Abstract
The well-known method of Iterated Defect Correction (IDeC) is based on the following idea: Compute a simple, basic approximation and form its defect w.r.t. the given ODE via a piecewise interpolant. This defect is used to define an auxiliary, neighboring problem whose exact solution is known. Solving the neighboring problem with the basic discretization scheme yields a global error estimate. This can be used to construct an improved approximation, and the procedure can be iterated. The fixed point of such an iterative process corresponds to a certain collocation solution. We present a variety of modifications to this algorithm. Some of these have been proposed only recently, and together they form a family of iterative techniques, each with its particular advantages. These modifications are based on techniques like defect quadrature (IQDeC), defect interpolation (IPDeC), and combinations thereof. We investigate the convergence on locally equidistant and nonequidistant grids and show how superconvergent approximations can be obtained. Numerical examples illustrate our considerations. The application to stiff initial value problems will be discussed in Part II of this paper.
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References
R.P. Agarwal and P.J.Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications (Kluwer Academic, Dordrecht, 1993).
W. Auzinger, R. Frank and G. Kirlinger, Asymptotic error expansions for stiff equations: Applications, Computing 43 (1990) 223–253.
W. Auzinger, H. Hofstätter, W. Kreuzer and E. Weinmüller, Modified defect correction algorithms for ODEs. Part II: Stiff initial value problems, ANUM Preprint No. 2/03, Department of Applied Mathematics and Numerical Analysis, Vienna University of Technology (2003).
W. Auzinger, O. Koch and E. Weinmüller, Analysis of a new error estimate for collocation methods applied to singular boundary value problems, to appear in SIAM J. Numer. Anal.
W. Auzinger, O. Koch and E. Weinmüller, Efficient collocation schemes for singular boundary value problems, Numer. Algorithms 31 (2002) 5–25.
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, Contemporary Mathematics, Vol. 329 (Amer. Math. Soc., Providence, RI, 2003) pp. 43–50.
R. Frank and C. Überhuber, Iterated defect correction for differential equations, part I: Theoretical results, Computing 20 (1978) 207–228.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer, Berlin, 1987).
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed. (Springer, Berlin, 1996).
F.B. Hildebrand, Introduction to Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1974).
K.H. Schild, Gaussian collocation via defect correction, Numer. Math. 58 (1990) 369–386.
H.J. Stetter, The defect correction principle and discretization methods, Numer. Math. 29 (1978) 425–443.
P.E. Zadunaisky, On the estimation of errors propagated in the numerical integration of ODEs, Numer. Math. 27 (1976) 21–39.
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Auzinger, W., Hofstätter, H., Kreuzer, W. et al. Modified Defect Correction Algorithms for ODEs. Part I: General Theory. Numerical Algorithms 36, 135–155 (2004). https://doi.org/10.1023/B:NUMA.0000033129.73715.7f
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DOI: https://doi.org/10.1023/B:NUMA.0000033129.73715.7f