from simframe.integration.scheme import Scheme
import numpy as np
# Butcher coefficients
a10 = 1/2
a20, a21 = 1/256, 255/256
b0, b1, b2 = 1/512, 255/256, 1/512
bs0, bs1 = 1/256, 255/256
e0, e1, e2 = b0-bs0, b1-bs1, b2
c1 = 1/2
def _f_expl_2_fehlberg_adptv(x0, Y0, dx, *args, dYdx=None, econ=0.0324, eps=0.1, pgrow=-0.5, pshrink=-1., safety=0.9, **kwargs):
"""Explicit adaptive 2nd-order Fehlberg's method
Parameters
----------
x0 : Intvar
Integration variable at beginning of scheme
Y0 : Field
Variable to be integrated at the beginning of scheme
dx : IntVar
Stepsize of integration variable
dYdx : Field, optional, default : None
Current derivative. Will be calculated, if not set.
econ : float, optional, default : 0.0324
Error controll parameter for setting stepsize
eps : float, optional, default : 0.1
Desired maximum relative error
prgrow : float, optional, default : -0.5
Power for increasing step size
pshrink : float, optional, default : -1.
Power for decreasing stepsize
safety : float, optional, default : 0.9
Safety factor when changing step size
args : additional positional arguments
kwargs : additional keyworda arguments
Returns
-------
dY : Field
Delta of variable to be integrated
False if step size too large
Butcher tableau
---------------
0 | 0 0 0
1/2 | 1/2 0 0
1 | 1/256 255/256 0
-----|---------------------
| 1/512 255/256 1/512
| 1/256 255/256 0
"""
k0 = Y0.derivative(x0, Y0) if dYdx is None else dYdx
k1 = Y0.derivative(x0 + c1*dx, Y0 + a10*k0 * dx)
k2 = Y0.derivative(x0 + dx, Y0 + (a20*k0 + a21*k1)*dx)
Yscale = np.abs(Y0) + np.abs(dx*k0)
Yscale[Yscale == 0.] = 1.e100 # Deactivate for zero crossings
e = dx*(e0*k0 + e1*k1 + e2*k2)
emax = np.max(np.abs(e/Yscale)) / eps
# Integration successful
if emax <= 1.:
# Suggest new stepsize
dxnew = safety*dx*emax**pgrow if econ < emax else 5.*dx
x0.suggest(dxnew)
return dx*(b0*k0 + b1*k1 + b2*k2)
else:
# Suggest new stepsize
dxnew = np.maximum(safety*dx*emax**pshrink, 0.1*dx)
x0.suggest(dxnew)
return False
[docs]
class expl_2_fehlberg_adptv(Scheme):
"""Class for explicit adaptive 2nd-order Fehlberg's method"""
def __init__(self, *args, **kwargs):
super().__init__(_f_expl_2_fehlberg_adptv,
description="Explicit adaptive 2nd-order Fehlberg's method", *args, **kwargs)