Composite exponential approximations
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- by Arieh Iserles PDF
- Math. Comp. 38 (1982), 99-112 Request permission
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.References
- 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, DOI 10.1137/0714068
- John Donelson III. and Eldon Hansen, Cyclic composite multistep predictor-corrector methods, SIAM J. Numer. Anal. 8 (1971), 137–157. MR 282531, DOI 10.1137/0708018
- Byron L. Ehle, $A$-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973), 671–680. MR 331787, DOI 10.1137/0504057 B. L. Ehle, Some Results on Exponential Approximation and Stiff Equations, Report 77, Dept. of Math., Univ. of Victoria, Canada, 1974.
- Byron L. Ehle and Zdenek Picel, Two-parameter, arbitrary order, exponential approximations for stiff equations, Math. Comp. 29 (1975), 501–511. MR 375737, DOI 10.1090/S0025-5718-1975-0375737-7
- C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
- 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, DOI 10.1007/BF01436522
- Arieh Iserles, $A$-stability and dominating pairs, Math. Comp. 32 (1978), no. 141, 19–33. MR 464595, DOI 10.1090/S0025-5718-1978-0464595-0
- Arieh Iserles, Nonexponential fitting techniques for numerical solution of stiff equations, Utilitas Math. 17 (1980), 276–302. MR 583147
- Arieh Iserles, On the generalized Padé approximations to the exponential function, SIAM J. Numer. Anal. 16 (1979), no. 4, 631–636. MR 537277, DOI 10.1137/0716048
- Rolf Jeltsch, Stiff stability and its relation to $A_{0}$- and $A(0)$-stability, SIAM J. Numer. Anal. 13 (1976), no. 1, 8–17. MR 411174, DOI 10.1137/0713002
- Allan M. Krall, The root locus method: A survey, SIAM Rev. 12 (1970), 64–72. MR 260452, DOI 10.1137/1012002
- 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, DOI 10.1137/0714058
- 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, DOI 10.1137/0707002
- D. Morrison, Optimal mesh size in the numerical integration of an ordianry differential equation, J. Assoc. Comput. Mach. 9 (1962), 98–103. MR 134854, DOI 10.1145/321105.321115 S. P. Nørsett, Semi-Explicit Runge-Kutta Methods, Report 6, Dept. of Math., Univ. of Trondheim, Norway, 1974.
- Syvert P. Nørsett, Restricted Padé approximations to the exponential function, SIAM J. Numer. Anal. 15 (1978), no. 5, 1008–1029. MR 510733, DOI 10.1137/0715066
- 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, DOI 10.1007/bf01932291
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- Hans J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer Tracts in Natural Philosophy, Vol. 23, Springer-Verlag, New York-Heidelberg, 1973. MR 0426438
- 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
- G. Wanner, E. Hairer, and S. P. Nørsett, Order stars and stability theorems, BIT 18 (1978), no. 4, 475–489. MR 520756, DOI 10.1007/BF01932026
- Jet Wimp, On the zeros of a confluent hypergeometric function, Proc. Amer. Math. Soc. 16 (1965), 281–283. MR 173793, DOI 10.1090/S0002-9939-1965-0173793-8
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 38 (1982), 99-112
- MSC: Primary 65L05
- DOI: https://doi.org/10.1090/S0025-5718-1982-0637289-7
- MathSciNet review: 637289