Fitted modifications of Runge–Kutta–Nyström pairs of orders 7(5) for addressing oscillatory problems

Abstract

Runge-Kutta-Nyström pair of orders 7(5) using six stages per step have been discovered very recently. Here we modify four of its weights. The resulting method integrates exactly the harmonic oscillator 𝜓 ′′ = −𝜇 2 𝜓, 𝜇 ∈ R, which serves as model problem. The new weights are O(𝜇 2 ) perturbations of the original ones. Order reduction which is usually present in such modifications is avoided.Numerical results over standard six stages pairs justify our efforts.

Framework for solving dynamics of Ca2+ ion concentrations in liver cells numerically: Analysis of a non‐negativity‐preserving non‐standard finite‐difference method

“… • Since our proof of convergence in Theorem 4 does not depend on equidistant time‐step sizes, adaptive time‐stepping algorithms might be an interesting extension of our work as presented by Nüßlein et al [13] or by Söderlind [22]. • Higher order time‐stepping schemes can be addressed as another interesting complement to our work [23, 24]. Since Algorithm 2 is explicit, there might be a chance that we can apply standard techniques, comparable to applied in Runge–Kutta integration schemes, to construct higher order accurate schemes for system () which preserve non‐negativity for numerical solution of system ().…”