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Evolutionary derivation of Runge–Kutta pairs for addressing inhomogeneous linear problems

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Abstract

Two new Runge–Kutta (RK) pairs of orders 6(4) and 7(5) are presented for solving numerically the inhomogeneous linear initial value problems with constant coefficients. These new pairs use only six and eight stages per step respectively. Six stages are needed for conventional Runge–Kutta pairs of orders 5(4) while for such a pair of orders 6(5) we use eight stages. Thus, our proposal is an improvement and it is achieved since the set of order conditions is smaller in the case of interest here. Since traditional simplifications for derivation of Runge–Kutta methods do not apply for this reduced set, we proceed using the differential evolution technique for solving it. We finalize by performing tests over some relevant problems with very pleasant results.

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Authors and Affiliations

  1. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, People’s Republic of China

    T. E. Simos

  2. South Ural State University, 76, Lenin Avenue, Chelyabinsk, 454080, Russia

    T. E. Simos

  3. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Dongtong Road 705, Neijiang, 641100, China

    T. E. Simos

  4. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece

    T. E. Simos

  5. General Department, National and Kapodistrian University of Athens, Euripus Campus, GR34400, Greece

    Ch. Tsitouras

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Highly Cited Researcher (2001-2013 List, 2017 List and 2018 List), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of European Academy of Arts, Sciences and Humanities.

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Simos, T.E., Tsitouras, C. Evolutionary derivation of Runge–Kutta pairs for addressing inhomogeneous linear problems. Numer Algor 87, 511–525 (2021). https://doi.org/10.1007/s11075-020-00976-9

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