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## CLASSICAL EXPLICIT RUNGE-KUTTA INTEGRATOR
## (solvers/rk4.py)
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## Milan Rother 2024
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# IMPORTS ==============================================================================
from ._rungekutta import ExplicitRungeKutta
# SOLVERS ==============================================================================
[docs]
class RK4(ExplicitRungeKutta):
"""Classical four-stage, 4th order explicit Runge-Kutta method.
The most well-known Runge-Kutta method. It provides a good balance
between accuracy and computational cost for non-stiff problems. Not SSP.
Characteristics:
* Order: 4
* Stages: 4
* Explicit
* Fixed timestep only
* Widely used, good general-purpose explicit solver.
"""
def __init__(self, *solver_args, **solver_kwargs):
super().__init__(*solver_args, **solver_kwargs)
#number of stages in RK scheme
self.s = 4
#order of scheme
self.n = 4
#intermediate evaluation times
self.eval_stages = [0.0, 0.5, 0.5, 1.0]
#butcher table
self.BT = {0:[1/2],
1:[0.0, 1/2],
2:[0.0, 0.0, 1.0],
3:[1/6, 2/6, 2/6, 1/6]}
[docs]
def interpolate(self, r, dt):
k1, k2, k3, k4 = self.K[0], self.K[1], self.K[2], self.K[3]
b1, b2, b3, b4 = r*(1-r)**2/6, r**2*(2-3*r)/2, r**2*(3*r-4)/2, r**3/6
return self.x_0 + dt*(b1 * k1 + b2 * k2 + b3 * k3 + b4 * k4)