4.3.5. pde.pdes.kpz_interface module¶
The Kardar–Parisi–Zhang (KPZ) equation describing the evolution of an interface
- class KPZInterfacePDE(nu=0.5, lmbda=1, *, noise=0, bc='auto_periodic_neumann')[source]¶
Bases:
PDEBase
The Kardar–Parisi–Zhang (KPZ) equation
The mathematical definition is
\[\partial_t h = \nu \nabla^2 h + \frac{\lambda}{2} \left(\nabla h\right)^2 + \eta(\boldsymbol r, t)\]where \(h\) is the height of the interface in Monge parameterization. The dynamics are governed by the two parameters \(\nu\) and \(\lambda\), while \(\eta\) is Gaussian white noise, whose strength is controlled by the noise argument.
- Parameters
nu (float) – Parameter \(\nu\) for the strength of the diffusive term
lmbda (float) – Parameter \(\lambda\) for the strenth of the gradient term
noise (float) – Strength of the (additive) noise term
bc (Union[Dict[str, Union[Dict, str, BCBase]], Dict, str, BCBase, Tuple[Union[Dict, str, BCBase], Union[Dict, str, BCBase]], BoundaryAxisBase, Sequence[Union[Dict[str, Union[Dict, str, BCBase]], Dict, str, BCBase, Tuple[Union[Dict, str, BCBase], Union[Dict, str, BCBase]], BoundaryAxisBase]]]) – The boundary conditions applied to the field. Boundary conditions are generally given as a list with one condition for each axis. For periodic axis, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using a tuple of two conditions). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘natural’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.
- evolution_rate(state, t=0)[source]¶
evaluate the right hand side of the PDE
- Parameters
state (
ScalarField
) – The scalar field describing the concentration distributiont (float) – The current time point
- Returns
Scalar field describing the evolution rate of the PDE
- Return type
ScalarField