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An advanced version of local-global step size control for multistep methods

Kulikov, Gennady Yu; Shindin, S.K.

Proceedings of the International Conference on Computational Mathematics (ICCM 2002): Part II. ICMMG Publisher, Novosibirsk, Russia, ISBN 5-901548-10-8 (2002), 567-573

In a recent paper we presented a new procedure to control stepsize for linear multistep methods applied to index 1 differential-algebraic equations $$ x'(t)=gbigl(x(t),y(t),alpha(t)bigr), quad y(t)=fbigl(x(t),y(t),alpha(t)bigr) eqno (1) $$ where $tin [t_0,t_0+T]$, $x(t)in mbox{bf R}^m$, $y(t) in mbox{bf R}^n$, $alpha(t)in mbox{bf R}^l$, $alpha(t)$ is a known vector, $ g:Dsubset , mbox{bf R}^{m+n+l} to mbox{bf R}^m $, $ f:Dsubset , mbox{bf R}^{m+n+l} to mbox{bf R}^n $, and where initial conditions $x(t_0)=x^0$, $y(t_0)=y^0$ are consistent: $y^0=f(x^0,y^0,alpha(t_0))$. In contrast to the standard approach, the new error control mechanism was based on monitoring and controlling both the local and global errors of multistep formulas. As a result, such methods with the local-global stepsize control solve differential-algebraic equations with any prescribed accuracy (up to round-off errors). For implicit multistep methods we gave the minimum number of both full and modified Newton iterations allowing the iterative approximations to be correctly used in the procedure of the local-global stepsize control. We also discussed validity of simple iterations for high accuracy solving differential-algebraic equations of index~1. The only drawback of the local-global stepsize control was high execution time for some test problems. It was a consequence of recomputing the numerical solution as many times as the error control mechanism had required that. In order to decrease the execution time we develop a new advanced version of the local-global stepsize control. We show that the new local-global stepsize control is not only effective but it is also reliable in practice. Numerical tests support the theoretical results of the paper.