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. 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.

💡 Research Summary

The paper presents a geometric, structure‑preserving semi‑discrete formulation of Maxwell’s equations in three‑ and two‑dimensional spaces using the framework of Discrete Exterior Calculus (DEC). The authors begin by constructing a combinatorial model of ℝ³ as a chain complex C(3)=C₀⊕C₁⊕C₂⊕C₃, where each component is generated by tensor products of one‑dimensional chains. Basis elements for 0‑, 1‑, 2‑, and 3‑cells are introduced (points, edges, faces, and volumes) and indexed by integer triples (k,s,m). A forward shift operator τ defines the discrete boundary operator ∂, which mimics the continuous exterior derivative’s action on oriented cells.

The dual cochain complex K(3) is then defined, with cochains interpreted as discrete differential forms. The coboundary operator d_c:K^r→K^{r+1} is introduced via the duality relation ⟨a,d_cΩ⟩=⟨∂a,Ω⟩, yielding explicit finite‑difference formulas for 0‑, 1‑, and 2‑forms. The authors verify the fundamental property d_c∘d_c=0, establishing a discrete analogue of the de Rham complex.

A cup product ∪ and a discrete Hodge star * are defined on the basis elements. The star maps an r‑form to a (3−r)‑form but, unlike the continuous case, ** does not return the original form; instead it shifts all indices by one lattice step. This shift is highlighted as a key distinction between the discrete and continuous settings. The discrete Leibniz rule d_c(Ω∪Φ)=d_cΩ∪Φ+(−1)^rΩ∪d_cΦ and the relation d_c(*Ω)=*δ_cΩ are proved, where the codifferential δ_c is defined by δ_c = (−1)^{r+1} *^{-1} d_c *. Explicit expressions for δ_c acting on 0‑, 1‑, 2‑, and 3‑forms are derived, and the discrete Laplacian Δ_c = d_cδ_c + δ_c d_c is introduced.

With this geometric machinery in place, the electromagnetic fields are represented as discrete forms: the electric and magnetic field intensities E and H are 1‑forms, the electric displacement D and magnetic flux density B are 2‑forms, the current density J is a 2‑form, and the charge density Q is a 3‑form. Keeping time continuous, the semi‑discrete Maxwell system is written as

 d_cE = −∂_t B, d_cH = ∂_t D + J, d_cD = Q, d_cB = 0.

Using the explicit formulas for d_c, these become systems of coupled difference‑differential equations. For example, Faraday’s law translates into Δ_k E₂ − Δ_s E₁ = −∂t B{12}, and similar relations hold for the other components. The constitutive relations D = ε₀ *E and B = μ₀ *H are also expressed in the discrete setting via the star operator.

The authors demonstrate that essential physical properties are retained. The discrete version of Poynting’s theorem follows from the inner product (Φ,Ω)_V = ⟨V, Φ∪*Ω⟩, guaranteeing energy conservation. The equation d_cB = 0 ensures the absence of magnetic monopoles on the lattice, and the continuity equation d_cJ + ∂_t Q = 0 holds automatically, reflecting charge conservation. Gauge invariance is preserved because the discrete exterior derivative annihilates exact forms, mirroring the continuous theory.

The three‑dimensional framework is then reduced to two dimensions by restricting the chain complex, and the authors illustrate the method on a combinatorial torus (a periodic 2‑D lattice). On this torus the semi‑discrete Maxwell equations reduce to a finite system of first‑order linear ordinary differential equations. By constructing the corresponding coefficient matrices, they obtain an explicit expression for the general solution in terms of matrix exponentials and particular integrals, providing a closed‑form analytical benchmark that is rarely available for discrete electromagnetic models.

Compared with traditional lattice or finite‑element approaches that rely on Whitney forms or de Rham maps, the present scheme avoids those constructions while still preserving the algebraic structure of exterior calculus. The explicit handling of the index shift in the star operator prevents inadvertent violation of topological constraints at the discrete level. The analytical solution on the torus serves both as a verification tool and as a pedagogical example of how DEC can yield solvable semi‑discrete models.

In conclusion, the paper delivers a rigorous, geometry‑aware semi‑discrete Maxwell model that maintains the topological and metric properties of the continuous equations. It opens avenues for extending DEC‑based electromagnetic simulations to nonlinear media, complex boundaries, and higher‑dimensional problems, while offering a solid theoretical foundation for structure‑preserving numerical schemes.