"""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 TYPE_CHECKING
from ..fields import ScalarField
from ..grids.boundaries import set_default_bc
from ..tools.docstrings import fill_in_docstring
from .base import SDEBase, expr_prod
if TYPE_CHECKING:
from collections.abc import Callable
import numpy as np
from ..grids.boundaries.axes import BoundariesData
from ..tools.typing import NumericArray
[docs]
class KPZInterfacePDE(SDEBase):
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
default_bc = "auto_periodic_neumann"
"""Default boundary condition used when no specific conditions are chosen."""
@fill_in_docstring
def __init__(
self,
nu: float = 0.5,
lmbda: float = 1,
*,
bc: BoundariesData | None = None,
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 = set_default_bc(bc, self.default_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):
msg = "`state` must be ScalarField"
raise TypeError(msg)
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
[docs]
def make_pde_rhs_numba( # type: ignore
self, state: ScalarField
) -> Callable[[NumericArray, float], NumericArray]:
"""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.
"""
import numba as nb
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)
@nb.jit(signature)
def pde_rhs(state_data: NumericArray, 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}) # type: ignore
return result
return pde_rhs # type: ignore