Special extended Nyström tree theory for ERKN methods

- A systematic and uniform special extended Nyström tree (SEN-tree) theory.
- Order conditions for extended Runge-Kutta-Nyström (ERKN) methods.
- New simplifying assumptions for the coefficients of ERKN methods.
- Practical ERKN schemes with numerical illustrations.

The aim of this paper is to develop a unified special extended Nyström tree (SEN-tree) theory which provides a theoretical framework for the order conditions of multidimensional extended Runge-Kutta-Nyström (ERKN) methods proposed by X. Wu et al. (Wu et al., 2010). The new SEN tree theory is complete and consistent, which has overcome the drawback of the bi-coloured tree theory in H. Yang et al.'s work (Yang et al., 2009) where two “branch sets” have to be constructed for the true solutions and for the numerical solutions, respectively.