Characterization of Optimal Stepsize Sequences for Methods for Stiff Differential Equations

The amount of work needed for a numerical method to solve a given problem depends mainly on the stepsize sequence that is used. For a given method and a given problem consider all stepsize sequences that give a global error whose norm is everywhere less than a prescribed tolerance. By an optimal stepsize sequence we mean a sequence that requires fewer steps than any other sequence. A large part of this study contains characterizations of optimal stepsize sequences for some classes of problems. From the characterization of the optimal stepsize sequences for stiff systems one can conclude that the stepsize sequence obtained with a fixed bound on the local error per unit step is far from optimal. Existing numerical methods that use a variable stepsize choose their stepsizes on the basis of either a bound on the local error per step or on a bound on the local error per unit step. For systems of stiff differential equations the stepsize sequence obtained with the first criterion, in general, requires less work than the stepsize sequence obtained with the second. A convenient estimate of the global error for a class of methods for solution of systems of stiff differential equations is described. Further a relation (valid for sufficiently smooth stepsize functions) between a bound on the local error and the global error for fixed and variable order methods is established.