"""
Defines a Crank-Nicolson solver
.. 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.base import FieldBase
from ..pdes.base import PDEBase
from ..tools.typing import BackendType
from .base import ConvergenceError, SolverBase
[docs]
class CrankNicolsonSolver(SolverBase):
"""Crank-Nicolson solver"""
name = "crank-nicolson"
def __init__(
self,
pde: PDEBase,
*,
maxiter: int = 100,
maxerror: float = 1e-4,
explicit_fraction: float = 0,
backend: BackendType = "auto",
):
"""
Args:
pde (:class:`~pde.pdes.base.PDEBase`):
The instance describing the pde that needs to be solved
maxiter (int):
The maximal number of iterations per step
maxerror (float):
The maximal error that is permitted in each step
explicit_fraction (float):
Hyperparameter determinig the fraction of explicit time stepping in the
implicit step. `explicit_fraction == 0` is the simple Crank-Nicolson
scheme, while `explicit_fraction == 1` reduces to the explicit Euler
method. Intermediate values can improve convergence.
backend (str):
Determines how the function is created. Accepted values are 'numpy` and
'numba'. Alternatively, 'auto' lets the code decide for the most optimal
backend.
"""
super().__init__(pde, backend=backend)
self.maxiter = maxiter
self.maxerror = maxerror
self.explicit_fraction = explicit_fraction
def _make_single_step_fixed_dt(
self, state: FieldBase, dt: float
) -> Callable[[np.ndarray, float], None]:
"""return a function doing a single step with an implicit Euler scheme
Args:
state (:class:`~pde.fields.base.FieldBase`):
An example for the state from which the grid and other information can
be extracted
dt (float):
Time step of the implicit step
"""
if self.pde.is_sde:
raise RuntimeError("Cannot use implicit stepper with stochastic equation")
self.info["function_evaluations"] = 0
self.info["scheme"] = "implicit-euler"
self.info["stochastic"] = False
self.info["dt_adaptive"] = False
rhs = self._make_pde_rhs(state, backend=self.backend)
maxiter = int(self.maxiter)
maxerror2 = self.maxerror**2
α = self.explicit_fraction
# handle deterministic version of the pde
def crank_nicolson_step(state_data: np.ndarray, t: float) -> None:
"""compiled inner loop for speed"""
nfev = 0 # count function evaluations
# keep values at the current time t point used in iteration
state_t = state_data.copy()
rate_t = rhs(state_t, t)
# new state is weighted average of explicit and Crank-Nicolson steps
state_cn = state_t + dt / 2 * (rhs(state_data, t + dt) + rate_t)
state_data[:] = α * state_data + (1 - α) * state_cn
state_prev = np.empty_like(state_data)
# fixed point iteration for improving state after dt
for n in range(maxiter):
state_prev[:] = state_data # keep previous state to judge convergence
# new state is weighted average of explicit and Crank-Nicolson steps
state_cn = state_t + dt / 2 * (rhs(state_data, t + dt) + rate_t)
state_data[:] = α * state_data + (1 - α) * state_cn
# calculate mean squared error
err = 0.0
for j in range(state_data.size):
diff = state_data.flat[j] - state_prev.flat[j]
err += (diff.conjugate() * diff).real
err /= state_data.size
if err < maxerror2:
# fix point iteration converged
break
else:
with nb.objmode:
self._logger.warning(
"Crank-Nicolson step did not converge after %d iterations "
"at t=%g (error=%g)",
maxiter,
t,
err,
)
raise ConvergenceError("Crank-Nicolson step did not converge.")
nfev += n + 2
return crank_nicolson_step