A local-global stepsize control for multistep methods applied to semi-explicit index 1 differential-algebraic equations
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In this paper we develop a new procedure to control stepsize for linear multistep methods applied to semi-explicit index 1 differential-algebraic equations. In contrast to the standard approach, the error control mechanism presented here is 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 give 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 discuss validity of simple iterations for high accuracy solving differential-algebraic equations. Numerical tests support the theoretical results of the paper.
AMS Mathematics Subject Classification65L06
Key word and phrasesdifferential-algebraic equations multistep methods local and global errors estimates error control mechanism
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- 9.E. Hairer, G. Wanner,Solving ordinary differential equations II: Stiff and differentialalgebraic problems, Springer-Verlag, Berlin, 1991.Google Scholar
- 11.G.Yu. Kulikov,A method for the numerical solution of the autonomous Cauchy problem with an algebraic restriction of the phase variables, (in Russian) Vestn. MGU Ser. 1 Mat. Mekh., (1992), No. 1, 14–19;translation in Moscow Univ. Math. Bull.47 (1992), No. 1, 14–18.Google Scholar
- 12.G.Yu. Kulikov,The numerical solution of an autonomous Cauchy problem with an algebraic constraint on the phase variables (the nonsingular case), (in Russian) Vestn. MGU Ser. 1 Mat. Mekh., (1993), No. 3, 6–10;translation in Moscow Univ. Math. Bull.48 (1993), No. 3, 8–12.Google Scholar
- 14.G.Yu. Kulikov,The numerical solution of the Cauchy problem with algebraic constrains on the phase variables (with applications in medical cybernetics), Dissertation of Candidate of Sciences in Mathematics, Computing Center of Russian Academy of Sciences, Moscow, 1994.Google Scholar
- 15.G.Yu. Kulikov, P.G. Thomsen,Convergence and implementation of implicit RungeKutta methods for DAEs, Technical report 7/1996, IMM, Technical University of Denmark, Lyngby, 1996.Google Scholar
- 18.G.Yu. Kulikov,Stable local-global stepsize control for Runge-Kutta methods, preprint numerics No. 3/1997, Mathematical Sciences Div., Norwegian University of Science and Technology, Trondheim, 1997.Google Scholar
- 19.G.Yu. Kulikov,Numerical solution of the Cauchy problem for a system of differentialalgebraic equations with the use of implicit Runge-Kutta methods with nontrivial predictor, (in Russian) Zh. Vychisl. Mat. Mat. Fiz.38 (1998), No. 1, 68–84;translation in Comp. Maths Math. Phys.38 (1998), No. 1, 64–80.MathSciNetGoogle Scholar
- 20.A. Kværnø,The order of Runge-Kutta methods applied to semi-explicit DAEs of index 1, using Newton-type iterations to compute the internal stage values, Technical report 2/1992, Mathematical Sciences Div., Norwegian Institute of Technology, Trondheim, 1992.Google Scholar