Composite exponential approximations
Author:
Arieh Iserles
Journal:
Math. Comp. 38 (1982), 99-112
MSC:
Primary 65L05
DOI:
https://doi.org/10.1090/S0025-5718-1982-0637289-7
MathSciNet review:
637289
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Abstract | References | Similar Articles | Additional Information
Abstract: The Composite Exponential Approximations (CEA) arise in a natural way when one investigates the stability and order properties of a combination of several methods for the numerical solution of ordinary differential equations, sequentially implemented with different step-lengths. Some general results on the order, acceptability and exponential fitting properties of CEA are derived. The composite Padé approximations and N-approximations are explored in detail.
- [1] Roger Alexander, Diagonally implicit Runge-Kutta methods for stiff o.d.e.’s, SIAM J. Numer. Anal. 14 (1977), no. 6, 1006–1021. MR 458890, https://doi.org/10.1137/0714068
- [2] John Donelson III. and Eldon Hansen, Cyclic composite multistep predictor-corrector methods, SIAM J. Numer. Anal. 8 (1971), 137–157. MR 282531, https://doi.org/10.1137/0708018
- [3] Byron L. Ehle, 𝐴-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973), 671–680. MR 331787, https://doi.org/10.1137/0504057
- [4] B. L. Ehle, Some Results on Exponential Approximation and Stiff Equations, Report 77, Dept. of Math., Univ. of Victoria, Canada, 1974.
- [5] Byron L. Ehle and Zdenek Picel, Two-parameter, arbitrary order, exponential approximations for stiff equations, Math. Comp. 29 (1975), 501–511. MR 375737, https://doi.org/10.1090/S0025-5718-1975-0375737-7
- [6] C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
- [7] H. Greenspan, W. Hafner, and M. Ribarič, On varying stepsize in numerical integration of first order differential equations, Numer. Math. 7 (1965), 286–291. MR 231536, https://doi.org/10.1007/BF01436522
- [8] Arieh Iserles, 𝐴-stability and dominating pairs, Math. Comp. 32 (1978), no. 141, 19–33. MR 464595, https://doi.org/10.1090/S0025-5718-1978-0464595-0
- [9] Arieh Iserles, Nonexponential fitting techniques for numerical solution of stiff equations, Utilitas Math. 17 (1980), 276–302. MR 583147
- [10] Arieh Iserles, On the generalized Padé approximations to the exponential function, SIAM J. Numer. Anal. 16 (1979), no. 4, 631–636. MR 537277, https://doi.org/10.1137/0716048
- [11] Rolf Jeltsch, Stiff stability and its relation to 𝐴₀- and 𝐴(0)-stability, SIAM J. Numer. Anal. 13 (1976), no. 1, 8–17. MR 411174, https://doi.org/10.1137/0713002
- [12] Allan M. Krall, The root locus method: A survey, SIAM Rev. 12 (1970), 64–72. MR 260452, https://doi.org/10.1137/1012002
- [13] Bengt Lindberg, Characterization of optimal stepsize sequences for methods for stiff differential equations, SIAM J. Numer. Anal. 14 (1977), no. 5, 859–887. MR 519728, https://doi.org/10.1137/0714058
- [14] Werner Liniger and Ralph A. Willoughby, Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Numer. Anal. 7 (1970), 47–66. MR 260181, https://doi.org/10.1137/0707002
- [15] D. Morrison, Optimal mesh size in the numerical integration of an ordianry differential equation, J. Assoc. Comput. Mach. 9 (1962), 98–103. MR 134854, https://doi.org/10.1145/321105.321115
- [16] S. P. Nørsett, Semi-Explicit Runge-Kutta Methods, Report 6, Dept. of Math., Univ. of Trondheim, Norway, 1974.
- [17] Syvert P. Nørsett, Restricted Padé approximations to the exponential function, SIAM J. Numer. Anal. 15 (1978), no. 5, 1008–1029. MR 510733, https://doi.org/10.1137/0715066
- [18] Syvert P. Nørsett and Arne Wolfbrandt, Attainable order of rational approximations to the exponential function with only real poles, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), no. 2, 200–208. MR 447900, https://doi.org/10.1007/bf01932291
- [19] Earl D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR 0107725
- [20] Hans J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer-Verlag, New York-Heidelberg, 1973. Springer Tracts in Natural Philosophy, Vol. 23. MR 0426438
- [21] Richard S. Varga, On higher order stable implicit methods for solving parabolic partial differential equations, J. Math. and Phys. 40 (1961), 220–231. MR 140191
- [22] G. Wanner, E. Hairer, and S. P. Nørsett, Order stars and stability theorems, BIT 18 (1978), no. 4, 475–489. MR 520756, https://doi.org/10.1007/BF01932026
- [23] Jet Wimp, On the zeros of a confluent hypergeometric function, Proc. Amer. Math. Soc. 16 (1965), 281–283. MR 173793, https://doi.org/10.1090/S0002-9939-1965-0173793-8
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Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1982-0637289-7
Article copyright:
© Copyright 1982
American Mathematical Society