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

**Vladislav N. Kovalnogov 1, Ruslan V. Fedorov 1, Andrey V. Chukalin 1, Theodore E. Simos 1,2,3,4,5,6,\* and Charalampos Tsitouras 7,8**


**Citation:** Kovalnogov, V.N.; Fedorov, R.V.; Chukalin, A.V.; Simos, T.E.; Tsitouras, C. Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions. *Mathematics* **2021**, *9*, 2306. https://doi.org/10.3390/ math9182306

Academic Editor: Alicia Cordero Barbero

Received: 13 August 2021 Accepted: 16 September 2021 Published: 18 September 2021

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**Abstract:** The purpose of the present work is to construct a new Runge–Kutta pair of orders five and four to outperform the state-of-the-art in these kind of methods when addressing problems with periodic solutions. We consider the family of such pairs that the celebrated Dormand–Prince pair also belongs. The chosen family comes with coefficients that all depend on five free parameters. These latter parameters are tuned in a way to furnish a new method that performs best on a couple of oscillators. Then, we observe that this trained pair outperforms other well known methods in the relevant literature in a standard set of problems with periodic solutions. This is remarkable since no special property holds such as high phase-lag order or an extended interval of periodicity.

**Keywords:** initial value problem; oscillatory problems; Runge–Kutta; differential evolution

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