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W. Auzinger, R. Frank:
"Asymptotic error expansions for stiff equations: The implicit midpoint rule";
Report for Technical Report No.77/88, Institut für Angewandte und Numerische Mathematik; 1988.

This is a study of the structure of the global discretization
error of the implicit midpoint rule applied to nonlinear stiff initial value problems. Those special cases where an asymptotically correct expansion in even powers of the stepsize $h$ exists have been described in an earlier paper. In the general stiff case the global error does not admit a "pure" asymptotic expansion but there occur dominant oscillating terms at the h^2-level. The amplitude of these oscillations shows a very smooth behavior and, consequently, the error structure is sufficiently regular to guarantee a satisfactory efficiency of acceleration algorithms like extrapolation or defect correction. These algorithmic applications are discussed in a separate paper.

http://www.math.tuwien.ac.at/~winfried/papers/aeimp.pdf