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 B-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 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.