Analysis of the dynamics of local error control via a piecewise continuous residual
- 48 Downloads
Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm  when applied to a variety of problems.
Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way.
AMS subject classification34C35 34D05 65L07 65L20 65L50
Key wordsError control continuous interpolants dissipativity contractivity gradient systems
Unable to display preview. Download preview PDF.
- 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.Google Scholar
- 6.J. K. Hale,Asymptotic Behaviour of Dissipative Systems, AMS Mathematical Surveys and Monographs 25, Rhode Island, 1988.Google Scholar
- 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.Google Scholar
- 9.H. Lamba and A. M. Stuart,Convergence results for the matlab ode23 routine, Preprint, 1997.Google Scholar
- 10.The Math Works, Inc.,MATLAB User's Guide, Natick, Massachusetts, 1992.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 15.A. M. Stuart and A. R. Humphries,Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.Google Scholar