A discrete exterior calculus and electromagnetic theory on a lattice

A highly accurate FDTD formulation was developed on an overlapped cubic grid that greatly reduces numerical dispersion errors. However, errors in in the FDTD method arise not only from numerical dispersion, but from geometrical modelling as well. Although representing a significant progress in addressing the numerical dispersion problem, it is still confined to a cubic grid with the subsequent "stair-casing" geometric approximations that it entails. The material presented represents a fundamentally new paradigm for finite-difference methods which hopes to address both issues of numerical dispersion and geometrical modelling. It involves a rigorous mathematical framework based on concepts from topology and differential geometry. Particularly, it involves a construction of a discrete analog to the calculus of differential forms. It should be noted that the use of differential forms, and their lattice counterparts, is well known within the field of algebraic topology. However, the original contribution here lies in the introduction of a metric onto the lattice. It is with the metric that the adjoint exterior derivative may be defined, which is required for most physical systems not the least of importance being Maxwell's equations.