On Aug 25, 2008, at 12:13 PM, Jakub Chaloupka wrote: > I am writing a diploma thesis which mentions FDTD and I would like > to refer the Meep with its capabilities. May I ask you, how does the > Meep solves problematics of numerical dispersion. Do you use any > specific method to reduce it?
Numerical dispersion is one of those things that all the books on FDTD and finite-difference methods love to analyze to death, because it can be easily analyzed and quantified analytically for a homogeneous medium. However, in practice I think it is rarely the dominant source of error in photonics calculations, because there is hardly any point in using FDTD for homogeneous media --- you are almost always interested in inhomogeneities, and these lead to much bigger errors. Like (almost) every other error associated with the finite resolution, numerical dispersion decreases with resolution, so you can deal with it by increasing the resolution until convergence is obtained to the desired accuracy. In particular, the errors from numerical dispersion go quadratically with resolution (in the ordinary center-difference FDTD scheme). On the other hand, if you use ordinary FDTD finite differences, the errors introduced by discretization of material interfaces go linearly with the resolution, so they are almost always dominant. Because of this, our effort focused on the material- interface error, which we can partially correct for by using an appropriate subpixel averaging scheme, as described in the Optics Letters paper cited by the Meep web page. (Another sometimes-dominant source of error in FDTD is boundary reflections, in cases where a true PML is not available; see our recent Optics Express paper http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-15-11376) There are ways to try to reduce numerical dispersion, of course. e.g. reducing spatial dispersion via a hexagonal grid, or using higher- order FDTD schemes. But we don't implement any of these because, at least in the cases we care about in our work, such schemes don't address the dominant sources of error and hence achieve little. Steven _______________________________________________ meep-discuss mailing list [email protected] http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/meep-discuss
