Integrators

Here, we collect the documentations of the ordinary-differential-equations (ODE) integrators that are implemented in PyGRO.

A series of different integration schemes is pre-built in PyGRO. These can be chosen at the moment of defining a GeodesicEngine.

In particular, we have implemented a series of adaptive step-size explicit Runge-Kutta methods:

• Runge-Kutta-Fehlberg4(5): (integrator = "rkf45") embedded method from the Runge-Kutta family of the 4th order with error estiamtion of the 5th order. The implemented version is based on [1].

• Dormand-Prince5(4): (integrator = "dp45") embedded method of the 5th order with error estiamtion of the 4th order. The implemented version is based on [2]. It is the default choice in PyGRO when no integrator argument is passed to the GeodesicEngine.

• Cash-Karp: (integrator = "ck45") embedded method of the 4th order with error estiamtion of the 5th order. The implemented version is based on [3].

• Runge-Kutta-Fehlberg7(8): (integrator = "rkf78") embedded method from the Runge-Kutta family of the 7th order with error estiamtion of the 8th order. The implemented version is based on [4].

All the implemented methods refer to the general class of ExplicitAdaptiveRungeKuttaIntegrator, whose docuemntation is reported here:

class pygro.integrators.ExplicitAdaptiveRungeKuttaIntegratorView on GitHub

Base class for explicit adaptive step-size integrators of the Runge-Kutta family.

The local error is estimated as the difference between the two approximations at higher and lower order:

\[E = \| y_{\text{high}} - y_{\text{low}} \|\]

where \(\| \cdot \|\) typically represents a norm, such as the maximum norm or the Euclidean norm (the one we use).

The step size is adapted based on a defined error tolerance, which ensures the solution’s accuracy, computed as:

(1)\[\text{tol} = \text{tol}_\text{abs} + \text{tol}_\text{rel} \cdot \| y \|\]

The tolerance acts as a threshold for acceptable error in the solution. If \(E \leq \text{tol}\), the step is accepted; otherwise, it is rejected, and a smaller step size is used.

The constructor for the ExplicitAdaptiveRungeKuttaIntegrator object accepts the following arguments that can be directly passed to the integrate() method of the GeodesicEngine object upon integration:

Parameters:

• accuracy_goal (int) – the exponent that defines the integration absolute tolerance atol = 10**(-accuracy_goal), \(\text{tol}_\text{abs}\), in formula (1).

• precision_goal (int) – the exponent that defines the integration relative tolerance rtol = 10**(-precision_goal), \(\text{tol}_\text{rel}\), in formula (1).

• hmax (float) – maximum acceptable step size (default is hmax = 1e+16).

• hmin (float) – minimum acceptable step size (default is hmin = 1e-16).

• max_iter (int) – maxium namber of step-size adaptation attempts before rising an error. If this error is reached, try to use smaller accuracy_goal and/or precision_goal or to reduce the initial step size of the integration (initial_step in the integrate() method).

interpolator_classView on GitHub

alias of HermiteCubicInterpolator

In the future, we also plan to implement implicit and symplectic integrations schemes.

References

[1]

Fehlberg, E (1964). “New high-order Runge-Kutta formulas with step size control for systems of first and second-order differential equations”. Zeitschrift für Angewandte Mathematik und Mechanik. 44 (S1): T17 - T29. doi:10.1002/zamm.19640441310.

[2]

Dormand, J.R.; Prince, P.J. (1980). “A family of embedded Runge-Kutta formulae”. Journal of Computational and Applied Mathematics. 6 (1): 19-26. doi:10.1016/0771-050X(80)90013-3

[3]

10. 18. Cash, A. H. Karp. (1990). “A variable order Runge-Kutta method for initial value problems with rapidly varying right hand sides”, ACM Transactions on Mathematical Software 16: 201-222. doi:10.1145/79505.79507

[4]

Fehlberg, Erwin (1968) “Classical fifth-, sixth-, seventh-, and eighth-order Runge-Kutta formulas with stepsize control”. NASA Technical Report 287. (PDF).