Abstract
We present an approach to the numerical integration of ordinary differential equations based on the algebraic theory of Butcher (Math. Comp. 26, 79–106, 1972) and the 
-series theory of Hairer and Wanner (Computing 13, 1–15, 1974). We clarify the differences of these two approaches by equating the elementary weight functions and showing the differences of the composition rules. By interpreting the elementary weight function as a mapping from input values to output values and introducing some special mappings, we are able to derive the order conditions of several types of integration methods in a straight-forward way. The simplicity of the derivation is illustrated by linear multistep methods that use the second derivative as an input value, Runge-Kutta type methods that use the second as well as first derivatives, and general two-step Runge-Kutta methods. We derive new high stage-order methods in each example. In particular, we found a symmetric and stiffly-accurate method of order eight in the second example.
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Chan, T.M.H., Chan, R.P.K. A Simplified Approach to the Order Conditions of Integration Methods. Computing 77, 237–252 (2006). https://doi.org/10.1007/s00607-005-0151-1
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DOI: https://doi.org/10.1007/s00607-005-0151-1