A Technical Note: Two-Step PECE Methods for Approximating Solutions To First- and Second-Order ODEs

TL;DR

This paper introduces two-step PECE predictor/corrector methods for efficiently solving various classes of first- and second-order ODEs, with error control and initial step strategies.

Contribution

It presents novel two-step PECE methods tailored for different ODE problem classes, including compatible one-step methods and step size control algorithms.

Findings

Effective in solving systems of ODEs with controlled local error

Provides compatible one-step methods for initialization

Demonstrates improved accuracy and stability over existing methods

Abstract

Two-step predictor/corrector methods are provided to solve three classes of problems that present themselves as systems of ordinary differential equations (ODEs). In the first class, velocities are given from which displacements are to be solved. In the second class, velocities and accelerations are given from which displacements are to be solved. And in the third class, accelerations are given from which velocities and displacements are to be solved. Two-step methods are not self starting, so compatible one-step methods are provided to take that first step with. An algorithm is presented for controlling the step size so that the local truncation error does not exceed a specified tolerance.