Abstract
Multistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas (for stiff equations) require special provision for changing the stepsize. New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize for such methods. Error estimation and some of the important factors in stepsize selection and the selection of integration order are also considered.
Keywords
- Local Error
- Variable Order
- Multistep Method
- Divided Difference
- Differentiation Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract NAS7-100, sponsored by the National Aeronautics and Space Administration.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Krogh, F. T.: Algorithms for Changing the Stepsize Used by a Multistep Method. JPL Internal Document, Section 314 TM No. 275, Jet Propulsion Laboratory, Pasadena, Calif. (Oct. 1970). (To appear in the SIAM Journal on Numerical Analysis.)
Krogh, F. T.: A Variable Step Variable Order Multistep Method for the Numerical Solution of Ordinary Differential Equations. Information Processing 68 (Proceeding of the IFIP Congress 1968.) 194–199. North Holland Publishing Co., Amsterdam (1969).
Forrington, C. V. D.: Extensions of the Predictor-Corrector Method for the Solution of Systems of Ordinary Differential Equations. Computer Journal 4 (1961–62), 80–84.
Gabel, G. F.: A Predictor-Corrector Method Using Divided Differences. University of Toronto Department of Computer Science Technical Report No. 5 (October 1968).
Piotrowski, P.: Stability, Consistency and Convergence of Variable k-Step Methods for Numerical Integration of Large Systems of Ordinary Differential Equations. Conference on the Numerical Solution of Differential Equations, Lecture Notes in Mathematics 109, 221–227. Springer-Verlag, Berlin (1969).
Van Wyk, R.: Variable Mesh Multistep Methods for Ordinary Differential Equations. J. Comp. Physics 5 (1970), 244–264.
Brayton, R. K., Gustavson, F. G. and Hachtel, G. D.: A New Algorithm for Solving Differential-Algebraic Systems Using Implicit Backward Differentiation Formulae. Proceedings of the IEEE 60 (1972), 98–108.
Curtiss, C. F. and Hirschfelder, J. O.: Integration of Stiff Equations. Proc. of the Nat. Acad. of Sci. 38 (1952), 235–243.
Gear, C. W.: The Automatic Integration of Stiff Ordinary Differential Equations. Information Processing 68 (Proceeding of the IFIP Congress 1968) 187–193. North Holland Publishing Co., Amsterdam (1969).
Gear, C. W.: The Automatic Integration of Ordinary Differential Equations. Comm. of the ACM 14 (1971), 176–179.
Rutishauser, H.: Bemerkungen zur Numerischen Integration Gewöhnlicher Differential Gleichunger n-ter Ordnung. Numerische Mathematik 2 (1960), 263–279.
Krogh, F. T.: The Numerical Integration of Stiff Differential Equations. TRW Report No. 99900-6573-R000, TRW Systems, Redondo Beach, Calif. (March 1968).
Klopfenstein, R. W.: Numerical Differentiation Formulas for Stiff Systems of Ordinary Differential Equations. RCA Review 32, (1971), 447–462.
Shampine, L. F. and Gordon, M. K.: Local Error and Variable Order, Variable Step Adams Codes. (Submitted to the SIAM Journal on Numerical Analysis.)
Krogh, F. T.: VODQ/SVDQ/DVDQ-Variable Order Integrators for the Numerical Solution of Ordinary Differential Equations. TU Doc. No. CP-2308, NPO-11643, Jet Propulsion Laboratory, Pasadena, Calif. (May 1969).
Krogh, F. T.: On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations. JPL Internal Document, Section 314 TM No. 217 (revised), Jet Propulsion Laboratory, Pasadena, Calif. (Oct 1970). (To appear in the Journal of the ACM.)
Hull, T. E., Enright, W. H., Fellen, B. M. and Sedgwick, A. E.: Comparing Numerical Methods for Ordinary Differential Equations. SIAM J. Numerical Analysis 9 (1972), 603–637.
Gear, C. W.: The Numerical Integration of Ordinary Differential Equations. Math. of Comp. 21 (1967), 146–156.
Thomas, L. H.: The Integration of Ordinary Differential Systems. The Ohio State University Engineering Experiment Station News 24 (1952), 8–9, 31–32.
Krogh, F. T.: A Method for Simplifying the Maintenance of Software Which Consists of Many Versions. JPL Internal Document, Section 914 TM No. 314, Jet Propulsion Laboratory, Pasadena, Calif. (Sept. 1972).
Krogh, F. T.: An Integrator Design. JPL TM No. 33-479, Jet Propulsion Laboratory, Pasadena, Calif. (May 1971).
Krogh, F. T.: Opinions on Matters Connected with the Evaluation of Programs and Methods for Integrating Ordinary Differential Equations. SIGNUM Newsletter 7 No. 3 (Oct. 1972).
Krogh, F. T.: A Test for Instability in the Numerical Solution of Ordinary Differential Equations. Journal of the ACM 14 (1967), 351–354.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Springer-Verlag
About this paper
Cite this paper
Krogh, F.T. (1974). Changing stepsize in the integration of differential equations using modified divided differences. In: Bettis, D.G. (eds) Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations. Lecture Notes in Mathematics, vol 362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066584
Download citation
DOI: https://doi.org/10.1007/BFb0066584
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06602-6
Online ISBN: 978-3-540-37911-9
eBook Packages: Springer Book Archive
