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

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.

Midway reassignment of Runge–Kutta step length

“…In order to check the improvements offered by the new algorithm, we present in Tables 5–10 all the characteristics of the results. In the final column we present an efficiency measure according to the formula [9], $$$$ \mathrm{efficiency}=\mathrm{total}\kern0.4em \mathrm{steps}\cdotp {\left(\mathrm{end}\kern0.4em \mathrm{point}\kern0.4em \mathrm{error}\right)}^{1/5}. $$$$ …”

“…In order to check the improvements offered by the new algorithm, we present in Tables 5-10 all the characteristics of the results. In the final column we present an efficiency measure according to the formula [9], efficiency = total steps•(end point error) 1∕5 .…”

Medical Image Encryption Based on Josephus Traversing and Hyperchaotic Lorenz System

“…Based on the above parameters, using the classic four-order Runge-Kutta method after discretization [28] , the projections of hyperchaotic attractors on each plane are given in Fig. 2, where x n , y n , z n and w n represent the number of total state points in order to describe the pairwise relations of hyperchaotic systems after order reduction.…”

Derivation of Three-Derivative Two-Step Runge–Kutta Methods

“…The Runge-Kutta (RK) methods [1][2][3] are the most widely employed numerical schemes for solving ordinary differential equations (ODEs) and partial differential equations (PDEs). With the increasing demand for high-order accuracy methods, the classical high-order Runge-Kutta methods have exhibited some unavoidable shortcomings [4], such as the increased number of stages and computational time.…”