Coordinate Changes and Grid Generation

Before looking directly at circuits and signal flow diagrams in multiple dimensions, it is useful to introduce coordinate changes, which were first applied in the context of multidimensional wave digital filters in [122]. One might add that it is *useful*, but not strictly *necessary*, since it is possible to develop numerical integration algorithms along the same lines without any explicit reference to new coordinates [61]. It is, however, a very convenient way of understanding causality and grid generation issues, as well as generalizing the passivity concept to MD [48,85,131].

Some of the lumped circuit elements we have discussed so far we have seen to be *passive*--that is, they dissipate energy as time progresses, as do Kirchoff networks composed of connections of such elements (by *Tellegen's Theorem* [136]). In the multidimensional setting, many systems possess a similar property; some measure of energy decreases as a function of time. For example, the amplitude of the vibrations in a struck string or membrane will gradually decrease (or at least not increase) as a function of time. We have also seen that, for lumped circuits, application of the trapezoid rule translates this passivity property to a discrete equivalent. When attempting a discretization of a set of PDEs, however, we have to cope not only with the time direction but spatial ones as well, and passivity (usually a result of the conservative nature of the laws from which a system of equations is derived) does not in general hold with respect to space [51].

The idea of Fettweis and Nitsche [62] was to perform a coordinate transformation such that the new coordinates, generally a mixture of time and space, all contain a part of the physical time variable. Traveling in the positive direction along any of the new coordinates implies that one is also moving forward in time (as well as in some spatial direction). More specifically, if

are the new coordinates, the authors provide the following conditions:

As a result,

- Structure of Coordinate Changes
- Coordinate Changes in (1+1)D
- Coordinate Changes in Higher Dimensions