Convergence results for the MATLAB ode23 routine
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We prove convergence results on finite time intervals, as the user-defined tolerance τ→0, for a class of adaptive timestepping ODE solvers that includes the ode23 routine supplied in MATLAB Version 4.2. In contrast to existing theories, these convergence results hold with error constants that are uniform in the neighbourhood of equilibria; such uniformity is crucial for the derivation of results concerning the numerical approximation of dynamical systems. For linear problems the error estimates are uniform on compact sets of initial data. The analysis relies upon the identification of explicit embedded Runge-Kutta pairs for which all but the leading order terms of the expansion of the local error estimate areO(∥f(u∥)2).
AMS subject classification34C35 65L07 65L20 65L50
Key wordsError control adaptivity convergence tolerance proportionality
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- 1.MATLAB, The Math Works Inc., Natick, MA.Google Scholar
- 7.H. Lamba,Dynamical systems and adaptive time-stepping ODE solvers. SCCM Technical Report, Stanford University, 1998.Google Scholar
- 8.H. Lamba and A. M. Stuart,Convergence proofs for numerical IVP software. SCCM Technical Report, Stanford University, 1998.Google Scholar
- 11.H. J. Stetter,Considerations concerning a theory for ODE solvers, in Numerical Treatment of Differential Equations, Lecture Notes in Mathematics 631, Springer-Verlag, Berlin, 1976.Google Scholar
- 12.H. J. Stetter,Tolerance proportionality in ODE codes, in Proceedings Second Conference on Numerical Treatment of Ordinary Differential Equations, Humboldt University, 1980.Google Scholar
- 15.A. M. Stuart and A. R. Humphries,Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.Google Scholar