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.
Highlights
The initial value problem (IVP) is Publisher’s Note: MDPI stays neutral y0 = f ( x, y), y( x0 ) = y0 with regard to jurisdictional claims in published maps and institutional affiliations.with x0 ∈ R, y, y0 ∈ Rm and f : R × Rm → Rm .Runge–Kutta (RK) pairs are amongst the most popular numerical methods for addressing (1)
C4 = 0.245431154837642, c7 = 0.8101151362080617, b2 = b3 = 0, b6 = −0.2872484367615202, b9 = 0 b b4 = 0.291009331941132, b b7 = 0.429687174664803, a32 = 0.077309649426606, a43 = 0.184073366128232, a53 = −0.430121641642955, a62 = 0, a65 = 2.155104563890663, a73 = 1.062724280290705, a76 = 0.045649127892262, a83 = 4.607278279969559, a86 = −0.399249451366671, Interpreting Table 1, we observe that DLMP6(5) was 151% and 116% more expensive than NEW6(5) for the first and second run, respectively
This paper is concerned with training the coefficients of a Runge–Kutta pair for addressing a certain kind of problem
Summary
We freely chose the coefficients c2 , c4 , c5 , c6 , c7 and b9 Pairs from this family have been proven to perform most efficiently in various classes of problems [17]. 1 2 3 4 T y1 , y1 , y1 , y1 , y2 = 1 y2 ,2 y2 ,3 y2 ,4 y2 , y3 , · · · , which represent the vectors approximating the solution at x1 , x2 , x3 , · · · This problem can be solved with an RK6(5) pair from the family we are interested in here. The optimization furnished six values for the free parameters It is difficult to believe a special performance could be obtained after seeing its traditional characteristics
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