from simframe.integration.scheme import Scheme
# Butcher coefficients
a10 = 0.4
a20, a21 = 0.29697761, 0.15875964
a30, a31, a32 = 0.21810040, -3.05096516, 3.83286476
b0, b1, b2, b3 = 0.17476028, -0.55148066, 1.20553560, 0.17118478
c1, c2 = 0.4, 0.45573725
def _f_expl_4_ralston(x0, Y0, dx, *args, dYdx=None, **kwargs):
"""Explicit 4th-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 0
0.4 | 0.4 0 0 0
0.45573725 | 0.29697761 0.15875964 0 0
1 | 0.21810040 -3.05096516 3.83286476 0
------------|----------------------------------------------
| 0.17476028 -0.55148066 1.20553560 0.17118478
"""
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 + a21*k1)*dx)
k3 = Y0.derivative(x0 + dx, Y0 + (a30*k0 + a31*k1 + a32*k2)*dx)
return dx*(b0*k0 + b1*k1 + b2*k2 + b3*k3)
[docs]
class expl_4_ralston(Scheme):
"""Class for explicit 4th-order Ralston's method"""
def __init__(self, *args, **kwargs):
super().__init__(_f_expl_4_ralston,
description="Explicit 4th-order Ralston's method", *args, **kwargs)