A Neural Network Type Approach for Constructing Runge–Kutta Pairs of Orders Six and Five That Perform Best on Problems with Oscillatory Solutions

Houssem Jerbi,Charalampos Tsitouras,Sondess Ben Aoun,Theodore E Simos,Mohamed Omri

doi:10.3390/math10050827

Abstract

We analyze a set of explicit Runge–Kutta pairs of orders six and five that share no extra properties, e.g., long intervals of periodicity or vanishing phase-lag etc. This family of pairs provides five parameters from which one can freely pick. Here, we use a Neural Network-like approach where these coefficients are trained on a couple of model periodic problems. The aim of this training is to produce a pair that furnishes best results after using certain intervals and tolerance. Then we see that this pair performs very well on a wide range of problems with periodic solutions.

Highlights

The Initial Value problem (IVP) is y = f (x, y), y(x0) = y0 (1)with x0 a real number, y, y ∈ Rm and f : R × Rm → Rm
For the above selection of free parameters, we obtained u1,NEW65 = 19.30 while for DLMP6(5) pair found in [13] we observe u1,DLMP65 = 71.09, i.e., u1,DLMP65 ≈ 3.68, u1,NEW65 meaning that the latter pair is about 268% more expensive than delivering the same accuracy
All the coefficients of this pair are expressed with respect to a set of free parameters

Summary

Introduction

With x0 a real number, y , y ∈ Rm and f : R × Rm → Rm. Runge–Kutta (RK) pairs are the most widely used numerical schemes for tackling problems (1). All currently known pairs (except those derived by Sharp and Smart [11]) effectively use eight stages (i.e., they use eight stages, with or without the first stage of the step) In such a case, the number of available free coefficients in A, b, b, c, is 44 or 45, depending on whether the FSAL (First Stage As Last) device is employed or not. We focus on a special kind of simplifying assumption that forms a family of solutions for the order conditions. This family of solutions (and in consequence RK pairs) was proposed simultaneously by Dormand et al [12] and Verner [13]. Methods with small phase-lag are ideal for use in situations with periodic solutions

Training the Coefficients

Findings

Conclusions