RKCK54 (Cash-Karp)

class pathsim.solvers.rkck54.RKCK54(*solver_args, **solver_kwargs)

Bases: ExplicitRungeKutta

Six-stage, 5th order explicit Runge-Kutta method by Cash and Karp.

Features an embedded 4th order method. The difference between the 5th and 4th order results provides a 5th order error estimate, typically used to control the step size while propagating the 5th order solution (local extrapolation). Known for better stability properties compared to RKF45.

Characteristics

• Order: 5 (Propagating solution)

• Embedded Order: 4

• Stages: 6

• Explicit

• Adaptive timestep

• Good stability, suitable for moderate-to-high accuracy requirements

When to Use

• Improved stability over RKF45: When RKF45 exhibits stability issues

• Moderate-to-high accuracy needs: 5th order for better accuracy than 3rd order methods

• Non-stiff problems: Excellent for smooth, non-stiff ODEs

• Alternative to RKDP54: Similar performance, sometimes better for specific problems

Note

RKDP54 is generally more efficient when first-same-as-last (FSAL) property is implemented (which it currently is not!), but RKCK54 can have better stability for certain problems. Both are excellent 5th order adaptive methods.

References

[1]

Cash, J. R., & Karp, A. H. (1990). “A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides”. ACM Transactions on Mathematical Software, 16(3), 201-222.

[2]

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). “Numerical Recipes: The Art of Scientific Computing” (3rd ed.). Cambridge University Press.

[3]

Hairer, E., Nørsett, S. P., & Wanner, G. (1993). “Solving Ordinary Differential Equations I: Nonstiff Problems”. Springer Series in Computational Mathematics, Vol. 8.