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  • CR: 5.17  (6)
  • AMS(MOS): 65 L05  (1)
  • watchdog strategy  (1)
  • 1
    Digitale Medien
    Digitale Medien
    Implementation issues in solving nonlinear equations for two-point boundary value problems (1990)
    Springer
    Computing 45 (1990), S. 17-37 
    Details
    ISSN: 1436-5057
    Schlagwort(e): 65L10 (boundary value problems) ; 68N99 (mathematical software) ; Boundary value problems ; nonlinear algebraic equations ; merit functions ; mathematical software ; watchdog strategy
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik
    Beschreibung / Inhaltsverzeichnis: Zusammenfassung Komplizierte numerische Methoden enthalten oft Teilprobleme, die sich leicht mathematisch formulieren lassen, die aber schwierig in Computerprogramme umgewandelt werden können. Mehrere solche Fragestellungen traten bei der Erstellung eines Newton-Verfahrens als Bestandteil eines Differenzenverfahrens für 2-Punkt Randwert-probleme auf. Wir beschreiben die praktischen und theoretischen Überlegungen, welche den Entscheidungen zugrunde liegen, die schließlich zum Computerprogramm führten. Insbesondere betonen wir dabei zwei “Watchdog”-Strategien, welche die Zuverlässigkeit verbessern und ein frühes Abbrechen der Newton-Iteration ermöglichen.
    Notizen: Abstract Complex numerical methods often contain subproblems that are easy to state in mathematical form, but difficult to translate into software. Several algorithmic isues of this nature arise in implementing a Newton iteration scheme as part of a finite-difference method for two-point boundary value problems. We describe the practical as well as theoretical considerations behind the decisions included in the final code, with special emphasis on two “watchdog” strategies designed to improve reliability and allow early termination of the Newton iterates.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02250582
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  • 2
    Digitale Medien
    Digitale Medien
    Split linear multistep methods for the numerical integration of stiff differential systems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 299-310 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389575
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  • 3
    Digitale Medien
    Digitale Medien
    On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 235-246 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396701
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  • 4
    Digitale Medien
    Digitale Medien
    On improving the absolute stability of local extrapolation (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 329-337 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396450
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  • 5
    Digitale Medien
    Digitale Medien
    On the design of high order exponentially fitted formulae for the numerical integration of stiff systems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 253-266 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65LO5 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Single step exponentially fitted integration formulae of orders 4 and 6 are derived. The final approximation to the required solution is obtained via a linear combination of asymptotically less accurate solutions obtained using conventional implicit integration formulae. This linear combination is performed in such a way that the final integration formula, as well as having an increased order of accuracy, integrates a certain pre-determined ordinary differential equation exactly. The algorithms developed are illustrated by means of some numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396654
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  • 6
    Digitale Medien
    Digitale Medien
    High order methods for the numerical integration of ordinary differential equations (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 385-409 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65 L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01398507
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  • 7
    Digitale Medien
    Digitale Medien
    High orderP-stable formulae for the numerical integration of periodic initial value problems (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 355-370 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy″=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400315
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