The Construction of Discrete Gauss Laws for Time Domain Schemes

The paper presents a method for constructing a suitable discrete Gauss law for a given discretization scheme for the Maxwell equations. Here, suitable means that if the discrete Gauss law is fulfilled initially the discrete time evolution given by the time domain scheme ensures that the discrete Gauss law is always fulfilled. Furthermore, conditions for the existence of such a law in the discrete sense are given. The construction of suitable Gauss laws and current interpolations is of importance in the context of Particle-In-Cell methods. In particular, the method is applied to the Longitudinal-Transversal Verlet-Leapfrog and the Alternating Direction Implicit scheme.