1.  Ordinary Differential Equations

This chapter is dedicated to ordinary differential equations of first and higher orders. Emphasis is put on equations which can be written in normal form. Among those a scheme is outlined to choose the optimum strategy for their integration. For equations which can not be directly integrated, the most effective numerical schemes are reviewed. Ordinary differential equations of higher order are reduced to systems of differential equations of first order. Special attention is directed to stiff systems which frequently occur with time scales of different orders of magnitude. Examples are frequent in chemical kinetics and in transport problems with phase changes due to the different orders of magnitudes of densities in the phases. Stiff systems require appropriate numerical treatments for correct integration.