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On modified equations for discretizations of ODEs

The theory of modified equations (MEs) for discretizations of ODEs is reconsidered. Obstructions to convergence of series expansions of MEs are pinpointed and alternative approaches are presented which provide more accurate descriptions of numerical approximations through MEs. We emphasize how structural assumptions on the ODE can be used to improve estimates. Then we give arguments for a slightly alternative approach based on time-dependent MEs which avoids the asymptotic nature traditionally associated with MEs. Some applications of the theory are also provided.