A new class of explicit two-step fourth order methods for y″ = f( t, y) with extended intervals of periodicity

Abstract

We introduce a new class of explicit two-step fourth order methods for the numerical integration of second order initial value problems: y″ = f( t, y), y( t 0) = y 0, y′( t 0) = y′ 0. We show the interesting result that a method of this family which is based on m + 2 evaluations of f when applied to the test equation: y′ = − λ 2 y, λ > 0, possesses an interval of periodicity of length very nearly 2(( m + 1)( m + 3)) 1 2 . We also note a class of explicit second order methods for which a method based on m + 1 evaluations of f possesses an interval of periodicity of length very nearly 2( m + 1).