On a semi-discrete model of Maxwell's equations in three and two dimensions
Volodymyr Sushch
TL;DR
The paper develops a geometric, structure-preserving semi-discrete formulation of Maxwell's equations by discretizing spatial variables with discrete exterior calculus while keeping time continuous, transforming the PDEs into a system of first-order ODEs. It constructs a 3D combinatorial framework with $C(3)$ and $K(3)$ and introduces discrete operators $d^c$, $\cup$, $\ast$, and $\delta^c$, deriving energy conservation and wave-type equations in the discrete setting. The 3D formulation is complemented by a 2D reduction on a combinatorial torus, where the Maxwell system yields a solvable nine-dimensional linear ODE with a explicit eigenstructure, providing a rigorous, analytic benchmark. Together, these results offer a robust, structure-preserving discretization of electromagnetism suitable for reliable simulations, including a tractable 2D torus example with closed-form solutions.
Abstract
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.