########################################################################################
##
## EMBEDDED DIAGONALLY IMPLICIT RUNGE KUTTA METHOD
## (solvers/esdirk54.py)
##
## Milan Rother 2024
##
########################################################################################
# IMPORTS ==============================================================================
from ._rungekutta import DiagonallyImplicitRungeKutta
# SOLVERS ==============================================================================
[docs]
class ESDIRK54(DiagonallyImplicitRungeKutta):
"""Seven-stage, 5th order ESDIRK method with embedded 4th order error
estimate. L-stable and stiffly accurate (ESDIRK5(4)7L[2]SA2).
Characteristics
---------------
* Order: 5 (propagating) / 4 (embedded)
* Stages: 7 (1 explicit, 6 implicit)
* Adaptive timestep
* L-stable, stiffly accurate
* Stage order 2
Note
----
The highest-accuracy L-stable single-step solver in this library before
the much more expensive ``ESDIRK85``. Use when tight tolerances are
needed on a stiff block diagram (e.g. multi-rate systems combining fast
electrical and slow thermal dynamics). At moderate tolerances,
``ESDIRK43`` achieves similar results with fewer implicit solves per
step.
References
----------
.. [1] Kennedy, C. A., & Carpenter, M. H. (2019). "Diagonally implicit
Runge-Kutta methods for stiff ODEs". Applied Numerical
Mathematics, 146, 221-244.
:doi:`10.1016/j.apnum.2019.07.008`
.. [2] Hairer, E., & Wanner, G. (1996). "Solving Ordinary Differential
Equations II: Stiff and Differential-Algebraic Problems". Springer
Series in Computational Mathematics, Vol. 14.
:doi:`10.1007/978-3-642-05221-7`
"""
def __init__(self, *solver_args, **solver_kwargs):
super().__init__(*solver_args, **solver_kwargs)
#number of stages in RK scheme
self.s = 7
#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, 46/125, 7121331996143/11335814405378, 49/353,
3706679970760/5295570149437, 347/382, 1.0
]
#butcher table
self.BT = {
0: None, #explicit first stage
1: [23/125, 23/125],
2: [791020047304/3561426431547, 791020047304/3561426431547, 23/125],
3: [-158159076358/11257294102345, -158159076358/11257294102345,
-85517644447/5003708988389, 23/125],
4: [-1653327111580/4048416487981, -1653327111580/4048416487981,
1514767744496/9099671765375, 14283835447591/12247432691556, 23/125],
5: [-4540011970825/8418487046959, -4540011970825/8418487046959,
-1790937573418/7393406387169, 10819093665085/7266595846747,
4109463131231/7386972500302, 23/125],
6: [-188593204321/4778616380481, -188593204321/4778616380481,
2809310203510/10304234040467, 1021729336898/2364210264653,
870612361811/2470410392208, -1307970675534/8059683598661, 23/125]
}
#coefficients for truncation error estimate
_A1 = [
-188593204321/4778616380481, -188593204321/4778616380481,
2809310203510/10304234040467, 1021729336898/2364210264653,
870612361811/2470410392208, -1307970675534/8059683598661, 23/125
]
_A2 = [
-582099335757/7214068459310, -582099335757/7214068459310,
615023338567/3362626566945, 3192122436311/6174152374399,
6156034052041/14430468657929, -1011318518279/9693750372484,
1914490192573/13754262428401
]
self.TR = [_a1 - _a2 for _a1, _a2 in zip(_A1, _A2)]