Optimal Second Derivative Methods for Stiff Systems

In implementing numerical methods for stiff ordinary differential equations several difficulties arise which are not usually encountered with methods designed for non-stiff equations. These difficulties include the choice of the basic formula, the choice of the iteration scheme to solve the implicit set of equations associated with each step, and the choice of a valid error estimate and step control strategy. We will introduce a class of second derivative multistep formulas suitable for stiff equations and show how they have been implemented in an efficient variable-order method.