Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case

Abstract

The structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems. The point is that, in general, the global error contains nonsmooth (oscillating) terms at the dominanth 2-level. However, it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor ? (where-1/? is the magnitude of the stiff eigenvalues). In these cases a "full" asymptotic error expansion exists in thestrongly stiff case (? sufficiently small compared to the stepsizeh). The general case (where the oscillating error components areO(h 2) and notO(?h 2)) and applications of our results (extrapolation and defect correction algorithims) will be studied in separate papers.