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

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.

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Notes

  1. The analysis in this paper can be carried out with respect to any given norm with analogous results. However, for simplicity we have chosen to work with inner product norms, later further specialized to the usual Euclidean norm.

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Acknowledgments

The substantial input from the reviewers is gratefully acknowledged.

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Correspondence to Gustaf Söderlind.

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Communicated by Anne Kværnø.

In memory of our colleague and friend Jan G. Verwer (1946–2011).

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Söderlind, G., Jay, L. & Calvo, M. Stiffness 1952–2012: Sixty years in search of a definition. Bit Numer Math 55, 531–558 (2015). https://doi.org/10.1007/s10543-014-0503-3

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  • DOI: https://doi.org/10.1007/s10543-014-0503-3

Keywords

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

Mathematics Subject Classification (2000)

  • 65L04
  • 65L05