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IMA Journal of Numerical Analysis 2003 23(2):197-220; doi:10.1093/imanum/23.2.197
© 2003 by Institute of Mathematics and its Applications
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Order conditions for a class of two-step methods for y'' = f (x, y)

John P. Coleman1

1 Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK

The theory of B-series is used to investigate the order of convergence of a general class of two-step hybrid methods for systems of differential equations of the special form y'' = f (x, y). The main result is a set of order conditions, analogous to those for Runge–Kutta methods, offering an alternative to the customary ad hoc Taylor expansions. A byproduct is a remarkably simple formula from which the order of dispersion of such methods is easily determined. Conditions under which the two-step methods are symmetric are established, and particular examples are considered.

Key Words: order conditions; y'' = f (x, y); two-step methods; B-series; hybrid methods; symmetric methods; phase lag

Received 26 February 2001. Revised 13 May 2002.


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