Higher-order Accuracy

It is known that for lumped systems (i.e., those described by sets of ODEs), A-stable methods [32,65,75] such as WDF network simulators can be at best second-order accurate; that is, the truncation error between the solution to the difference scheme and the exact solution will behave as $O(T^{2})$, where $T$ is the time step. In §3.13 and §4.10.5, however, we showed that it is possible to construct networks that behave as higher-order spatially accurate difference schemes, at least for the (1+1)D transmission line equations. The question remains open, however, as to whether it is possible to obtain higher-order time-accurate scattering methods. (One of the originators of the MDWD simulation method remarked [130] that he had spent an inordinate amount of time trying to design higher-order time accurate methods with no success.) It would be of great use to have a firm answer to this question, mainly because it would provide a clue to answering the question in the previous paragraph; rather, it would encourage us to rephrase it as: ``Is there any stable difference scheme which can not be written in a scattering form?''