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Published in: BIT Numerical Mathematics 2/2015

01-06-2015

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

Authors: Gustaf Söderlind, Laurent Jay, Manuel Calvo

Published in: BIT Numerical Mathematics | Issue 2/2015

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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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Footnotes
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.
 
Literature
1.
go back to reference Artemiev, S., Averina, T.: Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations. VSP, Utrecht (1997)CrossRefMATH Artemiev, S., Averina, T.: Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations. VSP, Utrecht (1997)CrossRefMATH
2.
go back to reference Brugnano, L., Mazzia, F., Trigiante, D.: Fifty years of stiffness. In: Simos, T.E. (ed.) Recent Advances in Computational and Applied Mathematics, pp. 1–21. Springer, Berlin (2011)CrossRef Brugnano, L., Mazzia, F., Trigiante, D.: Fifty years of stiffness. In: Simos, T.E. (ed.) Recent Advances in Computational and Applied Mathematics, pp. 1–21. Springer, Berlin (2011)CrossRef
3.
4.
go back to reference Cash, J.R.: Efficient numerical methods for the solution of stiff initial-value problems and differential-algebraic equations. Proc. R. Soc. Lond. A 459, 797–815 (2003)CrossRefMATHMathSciNet Cash, J.R.: Efficient numerical methods for the solution of stiff initial-value problems and differential-algebraic equations. Proc. R. Soc. Lond. A 459, 797–815 (2003)CrossRefMATHMathSciNet
6.
go back to reference Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Almqvist & Wiksells, Uppsala (1959)MATH Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Almqvist & Wiksells, Uppsala (1959)MATH
7.
go back to reference Dahlquist, G.: A numerical method for some ordinary differential equations with large Lipschitz constants. In: Morrell, A.J.H. (ed.) Proceedings of IFIP Congress. Information Processing 68, Edinburgh, UK, vol. 1, Mathematics, Software, pp. 183–186 (1968) Dahlquist, G.: A numerical method for some ordinary differential equations with large Lipschitz constants. In: Morrell, A.J.H. (ed.) Proceedings of IFIP Congress. Information Processing 68, Edinburgh, UK, vol. 1, Mathematics, Software, pp. 183–186 (1968)
8.
go back to reference Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. CWI Monographs, vol. 2. North-Holland, Amsterdam (1984)MATH Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. CWI Monographs, vol. 2. North-Holland, Amsterdam (1984)MATH
9.
go back to reference Ekeland, K., Owren, B., Øines, E.: Stiffness detection and estimation of dominant spectrum with explicit Runge-Kutta methods. ACM Trans. Math. Softw. 24, 368–382 (1998)CrossRefMATH Ekeland, K., Owren, B., Øines, E.: Stiffness detection and estimation of dominant spectrum with explicit Runge-Kutta methods. ACM Trans. Math. Softw. 24, 368–382 (1998)CrossRefMATH
10.
go back to reference Gear, C.W.: Numerical initial value problems in ordinary differential equations. Prentice Hall, Englewood Cliffs (1971)MATH Gear, C.W.: Numerical initial value problems in ordinary differential equations. Prentice Hall, Englewood Cliffs (1971)MATH
11.
go back to reference Hairer, E.; Wanner, G.: Solving ordinary differential equations II. Stiff and differential-algebraic problems, second revised edition. Comput. Math., vol. 14. Springer, Berlin (1996) Hairer, E.; Wanner, G.: Solving ordinary differential equations II. Stiff and differential-algebraic problems, second revised edition. Comput. Math., vol. 14. Springer, Berlin (1996)
13.
go back to reference Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, London (1973)MATH Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, London (1973)MATH
15.
go back to reference Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comp. 28, 145–162 (1974)CrossRefMathSciNet Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comp. 28, 145–162 (1974)CrossRefMathSciNet
16.
17.
go back to reference Shampine, L.: What is stiffness? In: Aiken, R.C. (ed.) Stiff Computation. Oxford University Press, New York (1985) Shampine, L.: What is stiffness? In: Aiken, R.C. (ed.) Stiff Computation. Oxford University Press, New York (1985)
19.
Metadata
Title
Stiffness 1952–2012: Sixty years in search of a definition
Authors
Gustaf Söderlind
Laurent Jay
Manuel Calvo
Publication date
01-06-2015
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 2/2015
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0503-3

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