Fully implicit one-step methods for neutral functional-differential equations

The author discusses the class of fully implicit one-step methods of any order for the numerical solution of neutral functional-differential equations. For judicious choices of the parameters these methods are NP-stable, which means that the numerical approximation to the solution Y of the linear test equation y´=ay(t)+by(t-d) + cy´(t-d), t⩾0, is bounded whenever Y is bounded. This property is an analogue of A-stability of ordinary differential equations. The local discretization error of these methods can be estimated by comparing two approximations of successive orders. This can be done in a very efficient way, and these methods can be implemented in variable-step mode with a step-changing strategy based on this estimate. Numerical results are presented that illustrate the high potential of fully implicit formulas