A modified Adams method for nonstiff and mildly stiff initial value problems

Abstract

Adams predictor-corrector methods, and explicit Runge–Kutta formulas, have been widely used for the numerical solution of nonstiff initial value problems. Both of these classes of methods have certain drawbacks, however, and it has long been the aim of numerical analysts to derive a class of formulas that has the advantages of both Adams and Runge–Kutta methods and the disadvantages of neither! In this paper we derive a class of modified Adams formulas that attempts to achieve this aim. When used in a certain precisely defined predictor-corrector mode, these new formulas require three function evaluations per step, but have much better stability than Adams formulas. This improved stability makes the modified Adams formulas particularly effective for mildly stiff problems, and some numerical evidence of this is given. We also consider the performance of the new class of methods on the well-known DETEST test set to show their potential on general nonstiff initial value problems.