Local interpolants for Numerov‐type methods and their implementation in variable step schemes

Abstract

The classical explicit fourth-order Numerov-type method is considered. The equations of condition for deriving the corresponding interpolants are given.Then using a local error estimation, we may construct a stable variable step scheme. Applying this new implementation in a set of problems, we get very pleasant results. KEYWORDS explicit hybrid Numerov, variable step, ′′ = (x, ) MSC CLASSIFICATION 65L05; 65L06 PRELIMINARY DISCUSSIONWe are interested in the special form of the initial value problem (IVP) of second-orderwith ∶ R × R m → R m and 0 , ′ 0 ∈ R m . Here, y ′ is not present in (1). Math Meth Appl Sci. 2019;42:7047-7058. wileyonlinelibrary.com/journal/mma

Interpolants for sixth‐order Numerov‐type methods

“…After the current step is completed, we have available the data: This could led only to a O ( h 5 ) interpolant, which is not enough . Thus, we proceed to construct a three‐step interpolant.…”

Variable step‐size implementation of sixth‐order Numerov‐type methods

“…For establishing a variable step algorithm, we have to form an error estimation. A similar approach was used for the fourth‐order methods . We consider another vector and try to determine its coefficients in order to approximate the solution at the fourth order of accuracy, ie, locally of O ( h 6 ).…”