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## EXPLICIT STRONG STABILITY PRESERVING RUNGE-KUTTA INTEGRATOR
## (solvers/ssprk22.py)
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## Milan Rother 2024
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# IMPORTS ==============================================================================
from ._rungekutta import ExplicitRungeKutta
# SOLVERS ==============================================================================
[docs]
class SSPRK22(ExplicitRungeKutta):
"""Two-stage, 2nd order, Strong Stability Preserving (SSP) explicit Runge-Kutta method.
Also known as the explicit midpoint method or Heun's method. SSP methods are designed
to preserve stability properties (like total variation diminishing - TVD) when solving
hyperbolic PDEs, but are also effective general-purpose low-order explicit methods.
Characteristics:
* Order: 2
* Stages: 2
* Explicit (SSP)
* Fixed timestep only
* Good balance of simplicity, cost, and stability (for an explicit method).
"""
def __init__(self, *solver_args, **solver_kwargs):
super().__init__(*solver_args, **solver_kwargs)
#number of stages in RK scheme
self.s = 2
#order of scheme
self.n = 2
#intermediate evaluation times
self.eval_stages = [0.0, 1.0]
#butcher table
self.BT = {0:[1.0],
1:[1/2, 1/2]}
[docs]
def interpolate(self, r, dt):
k1, k2 = self.K[0], self.K[1]
b1, b2 = r*(2-r)/2, r**2/2
return self.x_0 + dt*(b1 * k1 + b2 * k2)