math:`h` is the height of the interface in Monge parameterization.
The dynamics are governed by the two parameters :math:`\nu` and
:math:`\lambda`, 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 = 0.5,
lmbda: float = 1,
*,
noise: float = 0,
bc: BoundariesData = "auto_periodic_neumann",
):
r"""
Args:
nu (float):
Parameter :math:`\nu` for the strength of the diffusive term
lmbda (float):
Parameter :math:`\lambda` for the strenth of the gradient term
noise (float):
Strength of the (additive) noise term
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
"""
super().__init__(noise=noise)
self.nu = nu
self.lmbda = lmbda
self.bc = bc
@property
def expression(self) -> str:
"""str: the expression of the right hand side of this PDE"""
return (
expr_prod(self.nu, "laplace(c)")
+ " + "
+ expr_prod(self.lmbda, "gradient_squared(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"
result = self.nu * state.laplace(bc=self.bc, args={"t": t})
result += self.lmbda * 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, lambda_value = self.nu, self.lmbda
laplace = state.grid.make_operator("laplace", bc=self.bc)
gradient_squared = state.grid.make_operator("gradient_squared", bc=self.bc)
@jit(signature)
def pde_rhs(state_data: np.ndarray, t: float):
"""compiled helper function evaluating right hand side"""
result = nu_value * laplace(state_data, args={"t": t})
result += lambda_value * gradient_squared(state_data, args={"t": t})
return result
return pde_rhs # type: ignore