from simframe.integration.scheme import Scheme
# Butcher coefficients
a10 = 1/3
a20 = -1/3
b0, b1, b2, b3 = 1/8, 3/8, 3/8, 1/8
c1, c2 = 1/3, 2/3
def _f_expl_4_38rule(x0, Y0, dx, *args, dYdx=None, **kwargs):
"""Explicit 4th-order 3/8 rule 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.
args : additional positional arguments
kwargs : additional keyworda arguments
Returns
-------
dY : Field
Delta of variable to be integrated
Butcher tableau
---------------
0 | 0 0 0 0
1/3 | 1/3 0 0 0
2/3 |-1/3 1 0 0
1 | 1 -1 1 0
-----|-----------------
| 1/8 3/8 3/8 1/8
"""
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 + c2*dx, Y0 + (a20*k0 + k1)*dx)
k3 = Y0.derivative(x0 + dx, Y0 + (k0 - k1 + k2)*dx)
return dx*(b0*k0 + b1*k1 + b2*k2 + b3*k3)
[docs]
class expl_4_38rule(Scheme):
"""Class for explicit 4th-order 3/8 rule method"""
def __init__(self, *args, **kwargs):
super().__init__(_f_expl_4_38rule,
description="Explicit 4th-order 3/8 rule method", *args, **kwargs)