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##
## EXPLICIT ADAPTIVE TIMESTEPPING RUNGE-KUTTA INTEGRATORS
## (solvers/rkv65.py)
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## Milan Rother 2024
##
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# IMPORTS ==============================================================================
from ._rungekutta import ExplicitRungeKutta
# SOLVERS ==============================================================================
[docs]
class RKV65(ExplicitRungeKutta):
"""9-stage 6-th order with embedded 5-th order Runge-Kutta method from Verner
with 6-th order truncation error estimate.
This is the 'most robust' 9, 6(5) pair of Jim Verner's Refuge for Runge-Kutta Pairs
URL: https://www.sfu.ca/~jverner/
"""
def __init__(self, *solver_args, **solver_kwargs):
super().__init__(*solver_args, **solver_kwargs)
#number of stages in RK scheme
self.s = 9
#order of scheme and embedded method
self.n = 6
self.m = 5
#flag adaptive timestep solver
self.is_adaptive = True
#intermediate evaluation times
self.eval_stages = [0.0, 9/50, 1/6, 1/4, 53/100, 3/5, 4/5, 1.0, 1.0]
#extended butcher table
self.BT = {0:[ 9/50],
1:[ 29/324, 25/324],
2:[ 1/16, 0, 3/16],
3:[ 79129/250000, 0, -261237/250000, 19663/15625],
4:[ 1336883/4909125, 0, -25476/30875, 194159/185250, 8225/78546],
5:[-2459386/14727375, 0, 19504/30875, 2377474/13615875, -6157250/5773131, 902/735],
6:[ 2699/7410, 0, -252/1235, -1393253/3993990, 236875/72618, -135/49, 15/22],
7:[ 11/144, 0, 0, 256/693, 0, 125/504, 125/528, 5/72],
8:[ 11/144, 0, 0, 256/693, 0, 125/504, 125/528, 5/72]}
#compute coefficients for truncation error
_A1 = [11/144, 0, 0, 256/693, 0, 125/504, 125/528, 5/72, 0]
_A2 = [28/477, 0, 0, 212/441, -312500/366177, 2125/1764, 0, -2105/35532, 2995/17766]
self.TR = [a-b for a, b in zip(_A1, _A2)]