Preview
In this chapter, we shall look at several families of integration algorithms that all have in common the fact that only a single function evaluation needs to be performed in every integration step, irrespective of the order of the algorithm. Both explicit and implicit varieties of this kind of algorithms exist and shall be discussed. As in the last chapter, we shall spend some time discussing the stability and accuracy properties of these families of integration algorithms.
Whereas step-size and order control were easily accomplished in the case of the single-step techniques, these issues are much more difficult to tackle in the case of the multi-step algorithms. Consequently, their discussion must occupy a significant portion of this chapter.
The chapter starts out with mathematical preliminaries that shall simplify considerably the subsequent derivation of the multi-step methods.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
4.16 References
François E. Cellier and Hilding Elmqvist. Automated Formula Manipulation in Object-Oriented Continuous-System Modeling. IEEE Control Systems, 13(2):28–38, 1993.
François E. Cellier. Continuous System Modeling. Springer Verlag, New York, 1991. 755p.
Hilding Elmqvist. A Structured Model Language for Large Continuous Systems. PhD thesis, Dept. of Automatic Control, Lund Institute of Technology, Lund, Sweden, 1978.
Walter Gander and Dominik Gruntz. Derivation of Numerical Methods Using Computer Algebra. SIAM Review, 41(3):577–593, 1999.
C. William Gear. Numerical Initial Value Problems in Ordinary Differential Equations. Series in Automatic Computation. Prentice-Hall, Englewood Cliffs, N.J., 1971. 253p.
C. William Gear. Runge-Kutta Starters for Multistep Methods. ACM Trans. Math. Software, 6(3):263–279, 1980.
Curtis F. Gerald and Patrick O. Wheatley. Applied Numerical Analysis. Addison-Wesley, Reading, Mass., 6th edition, 1999. 768p.
Kjell Gustafsson. Control of Error and Convergence in ODE Solvers. PhD thesis, Dept. of Automatic Control, Lund Institute of Technology, Lund, Sweden, 1992.
Klaus Hermann. Solution of Stiff Systems Described by Ordinary Differential Equations Using Regression Backward Difference Formulae. Master’s thesis, Dept. of Electrical & Computer Engineering, University of Arizona, Tucson, Ariz., 1995.
John D. Lambert. Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley, New York, 1991. 304p.
William E. Milne. Numerical Solution of Differential Equations. John Wiley, New York, 1953. 275p.
Cleve Moler and Charles van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix. SIAM Review, 20(4):801–836, 1978.
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
(2006). Multi-step Integration Methods. In: Continuous System Simulation. Springer, Boston, MA. https://doi.org/10.1007/0-387-30260-3_4
Download citation
DOI: https://doi.org/10.1007/0-387-30260-3_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-26102-7
Online ISBN: 978-0-387-30260-7
eBook Packages: Computer ScienceComputer Science (R0)