A Symmetric $k$-Step Method for Direct Integration of Second Order Initial Value Problems of Ordinary Differential Equations

Abstract

The design and implementation analysis of a $k$-step linear multistep method for direct integration of second order initial value problems of ordinary differential equations without reformulation into first order systems is discussed. The derivation of the method and analysis of its basic properties are adopted from the Taylor series expansion and Dahlquist stability model test methods. The result when examined with step-number $k=6$ shows that the scheme is symmetric, consistent, zero-stable, and convergent.