2025/10/30 by Sushch, Volodymyr
#35Q61 #39A12 #39A70 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2510.26427
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. This 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 special case, the model is illustrated on a combinatorial two-dimensional torus, where the semi-discrete Maxwell's equations take the form of a system of first-order linear ordinary differential equations. An explicit expression for the general solution of this system is also derived.