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Composition constants for raising the orders of unconventional schemes for ordinary differential equations


Authors: William Kahan and Ren-Cang Li
Journal: Math. Comp. 66 (1997), 1089-1099
MSC (1991): Primary 34A50, 65L05
DOI: https://doi.org/10.1090/S0025-5718-97-00873-9
MathSciNet review: 1423077
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Abstract: Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.


References [Enhancements On Off] (What's this?)

  • 1. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Springer-Verlag, New York, second ed., 1993. MR 94c:65005
  • 2. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, New York, 1991. MR 92a:65016
  • 3. N. J. Higham, The accuracy of floating point summation, SIAM J. Sci. Comput., 14 (1993), pp. 783-799. MR 94a:65025
  • 4. W. Kahan, Relaxation methods for solving systems of ordinary differential equations, manuscript, CS Division, Department of EECS, University of California at Berkeley, Oct. 1977.
  • 5. height 2pt depth -1.6pt width 23pt, Analysis and application of simply compensated summation, work in progress, CS Division, Department of EECS, University of California at Berkeley, Oct. 1993.
  • 6. height 2pt depth -1.6pt width 23pt, Unconventional numerical methods for trajectory calculations, lectures notes, CS Division, Department of EECS, University of California at Berkeley, Oct. 1993.
  • 7. J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, New York, 1991. MR 92i:65114
  • 8. R.-C. Li, Raising the Orders of Unconventional Schemes for Ordinary Differential Equations, PhD thesis, Department of Mathematics, University of California at Berkeley, CA, 1995.
  • 9. R. I. McLachlan, On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM Journal on Scientific Computing, 16 (1995), pp. 151-168. MR 95j:65073
  • 10. J. M. Sanz-Serna and L. Abia, Order conditions for canonical Runge-Kutta schemes, SIAM Journal on Numerical Analysis, 28 (1991), pp. 1081-1096. MR 92e:65103
  • 11. M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, Journal of Mathematical Physics, 32 (1991), pp. 400-407. MR 92k:81096
  • 12. height 2pt depth -1.6pt width 23pt, General theory of higher-order decomposition of exponential operators and symplectic integrators, Physics Letters A, 165 (1992), pp. 387-395. MR 93b:22011
  • 13. H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A, 150 (1990), pp. 262-268. MR 91h:70014

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Additional Information

William Kahan
Affiliation: Computer Science Division and Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: wkahan@cs.berkeley.edu

Ren-Cang Li
Affiliation: Mathematical Science Section, Oak Ridge National Laboratory, P.O. Box 2008, Bldg. 6012, Oak Ridge, Tennessee 37831-6367
Email: na.rcli@na-net.ornl.gov

DOI: https://doi.org/10.1090/S0025-5718-97-00873-9
Keywords: Ordinary differential equations, reflexive methods, composition schemes, palindromic schemes
Received by editor(s): June 10, 1996
Additional Notes: The first author was supported in part by the Office of Naval Research contract N00014-90-J-1372 and National Science Foundation contract ASC-9005933.
The second author was supported in part by a Householder Fellowship in Scientific Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, Office of Energy Research, United States Department of Energy contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corporation.
Article copyright: © Copyright 1997 American Mathematical Society

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