Composite exponential approximations

Iserles, Arieh

doi:10.1090/S0025-5718-1982-0637289-7

Composite exponential approximations
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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

Some Results on Exponential Approximation and Stiff Equations

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

Semi-Explicit Runge-Kutta Methods

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