On a Semi-Discrete Model of Maxwell’s Equations in Three and Two Dimensions

In this paper, we develop a geometric, structure-preserving semi-discrete formulation of Maxwell’s equations in both three- and two-dimensional settings within the framework of discrete exterior calculus. The proposed approach preserves the intrinsic geometric and topological structures of the continuous theory while providing a consistent spatial discretization. We analyze the essential properties of the proposed semi-discrete model and compare them with those of the classical Maxwell’s equations. As a representative example, the framework is applied to a combinatorial two-dimensional torus, where the semi-discrete Maxwell system reduces to a set of first-order linear ordinary differential equations. An explicit expression for the general solution of this system is also derived.