Electromagnetic theory on a lattice

Abstract

A self-contained electromagnetic theory is derived on a regular lattice. The discretized form of integral and differential calculus, which is called discrete calculus, is used to describe this theory. It is shown that discrete forms of Gauss' theorem, Stokes' theorem, Green's theorem, and Huygens' principle can be derived. Moreover, many electromagnetic theorems can also be derived in this discretized world, for example, reciprocity theorem, uniqueness theorem, and Poynting's theorem. The preservation of these theorems and the conservation of charge imply that the use of this discretized form of Maxwell's equations for numerical simulation will not give rise to spurious solutions due to spurious charges.