Explicit, Eighth-Order, Four-Step Methods for Solving \(y^{\prime \prime }=f(x,y)\)

A family of explicit, eighth-order, four-step methods for the numerical solution of \(y^{\prime \prime }=f(x,y)\) is studied. This family is derived through an interpolatory approach after using three stages (i.e., function evaluations) per step. Three coefficients of the methods in this family remain free. Thus we may use them for achieving zero-stability, non-empty intervals of periodicity or absolute stability and reducing the phase lag. We may even construct a method that attains ninth algebraic order for scalar autonomous problems. A discussion is given about various numerical instabilities that are usually present in such type of multistep methods, and it is shown how to circumvent them. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.