Source code for pde.solvers.crank_nicolson

Defines a Crank-Nicolson solver

.. codeauthor:: David Zwicker <> 

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 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")["function_evaluations"] = 0["scheme"] = "implicit-euler"["stochastic"] = False["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