Abstract
As shown in part I of this paper and references therein, the classical method of Iterated Defect Correction (IDeC) can be modified in several nontrivial ways, extending the flexibility and range of applications of this approach. The essential point is an adequate definition of the defect, resulting in a significantly more robust convergence behavior of the IDeC iteration, in particular, for nonequidistant grids. The present part II is devoted to the efficient high-order integration of stiff initial value problems. By means of model problem investigation and systematic numerical experiments with a set of stiff test problems, our new versions of defect correction are systematically evaluated, and further algorithmic measures are proposed for the stiff case. The performance of the different variants under consideration is compared, and it is shown how strong coupling between non-stiff and stiff components can be successfully handled.
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References
W. Auzinger, A. Eder and R. Frank, Extending convergence theory for nonlinear stiff problems, part II, ANUM Preprint No. 22/01, Institite for Applied Mathematics and Numerical Analysis, Vienna University of Technology (2001).
W. Auzinger, R. Frank, H. Hofstätter and E. Weinmüller, Defektkorrektur zur numerischen Lösung steifer Anfangswertprobleme, Technical Report No. 131, Institut für Angewandte und Numerische Mathematik, TU Wien (2002).
W. Auzinger, R. Frank and G. Kirlinger, Asymptotic error expansions for stiff equations: Applications, Computing 43 (1990) 223–253.
W. Auzinger, R. Frank and G. Kirlinger, Extending convergence theory for nonlinear stiff problems, part I, BIT 36 (1996) 635–652.
W. Auzinger, R. Frank, W. Kreuzer and E. Weinmüller, Computergestützte Analyse verschiedener Varianten der Iterierten Defektkorrektur unter besonderer Berücksichtigung steifer Differentialgleichungen, ANUM Preprint No. 20/01, Department of Applied Mathematics and Numerical Analysis, Vienna University of Technology (2001).
W. Auzinger, H. Hofstätter, W. Kreuzer and E. Weinmüller, Modified defect correction algorithms for ODEs. Part I: General theory, Numer. Algorithms 36 (2004) 135–156.
W. Auzinger, O. Koch and E. Weinmüller, New variants of defect correction for boundary value problems in ordinary differential equations, in: Current Trends in Scientific Computing, eds. Z. Chen, R. Glowinski and K. Li, AMS Series in Contemporary Mathematics, Vol. 329 (Amer. Math. Soc., Providence, RI, 2003) pp. 43–50.
R. Frank and C.W. Ueberhuber, Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations, BIT 17 (1977) 146–159.
R. Frank and C.W. Ueberhuber, Iterated defect correction for differential equations, part I: Theoretical results, Computing 20 (1978) 207–228.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd ed. (Springer, Berlin, 1996).
H.R. Schwarz, Numerische Mathematik (Teubner, Stuttgart, 1997).
H.J. Stetter, The defect correction principle and discretization methods, Numer. Math. 29 (1978) 425–443.
H.J. Stetter, Personal communication.
P. Zadunaisky, On the estimation of errors propagated in the numerical integration of ODEs, Numer. Math. 27 (1976) 21–39.
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Communicated by J.C. Butcher
AMS subject classification
65L05
Supported by the Austrian Research Fund (FWF) grant P-15030.
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Auzinger, W., Hofstätter, H., Kreuzer, W. et al. Modified defect correction algorithms for ODEs. Part II: Stiff initial value problems. Numer Algor 40, 285–303 (2005). https://doi.org/10.1007/s11075-005-5327-4
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DOI: https://doi.org/10.1007/s11075-005-5327-4