Stiffness 19522012. Sixty years in search of a definition
(2015) In BIT Numerical Mathematics 55(2). p.531558 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/5265741
 author
 Söderlind, Gustaf ^{LU} ; Jay, Laurent and Manuel, Calvo
 organization
 publishing date
 2015
 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
 00063835
 DOI
 10.1007/s1054301405033
 language
 English
 LU publication?
 yes
 id
 a580a50e9f8948149fa77ec1353a442b (old id 5265741)
 date added to LUP
 20150529 12:48:25
 date last changed
 20170213 13:10:00
@article{a580a50e9f8948149fa77ec1353a442b, 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 = {00063835}, keyword = {Initial value problems,stability,logarithmic norms,stiffness,stiffness indicator,stiffness factor,reference time scale,step size}, language = {eng}, number = {2}, pages = {531558}, publisher = {Springer}, series = {BIT Numerical Mathematics}, title = {Stiffness 19522012. Sixty years in search of a definition}, url = {http://dx.doi.org/10.1007/s1054301405033}, volume = {55}, year = {2015}, }