"""
A simple diffusion equation
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from typing import Callable, Dict, Optional
import numba as nb
import numpy as np
from ..fields import FieldCollection, 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 WavePDE(PDEBase):
r""" A simple wave equation
The mathematical definition,
.. math::
\partial_t^2 u = c^2 \nabla^2 u
is implemented as two first-order equations:
.. math::
\partial_t u &= v \\
\partial_t v &= c^2 \nabla^2 u
where :math:`u` is the density field that and :math:`c` sets the wave speed.
"""
explicit_time_dependence = False
@fill_in_docstring
def __init__(self, speed: float = 1, bc: BoundariesData = "auto_periodic_neumann"):
"""
Args:
speed (float):
The speed :math:`c` of the wave
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
"""
super().__init__()
self.speed = speed
self.bc = bc
[docs] def get_initial_condition(self, u: ScalarField, v: Optional[ScalarField] = None):
"""create a suitable initial condition
Args:
u (:class:`~pde.fields.ScalarField`):
The initial density on the grid
v (:class:`~pde.fields.ScalarField`, optional):
The initial rate of change. This is assumed to be zero if the
value is omitted.
Returns:
:class:`~pde.fields.FieldCollection`:
The combined fields u and v, suitable for the simulation
"""
if v is None:
v = ScalarField(u.grid)
return FieldCollection([u, v], labels=["u", "v"])
@property
def expressions(self) -> Dict[str, str]:
"""dict: the expressions of the right hand side of this PDE"""
return {"u": "v", "v": expr_prod(self.speed**2, "∇²u")}
[docs] def evolution_rate( # type: ignore
self,
state: FieldCollection,
t: float = 0,
) -> FieldCollection:
"""evaluate the right hand side of the PDE
Args:
state (:class:`~pde.fields.FieldCollection`):
The fields :math:`u` and :math:`v` distribution
t (float):
The current time point
Returns:
:class:`~pde.fields.FieldCollection`:
Scalar field describing the evolution rate of the PDE
"""
assert isinstance(state, FieldCollection), "`state` must be FieldCollection"
assert len(state) == 2, "`state` must contain two fields"
u, v = state
u_t = v.copy()
v_t = self.speed**2 * u.laplace(self.bc, args={"t": t}) # type: ignore
return FieldCollection([u_t, v_t])
def _make_pde_rhs_numba( # type: ignore
self, state: FieldCollection
) -> Callable[[np.ndarray, float], np.ndarray]:
"""create a compiled function evaluating the right hand side of the PDE
Args:
state (:class:`~pde.fields.FieldCollection`):
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)
speed2 = self.speed**2
laplace = state.grid.make_operator("laplace", bc=self.bc)
@jit(signature)
def pde_rhs(state_data: np.ndarray, t: float):
"""compiled helper function evaluating right hand side"""
rate = np.empty_like(state_data)
rate[0] = state_data[1]
rate[1][:] = laplace(state_data[0], args={"t": t})
rate[1] *= speed2
return rate
return pde_rhs # type: ignore