Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

Abstract

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order.

New family for Runge‐Kutta‐Nyström pairs of orders 6(4) with coefficients trained to address oscillatory problems

“…For the minimization of the positive value $$$ u $$$, we used differential evolution algorithm 9 . We have already tried this approach and got some interesting results in producing pairs from the family of interest here for integrating orbits 10 . In this latter work, we trained the coefficients of a RK6(5) pair on a couple of Kepler orbits.…”

“…9 We have already tried this approach and got some interesting results in producing pairs from the family of interest here for integrating orbits. 10 In this latter work, we trained the coefficients of a RK6(5) pair on a couple of Kepler orbits. Then we observed very pleasant results over a set of Kepler orbits as well as on other known orbital problems.…”

A Neural Network Type Approach for Constructing Runge–Kutta Pairs of Orders Six and Five That Perform Best on Problems with Oscillatory Solutions

“…For the minimization process, we used the Differential Evolution technique [18,19]. We have already tried this approach and obtained some interesting results in producing pairs from the family of interest here for integrating orbits [20]. In this latter work we trained the coefficients of a RK6(5) pair on a couple of Kepler orbits.…”

Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions

“…Here, for the minimization of u 1 + u 2 we tried Differential Evolution [17]. We have already got positive results using this approach for methods in integrating of orbits [18,19]. In these latter works, we trained the coefficients of the methods on a Kepler orbit.…”