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

**Yu-Cheng Shen 1, Chia-Liang Lin 2, Theodore E. Simos 3,4,5,6,7,\* and Charalampos Tsitouras <sup>8</sup>**


**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.

**Keywords:** initial value problem; Kepler-type orbits; Runge–Kutta; differential evolution

**MSC:** 65L05; 65L06; 90C26; 90C30
