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W. Auzinger:
"On the error structure of the implicit Euler scheme applied to stiff systems of differential equations";
Computing, 43 (1989), 115 - 131.

In this paper we investigate the structure of the global discretization error of the implicit Euler scheme applied to systems of stiff differential equations, extending earlier work on this subject. We restrain our considerations to the linear, self-adjoint, constant coefficient case but we make no assumptions about the nature of the stiff spectrum; the stiff eigenvalues may be arbitrarily distributed on the negative real axis.

Our main result says that the global error of the implicit Euler scheme admits an asymptotic expansion in powers of the stepsize \tau which is not asymptotically correct in the conventional sense: Near the initial point t=0 the expansion is spoiled at the (\tau^2)-level by `irregular' error components which are, however, (algebraically) damped, such that away from t=0 the `pure' asymptotic expansion reappears. We present numerical experiments confirming this result.

Our considerations should be particularly helpful for a rigorous, quantitative analysis of the structure of the full (space & time) discretization error in the PDE (method of lines) context, and thus for a sound theoretical justification of extrapolation techniques for this important class of stiff problems.

http://dx.doi.org/10.1007/BF02241856