Abstract
The family of Numerov-type methods that effectively uses seven stages per step is considered. All the coefficients of the methods belonging to this family can be expressed analytically with respect to four free parameters. These coefficients are trained through a differential evolution technique in order to perform best in a wide range of Keplerian-type orbits. Then it is observed with extended numerical tests that a certain method behaves extremely well in a variety of orbits (e.g., Kepler, perturbed Kepler, Arenstorf, Pleiades) for various steplengths used by the methods and for various intervals of integration.
Highlights
We are interested in the particular version of the Initial Value Problem (IVP): y00 = f (t, y), y(t0 ) = y0, y0 (t0 ) = y00, Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
After choosing the free parameters c3, c4, c5, a64, we get a method named NEW8 and form Tables like those presented in the previous section
The optimization via differential evolution produced a number of quadruplets for the parameters, and we provide the chosen one, in double precision, below, c3 = −0.4821271178014236, c4 = −0.1599331990972641, c5 = −0.81752579390977, a64 = 2.118887522290334
Summary
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641100, China. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece.
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