########################################################################################
##
## EXPLICIT ADAPTIVE TIMESTEPPING RUNGE-KUTTA INTEGRATORS
## (solvers/rkck54.py)
##
## Milan Rother 2024
##
########################################################################################
# IMPORTS ==============================================================================
from ._rungekutta import ExplicitRungeKutta
# SOLVERS ==============================================================================
[docs]
class RKCK54(ExplicitRungeKutta):
"""Cash-Karp 5(4) pair. Six stages, 5th order with embedded 4th order
error estimate.
Designed to improve on the stability properties of the Fehlberg pair
(``RKF45``) while keeping the same stage count.
Characteristics
---------------
* Order: 5 (propagating) / 4 (embedded)
* Stages: 6
* Explicit, adaptive timestep
Note
----
Comparable to ``RKDP54`` in cost and accuracy for most non-stiff block
diagrams. Can exhibit slightly better stability on problems with
eigenvalues near the imaginary axis. Both pairs are solid 5th order
choices; ``RKDP54`` is the more commonly used default.
References
----------
.. [1] Cash, J. R., & Karp, A. H. (1990). "A variable order Runge-Kutta
method for initial value problems with rapidly varying right-hand
sides". ACM Transactions on Mathematical Software, 16(3), 201-222.
:doi:`10.1145/79505.79507`
.. [2] Hairer, E., Nørsett, S. P., & Wanner, G. (1993). "Solving
Ordinary Differential Equations I: Nonstiff Problems". Springer
Series in Computational Mathematics, Vol. 8.
:doi:`10.1007/978-3-540-78862-1`
"""
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, 1/5, 3/10, 3/5, 1, 7/8]
#extended butcher table
self.BT = {
0: [ 1/5],
1: [ 3/40, 9/40],
2: [ 3/10, -9/10, 6/5],
3: [ -11/54, 5/2, -70/27, 35/27],
4: [1631/55296, 175/512, 575/13824, 44275/110592, 253/4096],
5: [ 37/378, 0, 250/621, 125/594, 0, 512/1771]
}
#coefficients for local truncation error estimate
self.TR = [-277/64512, 0, 6925/370944, -6925/202752, -277/14336, 277/7084]