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Stiffness 1952-2012. Sixty years in search of a definition

Söderlind, Gustaf LU ; Jay, Laurent and Manuel, Calvo (2015) In BIT Numerical Mathematics 55(2). p.531-558
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Initial value problems, stability, logarithmic norms, stiffness, stiffness indicator, stiffness factor, reference time scale, step size
in
BIT Numerical Mathematics
volume
55
issue
2
pages
531 - 558
publisher
Springer
external identifiers
  • wos:000354704400009
  • scopus:84929712609
ISSN
0006-3835
DOI
10.1007/s10543-014-0503-3
language
English
LU publication?
yes
id
a580a50e-9f89-4814-9fa7-7ec1353a442b (old id 5265741)
date added to LUP
2015-05-29 12:48:25
date last changed
2017-02-13 13:10:00
@article{a580a50e-9f89-4814-9fa7-7ec1353a442b,
  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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.},
  author       = {Söderlind, Gustaf and Jay, Laurent and Manuel, Calvo},
  issn         = {0006-3835},
  keyword      = {Initial value problems,stability,logarithmic norms,stiffness,stiffness indicator,stiffness factor,reference time scale,step size},
  language     = {eng},
  number       = {2},
  pages        = {531--558},
  publisher    = {Springer},
  series       = {BIT Numerical Mathematics},
  title        = {Stiffness 1952-2012. Sixty years in search of a definition},
  url          = {http://dx.doi.org/10.1007/s10543-014-0503-3},
  volume       = {55},
  year         = {2015},
}