"""
The Kardar–Parisi–Zhang (KPZ) equation describing the evolution of an interface
.. 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 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,
*,
bc: BoundariesData = "auto_periodic_neumann",
noise: float = 0,
rng: np.random.Generator | None = None,
):
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
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.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, "∇²c") + " + " + expr_prod(self.lmbda, "|∇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
"""
if not isinstance(state, ScalarField):
raise ValueError("`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