Time-domain Green's functions for waves on simplicial complexes

Maxwell's laws of classical electrodynamics have exact counterparts on a purely discrete space, reformulated as discrete Stokes' theorems on simplicial complexes. We here address the formulation of discrete Green's theorems and discrete Green's functions in discrete-time on simplicial complexes in 3-space as the fully discretised versions of continuum wave equations that preserve Stokes' theorems in all sub-dimensions of the simplicial complexes. These Green's functions have the same applications in discrete wave systems as their continuum counterparts, but are pre-conditioned to live wholly on the discrete space. 3-complexes in space with discrete time-stepping are the natural setting for generalisations of the FDTD algorithm from Yee-cells to general unstructured meshes.