Stiff Problems — One-Step Methods

Abstract

This chapter introduces stiff (styv (Swedish first!), steif (German), stìfar (Icelandic), stijf (Dutch), raide (French), kankea (Finnish), rígido (Spanish), stiff (Italian), merev (Hungarian), rigid (Rumanian), tog (Slovenian), čvrst (Serbo-Croatian), tuhý (Czecho-Slovak), sztywny (Polish), stign (Breton), ЖecTKИЙ (Russian), TBΓbPA (Bulgarian), חישק (Hebrew), ڧﺴ (Arabic), ڛﯼ (Urdu), ﺽ (Persian), ض (Sanscrit), ښ (Hindi), (Chinese), (Japanese), cuong (Vietnam), ngumu (Swahili) ...) differential equations. While the intuitive meaning of stiff is clear to all specialists, much controversy is going on about its correct mathematical definition (see e.g. Aiken 1985, p. 360–363). The most pragmatical opinion is also historically the first one (Curtiss & Hirschfelder 1952): stiff equations are equations where certain implicit methods,in particular BDF, perform better, usually tremendously better, than explicit ones. The eigenvalues of the Jacobian ∂f /∂y certainly play a role in this decision, but quantities such as the dimension of the system, the smoothness of the solution or the integration interval are also important (Sections IV.1 and IV.2).