Consider a rectangular tapered waveguide, reminiscent of a horn antenna. Of course, the inclined PEC walls cannot accurately be represented by the Yee grid due to the staircase approximation. Here comes the CFDTD (or Yu-Mittra method) where in the cut-cells, the integral form of Faraday law for H field is applied, instead of the differential one. The nearby electric fields are multiplied by a factor
You cannot calculate fields across boundaries by interpolation. In FDTD objects are modelled as a consequence of field component placement in the Yee staggered arrangement that "carry" properties in their location and not the other way around. Averaging within homogenous regions is OK but generally not across interfaces. The E field on the deformed contour that is part of the PEC is assumed to be zero even if there is no value for it. The method simply introduce different length weights of at the E edges to simulate the different overall area that the B flux goes through instead of the area of the full face of a cell. In simpler PEC interfaces you set E to zero anyway so it should be zero as defined. But maybe I misunderstood the question ;-) I think normal FDTD is better explained with the integral equations than the differential ones.
Antonios Giannopoulos thank you for your answer. You confirm me what I initially thought. For some reason, I expected that the tangential field on the boundary would be zero or close to it after interpolating neighbouring nodes. The field is not explicitly set equal to zero there, since the boundary is not on any of the main axes. However, the field is set to zero in the PEC, for the cells just after the PEC wall, resembling a staircase like situation. I found that CFDTD doesn't make a large difference with regard to the fields behaviour. The problem that I study is a horn antenna fed by a waveguide mode.
The improvement in CFDTD or other contour deforming methods for PEC structures comes - in my view - from a "better" approximation of the magnetic flux. I put this in quotes as this is what they are trying in essence to achieve but don't know how much and how accurately they do this. The better determination of the magnetic field which in essence is seen as being tangential (or as close as possibly be) to the metallic structure gives a better simulation of the surface current that exists on this metallic surface which in turn I suppose is responsible for what radiates back. Staircasing creates erroneous paths for these currents as the fluxes around them are less accurate and hence create the errors (primarily on phase) that we then see when the wavelengths become smaller and the errors in the paths more significant. I think that is the reason why these contour deforming methods extended the accuracy a little further in frequencies from the results you get with standard FDTD. So, more accurate H means better result and does not really has to do with determining the actual values of E as much. I think there have been papers long time ago about some of these concepts but nothing more recent or more conclusive and I don't think it is really a very well clarified topic. I might obviously be horribly wrong!
fdtd is really a finite volume method so there are many ways to implement boundaries/interfaces that cut through the rectangular grid. The useful ways out of the many ways to do this are the ones that maintain accuracy and do not alter the stability condition. Additionally, many ways that have been published are not stable even if the publication's results appear stable.
Stair-stepping is not bad, once you realize that your discretization now has to deal with another length scale in the problem ( via the slope of the stair stepped curve). A curved boundary, say PEC, that is too stair-stepped behaves like a diffraction grating and if the wavelengths in the domain are larger than the period of this grating then the stair-stepped boundary "homogenizes" to an impedance boundary. But as dxdx,dz--->0 this artifact goes away and you recover the boundary condition intended.
At the end of the day, if your interest is to actually solve a problem, just use a very fine discretization so stair-stepping is not a problem. There are fast clusters with huge caches and RAM to make this possible.
P. G. Petropoulos , thank you for the detailed answer. This text actually solved a lot of my questions.
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