Eighth-order, phase-fitted, four-step methods for solving y′′=f(x,y)

A family of explicit, eighth-order, four-step methods for the numerical solution of (Formula presented.) is considered. This family is derived through an interpolatory approach after using three stages (ie, function evaluations) per step. In the present work, we alter three of the coefficients of such a method in order to become phase fitted. We conclude with numerical tests over a set of problems justifying our effort of dealing with the new methods. © 2019 John Wiley & Sons, Ltd

Interpolants for sixth-order Numerov-type methods

The classical four-stage family of explicit sixth-order Numerov-type method is considered. We provide two kinds of interpolants: (a) a three-step interpolation based on all available data at mesh points and (b) a local interpolant (ie, two steps) that is constructed after solving scaled equations of condition. These latter equations are explained and provided here. Applying these interpolants in a set of tests, we conclude that they produce global errors of the same magnitude with the underlying method. © 2019 John Wiley & Sons, Ltd