Linear Multistep Numerical Methods for Ordinary Differential Equations

TL;DR

This paper reviews popular Linear Multistep methods for numerically solving Ordinary Differential Equations, discussing their derivation, stability, and practical advantages and disadvantages.

Contribution

It provides a comprehensive overview of key LM methods, including Adams-Bashforth, Adams-Moulton, and Backwards Differentiation Formulas, with insights into their stability and applicability.

Findings

Analysis of stability properties of LM methods

Comparison of advantages and disadvantages

Clarification of stability types and their implications

Abstract

A review of the most popular Linear Multistep (LM) Methods for solving Ordinary Differential Equations numerically is presented. These methods are first derived from first principles, and are discussed in terms of their order, consistency, and various types of stability. Particular varieties of stability that may not be familiar, are briefly defined first. The methods that are included are the Adams-Bashforth Methods, Adams-Moulton Methods, and Backwards Differentiation Formulas. Advantages and disadvantages of these methods are also described. Not much prior knowledge of numerical methods or ordinary differential equations is required, although knowledge of basic topics from calculus is assumed.