RK4

class pathsim.solvers.rk4.RK4(*solver_args, **solver_kwargs)

Bases: ExplicitRungeKutta

Classical four-stage, 4th order explicit Runge-Kutta method.

\[\begin{split}\begin{aligned} k_1 &= f(x_n,\; t_n) \\ k_2 &= f\!\left(x_n + \tfrac{h}{2}\,k_1,\; t_n + \tfrac{h}{2}\right) \\ k_3 &= f\!\left(x_n + \tfrac{h}{2}\,k_2,\; t_n + \tfrac{h}{2}\right) \\ k_4 &= f(x_n + h\,k_3,\; t_n + h) \\ x_{n+1} &= x_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4) \end{aligned}\end{split}\]

Characteristics

• Order: 4

• Stages: 4

• Explicit, fixed timestep

Note

The standard fixed-step explicit solver. Provides a good cost-to-accuracy ratio for non-stiff block diagrams where the timestep is known a priori (e.g. real-time or hardware-in-the-loop simulation with a fixed clock). Not suitable for stiff systems. When accuracy demands vary during a run, adaptive methods like RKDP54 are more efficient because they concentrate steps where the dynamics change rapidly.

References

[1]

Kutta, W. (1901). “Beitrag zur näherungsweisen Integration totaler Differentialgleichungen”. Zeitschrift für Mathematik und Physik, 46, 435-453.

[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`

interpolate(r, dt)

Interpolate solution after successful timestep as a ratio in the interval [t, t+dt].

This is especially relevant for Runge-Kutta solvers that have a higher order interpolant. Otherwise this is just linear interpolation using the buffered state.

Parameters:

• r (float) – ration for interpolation within timestep

• dt (float) – integration timestep

Returns:

x – interpolated state

Return type:

numeric, array[numeric]