Abstract
Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems.
Highlights
Preliminary DiscussionRemark that f k , zk ∈ Rm. Hairer [1], Chawla [2] and Cash [3] presented implicit Numerov-type techniques using off-step points for the first time around 40 years ago
Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
We are interested in the initial value problem (IVP) of the particular form: z00 = f (t, z), z(t0 ) = z0, z0 (t0 ) = z00, with regard to jurisdictional claims in published maps and institutional affiliations
Summary
Remark that f k , zk ∈ Rm. Hairer [1], Chawla [2] and Cash [3] presented implicit Numerov-type techniques using off-step points for the first time around 40 years ago. Tsitouras suggested a Runge–Kutta–Nyström (RKN)-style method [5] Just four steps are required to create a sixth-order method, whereas previous implementations required six function evaluations (see [6]). The main novelty here is that we will train the available free parameters in a wide set of relevant problems. For this training, we will use the differential evolution technique. It is believed that by using this methodology, we will conclude with a method better tuned for oscillatory problems
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