Stability properties of Adams codes

The Adams methods are among the most effective [1, 6, 11] and popular for the solution of the initial value problem for nonstiff ordinary differential equations. The term Adams method includes quite a few possibilities. All involve a predictor (explicit, Adams-Bashforth) formula and a corrector (implicit, Adams-Moulton) formula. The orders of these formulas have to be specified, and they need not be the same. For a predictor of order k the only possibilities for a corrector which have been seriously considered are those of orders k and k + 1. The number of iterations to be made with the corrector must be specified. Alternatively, a variable number of iterations could be made with the object of solving the corrector equation "exactly." If a fixed number of iterations is made, one must decide whether or not to end the computation of the step with a final evaluation of the derivative. Among the very best codes based on Adams methods, only two methods are represented here [1, 6, 11]. Method I uses a corrector of order one higher than the predictor (also known as local extrapolation when viewed differently), corrects only once, and ends with a final evaluation. This method is represented by Krogh's DVDQ and by Shampine and Gordon's STEP. Method II iterates the corrector to "convergence" and hence is the Adams-Moulton method. It is represented by Gear's DIFSUB and its variants GEAR written by Hindmarsh and STIFF written by Kahaner and Sutherland [7]. Naturally the authors of the codes cited tried to select the best method from the class of Adams methods just described, but they came to different conclusions. Generally speaking, this author has seen no advantage to one method or the other. A striking exception reported in [11] occurs when the codes are confronted with a mildly stiff equation. It is of obvious practical value to understand such differences so as to improve the codes or to see that a choice of method (code?} appropriate to the application is made.