A review of recent developments in solving ODEs

Abstract

Mathematical models when simulating the behavior of physical, chemical, and biological systems often include one or more ordinary differential equations (ODEs). To study the system behavior predicted by a model, these equations are usually solved numerically. Although many of the current methods for solving ODEs were developed around the turn of the century, the past 15 years or so has been a period of intensive research. The emphasis of this survey is on the methods and techniques used in software for solving ODEs. ODEs can be classified as stiff or nonstiff, and may be stiff for some parts of an interval and nonstiff for others. We discuss stiff equations, why they are difficult to solve, and methods and software for solving both nonstiff and stiff equations. We conclude this review by looking at techniques for dealing with special problems that may arise in some ODEs, for example, discontinuities. Although important theoretical developments have also taken place, we report only those developments which have directly affected the software and provide a review of this research. We present the basic concepts involved but assume that the reader has some background in numerical computing, such as a first course in numerical methods.