Abstract
We consider the scalar autonomous initial value problem as solved by an explicit Runge–Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge–Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.
Highlights
The Initial Value Problem (IVP) is given as published maps and institutional affilx0 x ( t0 ) iations.Licensee MDPI, Basel, Switzerland. = f (t, x ), = x0 (1)m and f : IR m → IR m. with t, t0 ∈ IR, · x, x 0 ∈ IR × IR
Runge–Kutta pairs are well suited for efficiently approximating the solution of nonstiff problems of the form in (1)
We concentrated on scalar autonomous problems and an extensively studied family of Runge–Kutta pairs of orders 5 and 6
Summary
Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems. 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. Administration of Businesses and Organizations Department, Hellenic Open University, 26335 Patras, Greece. E. Simos, 10 Konitsis Street, 17564 Athens, Greece
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