from simframe.integration.scheme import Scheme
# Butcher coefficients
a10 = 1/2
a21 = 3/4
b0, b1, b2 = 2/9, 1/3, 4/9
c1, c2 = 1/2, 3/4
def _f_expl_3_ralston(x0, Y0, dx, *args, dYdx=None, **kwargs):
"""Explicit 3rd-order Ralston'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.
args : additional positional arguments
kwargs : additional keyworda arguments
Returns
-------
dY : Field
Delta of variable to be integrated
Butcher tableau
---------------
0 | 0 0 0
1/2 | 1/2 0 0
3/4 | 0 3/4 0
-----|-------------
| 2/9 1/3 4/9
"""
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 + a21*k1*dx)
return dx*(b0*k0 + b1*k1 + b2*k2)
[docs]
class expl_3_ralston(Scheme):
"""Class for explicit 3rd-order Ralston's method"""
def __init__(self, *args, **kwargs):
super().__init__(_f_expl_3_ralston,
description="Explicit 3rd-order Ralston's method", *args, **kwargs)