Source code for pde.pdes.kpz_interface

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

.. codeauthor:: David Zwicker <> 

from typing import Callable

import numba as nb
import numpy as np

from ..fields import ScalarField
from ..grids.boundaries.axes import BoundariesData
from import fill_in_docstring
from 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): Variance of the (additive) noise term bc: The boundary conditions applied to the field. {ARG_BOUNDARIES} """ super().__init__(noise=noise) = 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(, "∇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 """ assert isinstance(state, ScalarField), "`state` must be ScalarField" result = * 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( signature = arr_type(arr_type, nb.double) nu_value, lambda_value =, 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