Citations: Eine effiziente Ordnungs- und Schrittweitensteuerung unter Verwendung von Fehlerformeln fur variable Gitter und ihre Realisierung in Mehrschrittverfahren vom BDF-Typ

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Bleser, G. "Eine effiziente Ordnungs- und Schrittweitensteuerung unter Verwendung von Fehlerformeln fur variable Gitter und ihre Realisierung in Mehrschrittverfahren vom BDFTyp. " Diplomarbeit, Universitat Bonn (1986).

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New Stepsize Estimators for Linear Multistep Methods - Willé (2000) (Correct)

....be solved by Newton iteration. For an introduction to BDF methods see [6, 7, 8] However, substituting the true values fy(t i )g for fe y i g in (11) and Y k;n 1 (t n 1 ) we merely note that the local truncation error n 1 b Y k;n 1 (t n 1 ) f(t n 1 ; e y(t n 1 ) n 1 7 may be written [1, 5] n 1 = h Delta n 1 (t n 1 ) Delta y[t n 1 ; t n ; t n 1 Gammak ] where y[t n 1 ; t n . t n 1 Gammak ] is the Newton divided difference computed from the values fe y n 1 ; e y n ; e y n 1 Gammak g and n 1 (t) t Gamma t n 1 Gammal ) which thus implies n 1 ....

Bleser, G. "Eine effiziente Ordnungs- und Schrittweitensteuerung unter Verwendung von Fehlerformeln fur variable Gitter und ihre Realisierung in Mehrschrittverfahren vom BDFTyp. " Diplomarbeit, Universitat Bonn (1986).

Experiments in stepsize control for Adams linear multistep methods - Willé (2000) (Correct)

....of suitable orders and stepsizes is non trivial and has been the subject of much research. Given suitable methods to compute and advance the integration formulae, it can make the difference between a good and a bad code. A number of different approaches have been suggested for this problem [1, 2, 4, 5, 8, 9, 10, 11]. This work concentrates on and extends two of these: one implemented by Shampine in his code RDEAM [11] and a second, simpler, version based on the difference between a predictor and corrector formula. Communicated by Prof. C. T. H. Baker, this document originally appeared as an IWR preprint ....

....By retaining the notation and the variables of schemes used to advance the integration, the broader mathematical context is lost. The dependence of h is spread in a complicated way throughout several terms and extensive manipulation and assumptions are needed to make even the smallest advances [1, 4, 9]. Confronted with this problem, it is natural to take one step back. Recurrence relations (c.f. for g k 1;1 and g k;1 ) rescaling ratios (fi k 1 ) and heuristics are clearly important for the practical implementation of numerical codes but distract from the basic principles underlying their ....

Bleser, G. "Eine effiziente Ordnungs- und Schrittweitensteuerung unter Verwendung von Fehlerformeln fur variable Gitter und ihre Realisierung in Mehrschrittverfahren vom BDF-Typ." Diplomarbeit, University of Bonn (1986).

New Stepsize Estimators for Linear Multistep Methods - Wille (1994) (Correct)

....by Newton iteration. For an introduction to BDF methods see [6, 7, 8] However, substituting the true values fy(t i )g for fe y i g in (11) and Y 0 k;n 1 (t n 1 ) we merely note that the local truncation error n 1 b Y 0 k;n 1 (t n 1 ) f(t n 1 ; e y(t n 1 ) n 1 7 may be written [1, 5] n 1 = h Delta 0 n 1 (t n 1 ) Delta y[t n 1 ; t n ; t n 1 Gammak ] where y[t n 1 ; t n . t n 1 Gammak ] is the Newton divided difference computed from the values fe y n 1 ; e y n ; e y n 1 Gammak g and n 1 (t) k Y l=0 (t Gamma t n 1 Gammal ) which thus ....

Bleser, G. "Eine effiziente Ordnungs- und Schrittweitensteuerung unter Verwendung von Fehlerformeln fur variable Gitter und ihre Realisierung in Mehrschrittverfahren vom BDFTyp. " Diplomarbeit, Universitat Bonn (1986).

Experiments in stepsize control for Adams linear multistep methods - Wille (1994) (Correct)