Analysis of the dynamics of local error control via a piecewise continuous residual

BIT Numerical Mathematics

Volumn 38, Issue 1, 1998, Pages 44-57

  Higham, D J ^a   Stuart, A M ^b  

^a UNIVERSITY OF STRATHCLYDE   (United Kingdom)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Continuous interpolants; Contractivity; Dissipativity; Error control; Gradient systems

Indexed keywords

EID: 0042240465     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02510916     Document Type: Article

References (17)

1. M. A. Aves, D. F. Griffiths and D. J. Higham, Does error control suppress spuriosity? SIAM J. Num. Anal., 34 (1997), pp. 756-778.
2. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Applied Mathematics 25, North-Holland, New York, 1992.
3. K. Dekker and J. G Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, 1984.
4. D. F. Griffiths, The dynamics of some linear multistep methods with step-size control, in Numerical Analysis 1987, D. F. Griffiths and G. A. Watson eds., Longman, 1988, pp. 115-134.
5. E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Springer-Verlag, New York, 1987.
6. J. K. Hale, Asymptotic Behaviour of Dissipative Systems, AMS Mathematical Surveys and Monographs 25, Rhode Island, 1988.
7. G. Hall, Equilibrium states of Runge-Kutta schemes, ACM Trans. Math. Software, 11 (1985), pp. 289-301.
8. D. J. Higham and A. M. Stuart, Analysis of the dynamics of local error control via a piecewise continuous residual, Stanford University Technical Report SCCM-95-03, 1995.
9. H. Lamba and A. M. Stuart, Convergence results for the MATLAB ode23 routine, Preprint, 1997.
10. The Math Works, Inc., MATLAB User's Guide, Natick, Massachusetts, 1992.
11. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
12. L. F. Shampine, Tolerance proportionality in ODE codes, in Numerical Methods for Ordinary Differential Equations (Proceedings), A. Bellen, C. W. Gear and E. Russo eds., Springer-Verlag, Lecture Notes 1386, 1987, pp. 118-136.
13. H. J. Stetter, Tolerance proportionality in ODE-codes, in Proc. Second Conf. on Numerical Treatment of Ordinary Differential Equations, R. März ed., Seminarberichte 32, Humboldt University, Berlin, 1980.
14. D. Stoffer and K. Nipp, Invariant curves for variable step-size integrators, BIT, 31 (1991), pp. 169-180 and (Erratum) BIT, 32 (1992), pp. 367-368.
15. A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.
16. A. M. Stuart and A. R. Humphries, The essential stability of local error control for dynamical systems, SIAM J. Num. Anal., 32 (1995), pp. 1940-1971.
17. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1989.