Fifty Years of Stiffness

TL;DR

This paper reviews the evolution of the concept of stiffness in differential problems over fifty years and offers a formal, comprehensive definition to aid in the development of robust numerical methods.

Contribution

It provides a historical overview and introduces a precise, unified definition of stiffness that consolidates previous diverse notions.

Findings

Historical evolution of stiffness concepts

A unified formal definition of stiffness

Implications for numerical method development

Abstract

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.