Review Paper: Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations

Abstract

In recent years, after a prolonged period of intense activity, the study of numerical methods for solving stiff initial-value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity. There now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems. In this paper, we sketch some of the main theory which underpins stiff integration methods and we use this to describe, and put into context, some of the best codes currently available. By referencing only codes which have been thoroughly tested and are widely available, our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type. An additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated.