BIT Numerical Mathematics

, Volume 55, Issue 2, pp 531–558 | Cite as

Stiffness 1952–2012: Sixty years in search of a definition

Article

Abstract

Although stiff differential equations is a mature area of research in scientific computing, a rigorous and computationally relevant characterization of stiffness is still missing. In this paper, we present a critical review of the historical development of the notion of stiffness, before introducing a new approach. A functional, called the stiffness indicator, is defined terms of the logarithmic norms of the differential equation’s vector field. Readily computable along a solution to the problem, the stiffness indicator is independent of numerical integration methods, as well as of operational criteria such as accuracy requirements. The stiffness indicator defines a local reference time scale\(\Delta t\), which may vary with time and state along the solution. By comparing \(\Delta t\) to the range of integration \(T\), a large stiffness factor\(T/\Delta t\) is a necessary condition for stiffness. In numerical computations, \(\Delta t\) can be compared to the actual step size \(h\), whose stiffness factor \(h/\Delta t\) depends on the choice of integration method. Thus \(\Delta t\) embodies the mathematical aspects of stiffness, while \(h\) accounts for its numerical and operational aspects.To demonstrate the theory, a number of highly nonlinear test problems are solved. We show, inter alia, that the stiffness indicator is able to distinguish the complex and rapidly changing behavior at (locally unstable) turning points, such as those observed in the van der Pol and Oregonator equations. The new characterization is mathematically rigorous, and in full agreement with observations in practical computations.

Keywords

Initial value problems Stability Logarithmic norms Stiffness Stiffness indicator Stiffness factor  Reference time scale Step size 

Mathematics Subject Classification (2000)

65L04 65L05 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Centre for Mathematical Sciences, Lund UniversityLundSweden
  2. 2.Department of MathematicsThe University of IowaIowa CityUSA
  3. 3.Departamento de Matemática AplicadaPza. San Francisco s/n, Universidad de ZaragozaZaragozaSpain

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