Stiffness in numerical initial-value problems

This paper reviews various aspects of stiffness in the numerical solution of initial-value problems for systems of ordinary differential equations. In the literature on numerical methods for solving initial value problems the term “stiff” has been used by various authors with quite different meanings, which often causes confusion. This paper attempts to clear up this confusion by reviewing some of these meanings and by giving a distinct difinition of a “stiff situation”. Further, the paper reviews classical as well as recent estimates, from the literature, of the Newton stopping error relevant to implicit step-by-step methods. These estimates illustrate the fact that the theoretical analysis of numerical procedures in the stiff situation generally requires more subtle arguments than in the nonstiff case. They also illustrate the interesting fact that classical error estimates (derived without taking stiffness into account) can be highly relevant in certain stiff situations while being deceptive in others. The paper concludes by presenting various open problems, and putting forward a conjecture, pertinent to the theoretical analysis of step-by-step methods in the stiff situation.