Numerical analysis 2000. ODE and integral equations

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2] R. Alexander, Diagonally implicit Runge–Kutta methods for sti ODEs, SIAM J. Numer. Anal. 14 (1977) 1006–1021. [3] R. Alt, Deux thÃeorÂems sur la A-stabilitÃe des schÃemas de Runge–Kutta simplement implicites, Rev. Francais d’Automat. Recherche OpÃerationelle SÃer. R-3 6 (1972) 99–104. [4] D. M. M. Zahar, The automatic solution of ordinary di erential equations by the method of Taylor series, Comput. J. 14 (1971) 243–248. [5] F. C. Adams, An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of uid, with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, Cambridge University Press, Cambridge, 1883.

Cyclic composite methods It is remarkable that even though a number of individual linear multistep methods may be unstable, it is possible to use them cyclically to obtain a method which, overall, is stable. C. Butcher / Journal of Computational and Applied Mathematics 125 (2000) 1–29 21 The method is used cyclically in the sense that in each of three steps, the ÿrst uses (22), the second uses (23) and the third uses (24) (which in this particular example happens to be the same as the ÿrst member of the cycle).

2. C. Butcher / Journal of Computational and Applied Mathematics 125 (2000) 1–29 Fig. 4. An extract from the order star paper. and · denotes an inner product, with · the corresponding norm. It is easy to see that two exact solutions to (26) possess the property that y(x) − z(x) 6 y(x0 ) − z(x0 ) ; for x¿x0 : (28) The aim is now to ÿnd conditions on a method such that a discrete analogue of (28) holds. C. Butcher / Journal of Computational and Applied Mathematics 125 (2000) 1–29 where     Yn =  25  yn yn−1 ..

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