Source code for pathsim.solvers.rkf45
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## EXPLICIT ADAPTIVE TIMESTEPPING RUNGE-KUTTA INTEGRATORS
## (solvers/rkf45.py)
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## Milan Rother 2024
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# IMPORTS ==============================================================================
from ._rungekutta import ExplicitRungeKutta
# SOLVERS ==============================================================================
[docs]
class RKF45(ExplicitRungeKutta):
"""Runge-Kutta-Fehlberg 4(5) pair. Six stages, 4th order propagation with
5th order error estimate.
The historically first widely-used embedded pair for automatic step-size
control. The 4th order solution is propagated; the difference to the 5th
order solution provides a local error estimate.
Characteristics
---------------
* Order: 4 (propagating) / 5 (error estimate)
* Stages: 6
* Explicit, adaptive timestep
Note
----
Largely superseded by the Dormand-Prince (``RKDP54``) and Cash-Karp
(``RKCK54``) pairs, which achieve better accuracy per function evaluation
on most problems. Still useful for reproducing legacy results or when
comparing against published benchmarks that used RKF45.
References
----------
.. [1] Fehlberg, E. (1969). "Low-order classical Runge-Kutta formulas
with stepsize control and their application to some heat transfer
problems". NASA Technical Report TR R-315.
.. [2] Fehlberg, E. (1970). "Klassische Runge-Kutta-Formeln vierter und
niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung
auf Wärmeleitungsprobleme". Computing, 6(1-2), 61-71.
:doi:`10.1007/BF02241732`
"""
def __init__(self, *solver_args, **solver_kwargs):
super().__init__(*solver_args, **solver_kwargs)
#number of stages in RK scheme
self.s = 6
#order of scheme and embedded method
self.n = 5
self.m = 4
#flag adaptive timestep solver
self.is_adaptive = True
#intermediate evaluation times
self.eval_stages = [0.0, 1/4, 3/8, 12/13, 1, 1/2]
#extended butcher table
self.BT = {
0: [ 1/4],
1: [ 3/32, 9/32],
2: [1932/2197, -7200/2197, 7296/2197],
3: [ 439/216, -8, 3680/513, -845/4104],
4: [ -8/27, 2, -3554/2565, 1859/4104, -11/40],
5: [ 25/216, 0, 1408/2565, 2197/4104, -1/5, 0]
}
#coefficients for local truncation error estimate
self.TR = [1/360, 0, -128/4275, -2197/75240, 1/50, 2/55]