caa a22 4500 605497192 CHVBK 20210128100540.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s10543-014-0503-3 doi (NATIONALLICENCE)springer-10.1007/s10543-014-0503-3 Stiffness 1952-2012: Sixty years in search of a definition [Elektronische Daten] [Gustaf Söderlind, Laurent Jay, Manuel Calvo] 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. Springer Science+Business Media Dordrecht, 2014 Initial value problems nationallicence Stability nationallicence Logarithmic norms nationallicence Stiffness nationallicence Stiffness indicator nationallicence Stiffness factor nationallicence Reference time scale nationallicence Step size nationallicence Söderlind Gustaf Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden aut Jay Laurent Department of Mathematics, The University of Iowa, 14 MacLean Hall, 52242-1419, Iowa City, IA, USA aut Calvo Manuel Departamento de Matemática Aplicada, Pza. San Francisco s/n, Universidad de Zaragoza, 50009, Zaragoza, Spain aut BIT Numerical Mathematics Springer Netherlands 55/2(2015-06-01), 531-558 0006-3835 55:2<531 2015 55 10543 https://doi.org/10.1007/s10543-014-0503-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10543-014-0503-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Söderlind Gustaf Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden aut NATIONALLICENCE 700 1- Jay Laurent Department of Mathematics, The University of Iowa, 14 MacLean Hall, 52242-1419, Iowa City, IA, USA aut NATIONALLICENCE 700 1- Calvo Manuel Departamento de Matemática Aplicada, Pza. San Francisco s/n, Universidad de Zaragoza, 50009, Zaragoza, Spain aut NATIONALLICENCE 773 0- BIT Numerical Mathematics Springer Netherlands 55/2(2015-06-01), 531-558 0006-3835 55:2<531 2015 55 10543