Stiffness 1952-2012: Sixty years in search of a definition
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| 024 | 7 | 0 | |a 10.1007/s10543-014-0503-3 |2 doi |
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| 245 | 0 | 0 | |a Stiffness 1952-2012: Sixty years in search of a definition |h [Elektronische Daten] |c [Gustaf Söderlind, Laurent Jay, Manuel Calvo] |
| 520 | 3 | |a 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 $$\Delta t$$ Δ t , which may vary with time and state along the solution. By comparing $$\Delta t$$ Δ t to the range of integration $$T$$ T , a large stiffness factor $$T/\Delta t$$ T / Δ t is a necessary condition for stiffness. In numerical computations, $$\Delta t$$ Δ t can be compared to the actual step size $$h$$ h , whose stiffness factor $$h/\Delta t$$ h / Δ t depends on the choice of integration method. Thus $$\Delta t$$ Δ t embodies the mathematical aspects of stiffness, while $$h$$ 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. | |
| 540 | |a Springer Science+Business Media Dordrecht, 2014 | ||
| 690 | 7 | |a Initial value problems |2 nationallicence | |
| 690 | 7 | |a Stability |2 nationallicence | |
| 690 | 7 | |a Logarithmic norms |2 nationallicence | |
| 690 | 7 | |a Stiffness |2 nationallicence | |
| 690 | 7 | |a Stiffness indicator |2 nationallicence | |
| 690 | 7 | |a Stiffness factor |2 nationallicence | |
| 690 | 7 | |a Reference time scale |2 nationallicence | |
| 690 | 7 | |a Step size |2 nationallicence | |
| 700 | 1 | |a Söderlind |D Gustaf |u Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden |4 aut | |
| 700 | 1 | |a Jay |D Laurent |u Department of Mathematics, The University of Iowa, 14 MacLean Hall, 52242-1419, Iowa City, IA, USA |4 aut | |
| 700 | 1 | |a Calvo |D Manuel |u Departamento de Matemática Aplicada, Pza. San Francisco s/n, Universidad de Zaragoza, 50009, Zaragoza, Spain |4 aut | |
| 773 | 0 | |t BIT Numerical Mathematics |d Springer Netherlands |g 55/2(2015-06-01), 531-558 |x 0006-3835 |q 55:2<531 |1 2015 |2 55 |o 10543 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10543-014-0503-3 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10543-014-0503-3 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Söderlind |D Gustaf |u Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Jay |D Laurent |u Department of Mathematics, The University of Iowa, 14 MacLean Hall, 52242-1419, Iowa City, IA, USA |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Calvo |D Manuel |u Departamento de Matemática Aplicada, Pza. San Francisco s/n, Universidad de Zaragoza, 50009, Zaragoza, Spain |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t BIT Numerical Mathematics |d Springer Netherlands |g 55/2(2015-06-01), 531-558 |x 0006-3835 |q 55:2<531 |1 2015 |2 55 |o 10543 | ||