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

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. 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.

1. Introduction

The construction of discrete models that preserve the geometric structure of mathematical physics problems is fundamental to achieving reliable and physically consistent numerical simulations of differential equations. The present work continues our series of studies [1,2,3] in which discrete analogues of several fundamental equations of mathematical physics were developed using a geometric discretization framework based on discrete exterior calculus. The main idea of this approach originates from the work of Dezin [4].

In this paper, we investigate a discrete formulation of Maxwell’s equations. The system of Maxwell’s equations possesses a remarkably rich history of solution methodologies, making it impossible to adequately account for all contributions. Our focus is on discrete models of Maxwell’s equations considered strictly from a mathematical perspective, with particular emphasis on discrete constructions that employ the calculus of differential forms. Differential form notation is known to simplify certain calculations and to clarify several features of electromagnetic fields. In this study, we introduce a discrete–continuous counterpart of Maxwell’s equations, wherein the spatial variables are discretized while the time variable is retained in continuous form. The resulting semi-discrete model is represented by a system of first-order linear ordinary differential equations. We develop discrete versions of Maxwell’s equations in both three- and two-dimensional spatial settings with time dependence.

Numerous studies have addressed the problem of discretizing electromagnetic theory within the framework of the exterior calculus of differential forms (see, for example, [5,6,7,8,9,10] and the references therein). Some of these approaches are based on lattice discretization schemes [7,9,10]. Formulating Maxwell’s equations in the language of differential forms [11] and employing discrete exterior calculus as the computational foundation have led to significant advancements in numerical methods based on finite element and finite difference techniques [5,6,12,13,14,15]. Numerical computations in the finite element exterior calculus method [16] are typically based on Whitney forms [17]. To solve Maxwell’s equations numerically, several methods have been proposed, such as mixed finite-volume methods in the two-dimensional domains [18], fast Maxwell solvers based on exact discrete eigen-decompositions in rectangular domains [19], and methods based on discrete exterior calculus [20,21]. The discretization scheme considered in the present paper, however, does not employ Whitney forms, nor does it use the Whitney or de Rham maps between cochains and differential forms [10]. Nevertheless, the essential structure of exterior calculus is preserved in the discrete setting. Our approach differs from existing ones through its definitions of the discrete Hodge star and exterior product. The latter is constructed so that a Leibniz-type rule holds for the discrete analogue of the exterior derivative acting on products of discrete forms. Furthermore, the discrete counterparts of differential operators are expressed explicitly in terms of difference operators, which represents a further structural advantage of the proposed discrete model.

Let us briefly recall the key definitions involved in the standard three-dimensional formulation of Maxwell’s equations within the framework of exterior calculus. See, for example, [22] or [23] for details. In this formalism, electromagnetic fields and source quantities are described using differential forms: the 1-forms E and H represent the electric and magnetic field intensities, respectively; the 2-forms D and B correspond to the electric and magnetic flux densities; the 2-form J denotes the electric current density; and the 3-form Q represents the electric charge density. Following [22], Maxwell’s equations can then be written as:

$$dE=-{\displaystyle \frac{\partial B}{\partial t}},$$

(1)

$$dH={\displaystyle \frac{\partial D}{\partial t}}+J,$$

(2)

$$dD=Q,$$

(3)

$$dB=0,$$

(4)

where d denotes the exterior derivative. The constitutive relationships are given by

$$D={\epsilon }_0\ast E,$$

(5)

$$B={\mu }_0\ast H,$$

(6)

where ${\epsilon }_0$ and ${\mu }_0$ are the vacuum permittivity and permeability, respectively, and ∗ denotes the Hodge star acting in ${\mathbb{R}}^3$. In three dimensions, the Hodge star satisfies $\ast \ast =\mathrm{Id}$ for any forms. Therefore, Equations (5) and (6) can be equivalently written as

$$\ast D={\epsilon }_{0}E,$$

(7)

$$\ast B={\mu }_{0}H.$$

(8)

Poynting’s theorem, within the framework of differential forms, can be expressed as

$$d(E\wedge H)=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}(E\wedge D+B\wedge H)-E\wedge J,$$

(9)

where $E\wedge H$ is the Poynting energy flow form, ${\displaystyle \frac{1}{2}}E\wedge D$ and ${\displaystyle \frac{1}{2}}B\wedge H$ represent the electric and the magnetic densities, respectively, and $E\wedge J$ denotes the power density. See [22] for details. In the two-dimensional case, Maxwell’s equations retain the form of Equations (1)–(6), differing only in the interpretation of the Hodge star operator, which depends on the dimension [23].

The aim of this work is to develop a geometric, structure-preserving semi-discrete formulation of Maxwell’s equations in both three- and two-dimensional settings. Building upon our previous studies [3,24], we construct discrete analogues of Equations (3)–(6) on a model of the two-dimensional torus. In this framework, the original system of partial differential equations is transformed into a system of linear ordinary differential equations that can be solved analytically.

Poynting’s theorem, which embodies the conservation of electromagnetic energy, is intrinsically connected to the symmetry of electromagnetic fields. In our semi-discrete formulation, we derive a discrete analogue of this theorem and demonstrate that the proposed model preserves the fundamental symmetry properties of the continuous theory.

The rest of the paper is organized as follows. In Section 2, we describe the construction of a combinatorial model of ${\mathbb{R}}^3$, extending the combinatorial model of ${\mathbb{R}}^2$ presented in [24]. We introduce a cochain complex and define discrete analogues of the fundamental operations of exterior calculus. In Section 3, we establish a three-dimensional discrete counterpart of Maxwell’s equations while keeping time as a continuous variable. Here, we generalize the semi-discrete approach introduced in [1]. Furthermore, we examine the essential properties of the proposed semi-discrete model and compare them with those of the classical Maxwell’s equations. In Section 4, we reduce our semi-discrete model of the three-dimensional Maxwell’s equations to the two-dimensional case. Following [3,24], we consider the discrete Maxwell’s equations on a combinatorial torus as an illustrative example and derive an explicit expression for the general solution in this setting. Finally, Section 5 contains concluding remarks and outlines future work.

2. Background on a Discrete Model

A detailed construction of a combinatorial model for the two-dimensional Euclidean space ${\mathbb{R}}^2$ is given in [24]. In this section, we generalize that approach to the three-dimensional case. For the reader’s convenience, the notations introduced in this section is listed in the Abbreviations. The combinatorial model of ${\mathbb{R}}^3$ is defined as a three-dimensional chain complex

$$C\left(3\right)=C_0\left(3\right)\oplus C_1\left(3\right)\oplus C_2\left(3\right)\oplus C_3\left(3\right)$$

generated by the 0-, 1-, 2-, and 3-dimensional basis elements

$$\left\{x_{k,s,m}\right\},\{e_{k,s,m}^1,e_{k,s,m}^2,e_{k,s,m}^3\},\{e_{k,s,m}^{12},e_{k,s,m}^{13},e_{k,s,m}^{23}\},\mathrm{and}\left\{V_{k,s,m}\right\},$$

respectively, where $k,s,m\in \mathbb{Z}$. More precisely, each basis element of $C\left(3\right)$ can be represented as the following tensor products:

$$\begin{array}{cc}\hfill x_{k,s,m}& =x_k\otimes x_s\otimes x_m,V_{k,s,m}=e_k\otimes e_s\otimes e_m,\hfill \\ \hfill e_{k,s,m}^1& =e_k\otimes x_s\otimes x_m,e_{k,s,m}^2=x_k\otimes e_s\otimes x_m,e_{k,s,m}^3=x_k\otimes x_s\otimes e_m,\hfill \\ \hfill e_{k,s,m}^{12}& =e_k\otimes e_s\otimes x_m,e_{k,s,m}^{13}=e_k\otimes x_s\otimes e_m,e_{k,s,m}^{23}=x_k\otimes e_s\otimes e_m,\hfill \end{array}$$

where $x_k$ and $e_k$ are the 0- and 1-dimensional basis elements of the 1-dimensional chain complex C. Geometrically, the 0-dimensional elements $x_k$ can be interpreted as points on the real line, and the 1-dimensional elements $e_k$ as open intervals between those points. The complex C thus represents a combinatorial real line, and the full complex $C\left(3\right)$ can be written as the tensor product $C\left(3\right)=C\otimes C\otimes C$. On the chain complex $C\left(3\right)$, we define the boundary operator $\partial :C_r\left(3\right)\to C_{r-1}\left(3\right)$, $r=1,2,3$, as follows

$$\begin{array}{cc}\hfill \partial x_{k,s,m}& =0,\partial e_{k,s,m}^1=x_{\tau k,s,m}-x_{k,s,m},\hfill \\ \hfill \partial e_{k,s,m}^2& =x_{k,\tau s,m}-x_{k,s,m},\partial e_{k,s,m}^3=x_{k,s,\tau m}-x_{k,s,m},\hfill \\ \hfill \partial e_{k,s,m}^{12}& =e_{\tau k,s,m}^2-e_{k,s,m}^2-e_{k,\tau s,m}^1+e_{k,s,m}^1,\hfill \\ \hfill \partial e_{k,s,m}^{13}& =e_{\tau k,s,m}^3-e_{k,s,m}^3-e_{k,s,\tau m}^1+e_{k,s,m}^1,\hfill \\ \hfill \partial e_{k,s,m}^{23}& =e_{k,\tau s,m}^3-e_{k,s,m}^3-e_{k,s,\tau m}^2+e_{k,s,m}^2,\hfill \\ \hfill \partial V_{k,s,m}& =e_{k,s,\tau m}^{12}-e_{k,s,m}^{12}+e_{\tau k,s,m}^{23}-e_{k,s,m}^{23}-e_{k,\tau s,m}^{13}+e_{k,s,m}^{13}.\hfill \end{array}$$

(10)

Here, $\tau$ denotes the forward shift operator, i.e., $\tau k=k+1$. This definition extends linearly to arbitrary chains in the complex.

We now introduce the dual object to the chain complex $C\left(3\right)$, denoted by $K\left(3\right)=K^0\left(3\right)\oplus K^1\left(3\right)\oplus K^2\left(3\right)\oplus K^3\left(3\right)$, as defined in [24]. This dual complex has a structure analogous to that of $C\left(3\right)$ and consists of cochains with real-valued coefficients. Let the sets

$$\left\{x^{k,s,m}\right\},\{e_1^{k,s,m},e_2^{k,s,m},e_3^{k,s,m}\},\{e_{12}^{k,s,m},e_{13}^{k,s,m},e_{23}^{k,s,m}\},\mathrm{and}\left\{V^{k,s,m}\right\}$$

denote the basis elements of $K^0\left(3\right)$, $K^1\left(3\right)$, $K^2\left(3\right)$, and $K^2\left(3\right)$, respectively. Using these bases, cochains $\Phi \in K^0\left(3\right)$, $\Psi \in K^3\left(3\right)$, $A\in K^1\left(3\right)$, and $B\in K^2\left(3\right)$ can be expressed in component form as

$$\Phi =\sum _{k,s,m}{\Phi }_{k,s,m}{x}^{k,s,m},\Psi =\sum _{k,s,m}{\Psi }_{k,s,m}{V}^{k,s,m},$$

(11)

$$A=\sum _{k,s,m}(A_{k,s,m}^1{e}_1^{k,s,m}+A_{k,s,m}^2{e}_2^{k,s,m}+A_{k,s,m}^3{e}_3^{k,s,m}),$$

(12)

$$B=\sum _{k,s,m}(B_{k,s,m}^{12}{e}_{12}^{k,s,m}+B_{k,s,m}^{13}{e}_{13}^{k,s,m}+B_{k,s,m}^{23}{e}_{23}^{k,s,m}),$$

(13)

where ${\Phi }_{k,s,m},{\Psi }_{k,s,m},A_{k,s,m}^i,B_{k,s,m}^{ij}\in \mathbb{R}$ for all $k,s,m\in \mathbb{Z}$ and $i,j=1,2,3$. Following the terminology in [24], we refer to these cochains as forms or discrete forms.

For discrete forms (11)–(13), the pairing with the basis elements of $C\left(3\right)$ is defined by the following rule:

$$\begin{array}{cc}\hfill & \langle x_{k,s,m},\Phi \rangle ={\Phi }_{k,s,m},\langle V_{k,s,m},\Psi \rangle ={\Psi }_{k,s,m},\hfill \\ \hfill & \langle e_{k,s,m}^1,A\rangle =A_{k,s,m}^1,\langle e_{k,s,m}^2,A\rangle =A_{k,s,m}^2,\langle e_{k,s,m}^3,A\rangle =A_{k,s,m}^3,\hfill \\ \hfill & \langle e_{k,s,m}^{12},B\rangle =B_{k,s,m}^{12},\langle e_{k,s,m}^{13},B\rangle =B_{k,s,m}^{13},\langle e_{k,s,m}^{23},B\rangle =B_{k,s,m}^{23}.\hfill \end{array}$$

(14)

Let $\Omega \in K^r\left(3\right)$ and let $a\in C_{r+1}\left(3\right)$ be an $(r+1)$-chain. As in [24], the coboundary operator $d^c:K^r\left(3\right)\to K^{r+1}\left(3\right)$ is defined through the duality relation

$$\langle a,d^c\Omega \rangle =\langle \partial a,\Omega \rangle ,$$

(15)

where ∂ is given by (10). This operator can be regarded as a discrete analogue of the exterior derivative. Accordingly, for the forms (11)–(13), we have

$$d^c\Phi =\sum _{k,s,m}\left(\left({\Delta }_k{\Phi }_{k,s,m}\right)e_1^{k,s,m}+\left({\Delta }_s{\Phi }_{k,s,m}\right)e_2^{k,s,m}+\left({\Delta }_m{\Phi }_{k,s,m}\right)e_3^{k,s,m}\right),$$

(16)

$$\begin{array}{cc}\hfill d^{c}A=\sum _{k,s,m}& (({\Delta }_k{A}_{k,s,m}^2-{\Delta }_s{A}_{k,s,m}^1)e_{12}^{k,s,m}\hfill \\ & +({\Delta }_k{A}_{k,s,m}^3-{\Delta }_m{A}_{k,s,m}^1)e_{13}^{k,s,m}\hfill \\ \hfill & +({\Delta }_s{A}_{k,s,m}^3-{\Delta }_m{A}_{k,s,m}^2)e_{23}^{k,s,m}),\hfill \end{array}$$

(17)

$$d^{c}B=\sum _{k,s,m}({\Delta }_k{B}_{k,s,m}^{23}-{\Delta }_s{B}_{k,s,m}^{13}+{\Delta }_m{B}_{k,s,m}^{12})V^{k,s,m},$$

(18)

and we have $d^c\Psi =0$. Here, the operators ${\Delta }_k,{\Delta }_s$, and ${\Delta }_m$ are finite difference operators, defined by

$$\begin{array}{cc}\hfill {\Delta }_k{\Phi }_{k,s,m}& ={\Phi }_{\tau k,s,m}-{\Phi }_{k,s,m},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\Delta }_s{\Phi }_{k,s,m}& ={\Phi }_{k,\tau s,m}-{\Phi }_{k,s,m},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\Delta }_m{\Phi }_{k,s,m}& ={\Phi }_{k,s,\tau m}-{\Phi }_{k,s,m}.\hfill \end{array}$$

(21)

Note that for any r-form $\Omega \in K^r\left(3\right)$, the identity

$$d^c(d^c\Omega )=0$$

(22)

holds. This follows directly from (10), using the fact that $\partial \partial =0$, together with (15). For instance, substituting (17) into (18), we obtain

$$\begin{array}{cc}\hfill d^c\left(d^{c}A\right)& =\sum _{k,s,m}({\Delta }_k{\Delta }_s{A}_{k,s,m}^3-{\Delta }_k{\Delta }_m{A}_{k,s,m}^2-{\Delta }_s{\Delta }_k{A}_{k,s,m}^3\hfill \\ \hfill & +{\Delta }_s{\Delta }_m{A}_{k,s,m}^1+{\Delta }_m{\Delta }_k{A}_{k,s,m}^2-{\Delta }_m{\Delta }_s{A}_{k,s,m}^1)V^{k,s,m}=0.\hfill \end{array}$$

Finally, we extend the definitions of the ∪ product and the star operator, as introduced in [24], to the 3-dimensional complex $K\left(3\right)$, For the basis elements of $K\left(3\right)$, the ∪ product is defined as follows

$$x^{k,s,m}\cup x^{k,s,m}=x^{k,s,m},x^{k,s}\cup e_1^{k,s,m}=e_1^{k,s,m},x^{k,s,m}\cup e_2^{k,s,m}=e_2^{k,s,m},$$

$$x^{k,s,m}\cup e_3^{k,s,m}=e_3^{k,s,m},x^{k,s,m}\cup e_{12}^{k,s,m}=e_{12}^{k,s,m},x^{k,s,m}\cup e_{13}^{k,s,m}=e_{13}^{k,s,m},$$

$$x^{k,s,m}\cup e_{23}^{k,s,m}=e_{23}^{k,s,m},x^{k,s,m}\cup V^{k,s,m}=V^{k,s,m},V^{k,s,m}\cup x^{\tau k,\tau s,\tau m}=V^{k,s,m},$$

$$e_1^{k,s,m}\cup x^{\tau k,s,m}=e_1^{k,s,m},e_2^{k,s,m}\cup x^{k,\tau s,m}=e_2^{k,s,m},e_3^{k,s,m}\cup x^{k,s,\tau m}=e_3^{k,s,m},$$

$$e_{12}^{k,s,m}\cup x^{\tau k,\tau s,m}=e_{12}^{k,s,m},e_{13}^{k,s,m}\cup x^{\tau k,s,\tau m}=e_{13}^{k,s,m},e_{23}^{k,s,m}\cup x^{k,\tau s,\tau m}=e_{23}^{k,s,m},$$

$$e_1^{k,s,m}\cup e_2^{\tau k,s,m}=e_{12}^{k,s,m},e_1^{k,s,m}\cup e_3^{\tau k,s,m}=e_{13}^{k,s,m},e_2^{k,s,m}\cup e_1^{k,\tau s,m}=-e_{12}^{k,s,m},$$

$$e_2^{k,s,m}\cup e_3^{k,\tau s,m}=e_{23}^{k,s,m},e_3^{k,s,m}\cup e_1^{k,s,\tau m}=-e_{13}^{k,s,m},e_3^{k,s,m}\cup e_2^{k,s,\tau m}=-e_{23}^{k,s,m},$$

$$e_1^{k,s,m}\cup e_{23}^{\tau k,s,m}=V^{k,s,m},e_2^{k,s,m}\cup e_{13}^{k,\tau s,m}=-V^{k,s,m},e_3^{k,s,m}\cup e_{12}^{k,s,\tau m}=V^{k,s,m},$$

$$e_{12}^{k,s,m}\cup e_3^{\tau k,\tau s,m}=V^{k,s,m},e_{13}^{k,s,m}\cup e_2^{\tau k,s,\tau m}=-V^{k,s,m},e_{23}^{k,s,m}\cup e_1^{k,\tau s,\tau m}=V^{k,s,m}.$$

(23)

In all other cases, the product is defined to be zero. This operation extends to arbitrary forms by linearity. As shown in Chapter 3, Proposition 2 [4], for real-valued discrete forms, the discrete analogue of the Leibniz rule holds:

$$d^c(\Omega \cup \Phi )=d^c\Omega \cup \Phi +{(-1)}^r\Omega \cup d^c\Phi ,$$

(24)

where r is the degree of $\Omega$.

The star operator ∗: $K^r\left(3\right)\to K^{3-r}\left(3\right)$ is defined by the rule:

$$\begin{array}{cc}\hfill \ast x^{k,s,m}& =V^{k,s,m},\ast V^{k,s,m}=x^{\tau k,\tau s,\tau m},\hfill \\ \hfill \ast e_1^{k,s,m}& =e_{23}^{\tau k,s,m},\ast e_2^{k,s,m}=-e_{13}^{k,\tau s,m},\ast e_3^{k,s,m}=e_{12}^{k,s,\tau m},\hfill \\ \hfill \ast e_{12}^{k,s,m}& =e_3^{\tau k,\tau s,m},\ast e_{13}^{k,s,m}=-e_2^{\tau k,s,\tau m},\ast e_{23}^{k,s,m}=e_1^{k,\tau s,\tau m}.\hfill \end{array}$$

(25)

As before, this operation is extended to arbitrary forms by linearity. The operator ∗ exhibits properties analogous to those of the Hodge star operator and can therefore be regarded as its discrete analogue.

Remark 1.

For any discrete forms, the operation $\ast \ast$ results in a shift of all indices of the basis elements, unlike in the continuous case, where $\ast \ast A=A$ for any differential r-form A. For example, for a discrete 1-form, we have

$$\begin{array}{cc}\hfill \ast \ast A& =\sum _{k,s,m}(A_{k,s,m}^1{e}_1^{\tau k,\tau s,\tau m}+A_{k,s,m}^2{e}_2^{\tau k,\tau s,\tau m}+A_{k,s,m}^3{e}_3^{\tau k,\tau s,\tau m})\hfill \\ \hfill & =\sum _{k,s,m}(A_{\sigma k,\sigma s,\sigma m}^1{e}_1^{k,s,m}+A_{\sigma k,\sigma s,\sigma m}^2{e}_2^{k,s,m}+A_{\sigma k,\sigma s,\sigma m}^3{e}_3^{k,s,m}),\hfill \end{array}$$

where σ denotes a unit shift to the left, i.e., $\sigma k=k-1$. Note that this is one of the key differences between our discrete model and the continuous case.

Proposition 1.

For any r-form A we have

$$d^c(\ast \ast A)=\ast \ast d^{c}A.$$

(26)

Proof. 

The proof is a direct computation. Let A be a 1-form. By (17) and (25), it follows that

$$\begin{array}{cc}\hfill d^c(\ast \ast A)& =d^c\sum _{k,s,m}\left(A_{\sigma k,\sigma s,\sigma m}^1{e}_1^{k,s,m}+A_{\sigma k,\sigma s,\sigma m}^2{e}_2^{k,s,m}+A_{\sigma k,\sigma s,\sigma m}^3{e}_3^{k,s,m}\right)\hfill \\ \hfill & =\sum _{k,s,m}(({\Delta }_k{A}_{\sigma k,\sigma s,\sigma m}^2-{\Delta }_s{A}_{\sigma k,\sigma s,\sigma m}^1)e_{12}^{k,s,m}\hfill \\ \hfill & +({\Delta }_k{A}_{\sigma k,\sigma s,\sigma m}^3-{\Delta }_m{A}_{\sigma k,\sigma s,\sigma m}^1)e_{13}^{k,s,m}\hfill \\ \hfill & +({\Delta }_s{A}_{\sigma k,\sigma s,\sigma m}^3-{\Delta }_m{A}_{\sigma k,\sigma s,\sigma m}^2)e_{23}^{k,s,m})\hfill \\ \hfill & =\sum _{k,s,m}(({\Delta }_k{A}_{k,s,m}^2-{\Delta }_s{A}_{k,s,m}^1)e_{12}^{\tau k,\tau s,\tau m}\hfill \\ \hfill & +({\Delta }_k{A}_{k,s,m}^3-{\Delta }_m{A}_{k,s,m}^1)e_{13}^{\tau k,\tau s,\tau m}\hfill \\ \hfill & +({\Delta }_s{A}_{k,s,m}^3-{\Delta }_m{A}_{k,s,m}^2)e_{23}^{\tau k,\tau s,\tau m})\hfill \\ \hfill & =\sum _{k,s,m}(({\Delta }_k{A}_{k,s,m}^2-{\Delta }_s{A}_{k,s,m}^1)\ast \ast e_{12}^{k,s,m}\hfill \\ \hfill & +({\Delta }_k{A}_{k,s,m}^3-{\Delta }_m{A}_{k,s,m}^1)\ast \ast e_{13}^{k,s,m}\hfill \\ \hfill & +({\Delta }_s{A}_{k,s,m}^3-{\Delta }_m{A}_{k,s,m}^2)\ast \ast e_{23}^{k,s,m})\hfill \\ \hfill & =\ast \ast d^{c}A.\hfill \end{array}$$

Similarly, the identity can be derived for 0-forms and 2-forms. □

We define V to be the three-dimensional finite chain with unit coefficients, given by

$$V=\sum _{k=1}^N\sum _{s=1}^S\sum _{m=1}^M{V}_{k,s,m}.$$

(27)

The inner product of discrete forms over V is defined as

$${(\Phi ,\Omega )}_V=\langle V,\Phi \cup \ast \Omega \rangle ,$$

(28)

where $\Phi$ and $\Omega$ are discrete forms of the same degree. If the forms have different degrees, the product (28) is defined to be zero. Form (14) and (25), using the definition of the ∪ product, we obtain the following explicit expressions. For 0-forms or 3-forms of the Form (11), the inner product becomes

$${(\Phi ,\Omega )}_V=\sum _{k=1}^N\sum _{s=1}^S\sum _{m=1}^M{\Phi }_{k,s,m}{\Omega }_{k,s,m}.$$

For 1-forms as in (12), the inner product is given by

$${(\Phi ,\Omega )}_V=\sum _{k=1}^N\sum _{s=1}^S\sum _{m=1}^M({\Phi }_{k,s,m}^1{\Omega }_{k,s,m}^1+{\Phi }_{k,s,m}^2{\Omega }_{k,s,m}^2+{\Phi }_{k,s,m}^3{\Omega }_{k,s,m}^3)$$

and for 2-forms given by (13), it takes the form

$${(\Phi ,\Omega )}_V=\sum _{k=1}^N\sum _{s=1}^S\sum _{m=1}^M({\Phi }_{k,s,m}^{12}{\Omega }_{k,s,m}^{12}+{\Phi }_{k,s,m}^{13}{\Omega }_{k,s,m}^{13}+{\Phi }_{k,s,m}^{23}{\Omega }_{k,s,m}^{23}).$$

The next proposition introduces the adjoint operator of $d^c$ with respect to the inner product (28).

Proposition 2.

Let $\Phi \in K^r\left(3\right)$ and $\Omega \in K^{r+1}\left(3\right)$, where $r=0,1,2$. Then the following identity holds

$${(d^c\Phi ,\Omega )}_V=\langle \partial V,\Phi \cup \ast \Omega \rangle +{(\Phi ,{\delta }^c\Omega )}_V,$$

(29)

where

$${\delta }^c\Omega ={(-1)}^{r+1}{\ast }^{-1}{d}^c\ast \Omega$$

(30)

and ${\ast }^{-1}$ denotes the inverse of the discrete Hodge star operator ∗.

We now extend the definition (30) to the full cochain complex $K\left(3\right)$. It follows that the operator ${\delta }^c:K^{r+1}\left(3\right)\to K^r\left(3\right)$, defined by (29), constitutes the discrete analogue of the codifferential $\delta$.

Note, that by (16), since ${\ast }^{-1}\ast =\mathrm{Id}$, we have

$$\begin{array}{cc}\hfill {\ast }^{-1}{x}^{k,s,m}& =V^{\sigma k,\sigma s,\sigma m},{\ast }^{-1}{V}^{k,s,m}=x^{k,s,m},\hfill \\ \hfill {\ast }^{-1}{e}_1^{k,s,m}& =e_{23}^{k,\sigma s,\sigma m},{\ast }^{-1}{e}_2^{k,s,m}=-e_{13}^{\sigma k,s,\sigma m},{\ast }^{-1}{e}_3^{k,s,m}=e_{12}^{\sigma k,\sigma s,m},\hfill \\ \hfill {\ast }^{-1}{e}_{12}^{k,s,m}& =e_3^{k,s,\sigma m},{\ast }^{-1}{e}_{13}^{k,s,m}=-e_2^{k,\sigma s,m},{\ast }^{-1}{e}_{23}^{k,s,m}=e_1^{\sigma k,s,m}.\hfill \end{array}$$

(31)

Using these relations and (16), along with the definition of $d^c$, we can derive explicit expressions for the operator ${\delta }^c$ for various types of forms. Let $\Phi ,\Psi ,A$ and B be the forms given by (11)–(13). Then we obtain ${\delta }^c\Phi =0$, and

$${\delta }^c\Psi =\sum _{k,s,m}\left(\left({\Delta }_s{\Psi }_{k,\sigma s,m}\right)e_{13}^{k,s,m}-\left({\Delta }_m{\Psi }_{k,s,\sigma m}\right)e_{12}^{k,s,m}-\left({\Delta }_k{\Psi }_{\sigma k,s,m}\right)e_{23}^{k,s,m}\right),$$

(32)

$${\delta }^{c}A=\sum _{k,s,m}(-{\Delta }_k{A}_{\sigma k,s,m}^1-{\Delta }_s{A}_{k,\sigma s,m}^2-{\Delta }_m{A}_{k,s,\sigma m}^3)x^{k,s,m},$$

(33)

$$\begin{array}{cc}\hfill {\delta }^{c}B=\sum _{k,s,m}& (({\Delta }_s{B}_{k,\sigma s,m}^{12}+{\Delta }_m{B}_{k,s,\sigma m}^{13})e_1^{k,s,m}\hfill \\ & -({\Delta }_k{B}_{\sigma k,s,m}^{12}-{\Delta }_m{B}_{k,s,\sigma m}^{23})e_2^{k,s,m}\hfill \\ & -({\Delta }_k{B}_{\sigma k,s,m}^{13}+{\Delta }_s{B}_{k,\sigma s,m}^{23})e_3^{k,s,m}).\hfill \end{array}$$

(34)

The operator

$${\Delta }^c=d^c{\delta }^c+{\delta }^c{d}^c:K^r\left(3\right)\to K^r\left(3\right)$$

(35)

defines a discrete analogue of the Laplacian on the complex $K\left(3\right)$.

3. Discrete Maxwell’s Equations in 3D

In this section, we develop a spatial discretization framework for constructing a semi-discrete analogue of Maxwell’s equations in the three-dimensional case. The discrete model introduced in the previous section is employed to represent the spatial variables, while the temporal variable is treated continuously. Furthermore, we examine the principal properties of the resulting semi-discrete formulation and discuss its relationship with the classical Maxwell’s equations.

Let the discrete analogues of the electric and magnetic field intensities, flux densities, and the electric current and charge densities be defined by the following discrete forms:

$$E=\sum _{k,s,m}\left(E_{k,s,m}^1\left(t\right)e_1^{k,s,m}+E_{k,s,m}^2\left(t\right)e_2^{k,s,m}+E_{k,s,m}^3\left(t\right)e_3^{k,s,m}\right),$$

(36)

$$H=\sum _{k,s,m}\left(H_{k,s,m}^1\left(t\right)e_1^{k,s,m}+H_{k,s,m}^2\left(t\right)e_2^{k,s,m}+H_{k,s,m}^3\left(t\right)e_3^{k,s,m}\right),$$

(37)

$$D=\sum _{k,s,m}\left(D_{k,s,m}^{12}\left(t\right)e_{12}^{k,s,m}+D_{k,s,m}^{13}\left(t\right)e_{13}^{k,s,m}+D_{k,s,m}^{23}\left(t\right)e_{23}^{k,s,m}\right),$$

(38)

$$B=\sum _{k,s,m}\left(B_{k,s,m}^{12}\left(t\right)e_{12}^{k,s,m}+B_{k,s,m}^{13}\left(t\right)e_{13}^{k,s,m}+B_{k,s,m}^{23}\left(t\right)e_{23}^{k,s,m}\right),$$

(39)

$$J=\sum _{k,s,m}\left(J_{k,s,m}^{12}\left(t\right)e_{12}^{k,s,m}+J_{k,s,m}^{13}\left(t\right)e_{13}^{k,s,m}+J_{k,s,m}^{23}\left(t\right)e_{23}^{k,s,m}\right),$$

(40)

$$Q=\sum _{k,s,m}{Q}_{k,s,m}\left(t\right)V^{k,s,m}.$$

(41)

We assume that all components of the above discrete forms are smooth functions of the time variable t. For simplicity, we omit the dependence on t in the components of these forms in what follows.

The semi-discrete counterparts of Maxwell’s Equations (1)–(4), with time remaining continuous, are given by

$$d^{c}E=-{\displaystyle \frac{dB}{dt}},$$

(42)

$$d^{c}H={\displaystyle \frac{dD}{dt}}+J,$$

(43)

$$d^{c}D=Q,$$

(44)

$$d^{c}B=0,$$

(45)

where $d^c$ denotes the discrete exterior derivative and the discrete forms are as defined in (36)–(40). Note that the time derivative operates on the discrete two-form B (and analogously on D) as follows

$${\displaystyle \frac{dB}{dt}}=\sum _{k,s,m}\left({\displaystyle \frac{d{B}_{k,s,m}^{12}}{dt}}{e}_{12}^{k,s,m}+{\displaystyle \frac{d{B}_{k,s,m}^{13}}{dt}}{e}_{13}^{k,s,m}+{\displaystyle \frac{d{B}_{k,s,m}^{23}}{dt}}{e}_{23}^{k,s,m}\right).$$

Using (17), Equation (42)—a semi-discrete analogue of Faraday’s law—can be expressed in terms of difference–differential equations as follows:

$$\begin{array}{cc}\hfill {\Delta }_k{E}_{k,s,m}^2-{\Delta }_s{E}_{k,s,m}^1& =-{\displaystyle \frac{d{B}_{k,s,m}^{12}}{dt}},\hfill \\ \hfill {\Delta }_k{E}_{k,s,m}^3-{\Delta }_m{E}_{k,s,m}^1& =-{\displaystyle \frac{d{B}_{k,s,m}^{13}}{dt}},\hfill \\ \hfill {\Delta }_s{E}_{k,s,m}^3-{\Delta }_m{E}_{k,s,m}^2& =-{\displaystyle \frac{d{B}_{k,s,m}^{23}}{dt}}\hfill \end{array}$$

for all $k,s,m\in \mathbb{Z}$.

Similarly, Equation (43)—a semi-discrete analogue of Ampère’s law—is equivalent to the following system of difference–differential equations:

$$\begin{array}{cc}\hfill {\Delta }_k{H}_{k,s,m}^2-{\Delta }_s{H}_{k,s,m}^1& ={\displaystyle \frac{d{D}_{k,s,m}^{12}}{dt}}+J_{k,s,m}^{12},\hfill \\ \hfill {\Delta }_k{H}_{k,s,m}^3-{\Delta }_m{H}_{k,s,m}^1& ={\displaystyle \frac{d{D}_{k,s,m}^{13}}{dt}}+J_{k,s,m}^{13},\hfill \\ \hfill {\Delta }_s{H}_{k,s,m}^3-{\Delta }_m{H}_{k,s,m}^2& ={\displaystyle \frac{d{D}_{k,s,m}^{23}}{dt}}+J_{k,s,m}^{23}.\hfill \end{array}$$

Finally, using (18), Equation (44)—a semi-discrete analogue of Gauss’ law—and Equation (45)—a discrete analog of Gauss’s law for magnetism—can be represented as

$${\Delta }_k{D}_{k,s,m}^{23}-{\Delta }_s{D}_{k,s,m}^{13}+{\Delta }_m{D}_{k,s,m}^{12}=Q_{k,s,m},$$

and

$${\Delta }_k{B}_{k,s,m}^{23}-{\Delta }_s{B}_{k,s,m}^{13}+{\Delta }_m{B}_{k,s,m}^{12}=0.$$

Using (25), a discrete component-wise representation of the constitutive relations (5) and (6) can be formulated as

$$B_{k,s,m}^{12}={\mu }_0{H}_{k,s,\sigma m}^3,B_{k,s,m}^{13}=-{\mu }_0{H}_{k,\sigma s,m}^2,B_{k,s,m}^{23}={\mu }_0{H}_{\sigma k,s,m}^1,$$

(46)

$$D_{k,s,m}^{12}={\epsilon }_0{E}_{k,s,\sigma m}^3,D_{k,s,m}^{13}=-{\epsilon }_0{E}_{k,\sigma s,m}^2,D_{k,s,m}^{23}={\epsilon }_0{E}_{\sigma k,s,m}^1.$$

(47)

In a similar fashion, a discrete counterpart of the dual relations (7) and (8) takes the form

$$B_{\sigma k,\sigma s,m}^{12}={\mu }_0{H}_{k,s,m}^3,B_{\sigma k,s,\sigma m}^{13}=-{\mu }_0{H}_{k,s,m}^2,B_{k,\sigma s,\sigma m}^{23}={\mu }_0{H}_{k,s,m}^1,$$

(48)

$$D_{\sigma k,\sigma s,m}^{12}={\epsilon }_0{E}_{k,s,m}^3,D_{\sigma k,s,\sigma m}^{13}=-{\epsilon }_0{E}_{k,s,m}^2,D_{k,\sigma s,\sigma m}^{23}={\epsilon }_0{E}_{k,s,m}^1.$$

(49)

It is clear that Equations (46) and (47) are not equivalent to the corresponding Equations (48) and (49), as they are in the continuous case. This discrepancy arises from the definition of the ∗ operator given by (25) (see Remark 1). In the analysis that follows, we employ both sets of Equations (46)–(49).

Let us now present a counterpart of Poynting’s theorem (9) in the framework of discrete forms.

Proposition 3.

The following identity holds

$$d^c(E\cup H)=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(E\cup D+B\cup H)-E\cup J.$$

(50)

Proof. 

From (15) for the 1-form E, we have

$$d^c(E\cup H)=d^{c}E\cup H-E\cup d^{c}H.$$

(51)

By the definition of the ∪ product and using (25), we compute

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}(E\cup \ast E)={\displaystyle \frac{d}{dt}}\sum _{k,s,m}\left({\left(E_{k,s,m}^1\right)}^2+{\left(E_{k,s,m}^2\right)}^2+{\left(E_{k,s,m}^3\right)}^2\right)V^{k,s,m}\\ \hfill =\sum _{k,s,m}\left({\displaystyle \frac{d}{dt}}{\left(E_{k,s,m}^1\right)}^2+{\displaystyle \frac{d}{dt}}{\left(E_{k,s,m}^2\right)}^2+{\displaystyle \frac{d}{dt}}{\left(E_{k,s,m}^3\right)}^2\right)V^{k,s,m}\\ \hfill =\sum _{k,s,m}\left(2E_{k,s,m}^1{\displaystyle \frac{d{E}_{k,s,m}^1}{dt}}+2E_{k,s,m}^2{\displaystyle \frac{d{E}_{k,s,m}^2}{dt}}+2E_{k,s,m}^3{\displaystyle \frac{d{E}_{k,s,m}^3}{dt}}\right)V^{k,s,m}\\ \hfill =2E\cup \ast {\displaystyle \frac{dE}{dt}}=2E\cup {\displaystyle \frac{d\ast E}{dt}}.\end{array}$$

Therefore,

$$E\cup {\displaystyle \frac{d\ast E}{dt}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(E\cup \ast E).$$

(52)

Similarly, we obtain

$${\displaystyle \frac{dB}{dt}}\cup \ast B={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(B\cup \ast B).$$

(53)

From the semi-discrete Maxwell’s equation for the electric field (42), using (48) and (53), it follows that

$$d^{c}E\cup H=-{\displaystyle \frac{dB}{dt}}\cup H=-{\displaystyle \frac{1}{{\mu }_0}}{\displaystyle \frac{dB}{dt}}\cup \ast B=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{{\mu }_0}}{\displaystyle \frac{d}{dt}}(B\cup \ast B)=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(B\cup H).$$

Similarly, from the semi-discrete Maxwell’s equation for the magnetic field (43), using (47) and (52), we get

$$\begin{array}{cc}\hfill E\cup d^{c}H& =E\cup {\displaystyle \frac{dD}{dt}}+E\cup J={\epsilon }_{0}E\cup {\displaystyle \frac{d\ast E}{dt}}+E\cup J\hfill \\ & ={\displaystyle \frac{1}{2}}{\epsilon }_0{\displaystyle \frac{d}{dt}}(E\cup \ast E)+E\cup J={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(E\cup D)+E\cup J.\hfill \end{array}$$

Substituting these into Equation (51), we obtain the desired result (50). □

The relation (50) captures the conservation of electromagnetic energy in the discrete setting, mirroring the continuous Poynting theorem while being adapted to the algebraic and topological structure of discrete forms. It should be noted that only the present definition of the discrete Hodge star, in conjunction with the corresponding form of the constitutive relations, permits the derivation of discrete energy-conservation relations that closely parallel those of the continuous theory.

Using definition of the ∪-product and by (18), identity (50) can be written in component form as

$$\begin{array}{cc}\hfill & {\Delta }_m(E_{k,s,m}^1{H}_{\tau k,s,m}^2-E_{k,s,m}^2{H}_{k,\tau s,m}^1)\hfill \\ \hfill & -{\Delta }_s(E_{k,s,m}^1{H}_{\tau k,s,m}^3-E_{k,s,m}^3{H}_{k,s,\tau m}^1)\hfill \\ \hfill & +{\Delta }_k(E_{k,s,m}^2{H}_{k,\tau s,m}^3-E_{k,s,m}^3{H}_{k,s,\tau m}^2)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(E_{k,s,m}^1{D}_{\tau k,s,m}^{23}-E_{k,s,m}^2{D}_{k,\tau s,m}^{13}+E_{k,s,m}^3{D}_{k,s,\tau m}^{12})\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}(B_{k,s,m}^{12}{H}_{\tau k,\tau s,m}^3-B_{k,s,m}^{13}{H}_{\tau k,s,\tau m}^2+B_{k,s,m}^{23}{H}_{k,\tau s,\tau m}^1)\hfill \\ \hfill & -E_{k,s,m}^1{J}_{\tau k,s,m}^{23}+E_{k,s,m}^2{J}_{k,\tau s,m}^{13}-E_{k,s,m}^3{J}_{k,s,\tau m}^{12}.\hfill \end{array}$$

Let us now consider the semi-discrete Maxwell’s equations in the special case where the charge density is set to zero, i.e., $Q=0$. Hence, Equation (44) becomes homogeneous. Using (30) and (47), we compute

$${\delta }^{c}E=-{\ast }^{-1}{d}^c\ast E=-{\displaystyle \frac{1}{{\epsilon }_0}}{\ast }^{-1}{d}^{c}D.$$

Since $d^{c}D=0$, it follows that

$${\delta }^{c}E=0.$$

(54)

Applying ${\delta }^c$ to both sides of Equation (42) and using the identities (26) and (46), we obtain

$${\delta }^c{d}^{c}E=-{\displaystyle \frac{d\left({\delta }^{c}B\right)}{dt}}=-{\mu }_0{\displaystyle \frac{d}{dt}}\left({\ast }^{-1}{d}^c\ast \ast H\right)=-{\mu }_0\ast {\displaystyle \frac{d\left(d^{c}H\right)}{dt}}.$$

Using Equation (43), we then have

$${\delta }^c{d}^{c}E=-{\mu }_0\ast {\displaystyle \frac{d^{2}D}{d{t}^2}}-{\mu }_0\ast {\displaystyle \frac{dJ}{dt}}.$$

By (47), this yields

$${\delta }^c{d}^{c}E+{\mu }_0{\epsilon }_0{\displaystyle \frac{d^2(\ast \ast E)}{d{t}^2}}=-{\mu }_0\ast {\displaystyle \frac{dJ}{dt}}.$$

Taking into account (35) and (54) this equation can be rewritten in the form

$${\Delta }^{c}E+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2(\ast \ast E)}{d{t}^2}}=-{\mu }_0\ast {\displaystyle \frac{dJ}{dt}}.$$

(55)

Recall that ${\mu }_0{\epsilon }_0={\displaystyle \frac{1}{c^2}}$, where c is the speed of light in vacuum. Thus, Equation (55) represents a semi-discrete analogue of the wave equation for the electric field. Equation (60) is equivalent to the following system of the difference–differential equations

$$\begin{array}{cc}\hfill & -{\left({\Delta }_k\right)}^2{E}_{\sigma k,s,m}^1-{\left({\Delta }_s\right)}^2{E}_{k,\sigma s,m}^1-{\left({\Delta }_m\right)}^2{E}_{k,s,\sigma m}^1+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{E}_{\sigma k,\sigma s,\sigma m}^1}{d{t}^2}}\hfill \\ \hfill & =-{\mu }_0{\displaystyle \frac{d{J}_{k,\sigma s,\sigma m}^{23}}{dt}},\hfill \\ \hfill & -{\left({\Delta }_k\right)}^2{E}_{\sigma k,s,m}^2-{\left({\Delta }_s\right)}^2{E}_{k,\sigma s,m}^2-{\left({\Delta }_m\right)}^2{E}_{k,s,\sigma m}^2+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{E}_{\sigma k,\sigma s,\sigma m}^2}{d{t}^2}}\hfill \\ \hfill & ={\mu }_0{\displaystyle \frac{d{J}_{\sigma k,s,\sigma m}^{13}}{dt}},\hfill \\ \hfill & -{\left({\Delta }_k\right)}^2{E}_{\sigma k,s,m}^3-{\left({\Delta }_s\right)}^2{E}_{k,\sigma s,m}^3-{\left({\Delta }_m\right)}^2{E}_{k,s,\sigma m}^3+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{E}_{\sigma k,\sigma s,\sigma m}^3}{d{t}^2}}\hfill \\ \hfill & =-{\mu }_0{\displaystyle \frac{d{J}_{\sigma k,\sigma s,m}^{12}}{dt}},\hfill \end{array}$$

where ${\left({\Delta }_k\right)}^2={\Delta }_k{\Delta }_k$.

Let us introduce the semi-discrete counterparts of the electromagnetic potentials. For reference to the continuous setting, see, for example [22]. As in the continuous theory, a semi-discrete version of the wave equation for the discrete potentials can be derived from the semi-discrete Maxwell equations.In our discrete formulation, the existence of potentials does not require any additional assumptions beyond those already imposed on the complex $K\left(3\right)$. The structure of $K\left(3\right)$ guarantees that the fields that the fields considered here admit potentials. Since the discrete magnetic flux density B satisfies Equation (45), then by (22), there is a 1-form A such that

$$B=d^{c}A.$$

(56)

By analogy with the continuous case, this 1-form A is called the discrete magnetic vector potential. Substituting (56) into Equation (42) yields

$$d^c\left(E+{\displaystyle \frac{dA}{dt}}\right)=0.$$

It follows, according to (22), that the discrete electric 1-form E can be expressed as

$$E=-d^c\Phi -{\displaystyle \frac{dA}{dt}},$$

(57)

where $\Phi$ is a 0-form. We interpret $\Phi$ as the discrete scalar potential.

Proposition 4.

The 1-form E, given by (57), is invariant under the following transformation

$$A^{\prime }=A+d^c\Psi ,{\Phi }^{\prime }=\Phi -{\displaystyle \frac{d\Psi }{dt}}.$$

(58)

Proof. 

Assume that

$$E^{\prime }=-d^c{\Phi }^{\prime }-{\displaystyle \frac{d{A}^{\prime }}{dt}}.$$

(59)

Substituting (58) into (59) yields

$$E^{\prime }=-d^c\left(\Phi -{\displaystyle \frac{d\Psi }{dt}}\right)-{\displaystyle \frac{d}{dt}}\left(A+d^c\Psi \right)=-d^c\Phi +d^c\left({\displaystyle \frac{d\Psi }{dt}}\right)-{\displaystyle \frac{dA}{dt}}-{\displaystyle \frac{d}{dt}}(d^c\Psi ).$$

Since the time derivative and the discrete exterior derivative commute, the middle terms cancel, i.e.,

$$d^c\left({\displaystyle \frac{d\Psi }{dt}}\right)={\displaystyle \frac{d}{dt}}(d^c\Psi ).$$

Thus,

$$E^{\prime }=-d^c\Phi -{\displaystyle \frac{dA}{dt}}=E.$$

□

The transformation (58) serves as a semi-discrete analogue of a gauge transformation. The semi-discrete Maxwell’s equations remain invariant under this transformation. As in the continuous case, gauge invariance underlies the symmetry of the semi-discrete Maxwell’s equations, indicating that they are unaffected by transformations of the form (58). This symmetry allows the introduction of an analogue of the Lorentz gauge condition in the semi-discrete setting. In our hybrid discrete–continuous framework, a semi-discrete counterpart of the Lorentz gauge condition is given by

$$-{\delta }^{c}A+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d\Phi }{dt}}=0.$$

(60)

Recall that for a 1-form A, we have ${\delta }^{c}A=-{\ast }^{-1}{d}^c\ast A$. Applying (47), namely $D={\epsilon }_0\ast E$, and substituting (57) into Equation (43) we obtain

$$d^{c}H={\epsilon }_0{\displaystyle \frac{d\ast E}{dt}}+J={\epsilon }_0\ast {\displaystyle \frac{d}{dt}}\left(-d^c\Phi -{\displaystyle \frac{dA}{dt}}\right)+J.$$

From this, applying (48), i.e., $\ast B={\mu }_{0}H$, and by (56) we have

$$d^c\ast d^{c}A={\mu }_0{\epsilon }_0\ast \left(-d^c{\displaystyle \frac{d\Phi }{dt}}-{\displaystyle \frac{d^{2}A}{d{t}^2}}\right)+{\mu }_{0}J.$$

Acting with ${\ast }^{-1}$ on both sides and using the gauge condition (60) we obtain

$${\delta }^c{d}^{c}A=-d^c{\delta }^{c}A-{\mu }_0{\epsilon }_0{\displaystyle \frac{d^{2}A}{d{t}^2}}+{\mu }_0{\ast }^{-1}J.$$

Thus, using the notation (35), we arrive at a semi-discrete analogue of the wave equation for the potential 1-form A:

$${\Delta }^{c}A+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^{2}A}{d{t}^2}}={\mu }_0{\ast }^{-1}J.$$

(61)

Using the definitions of the operators $d^c,{\delta }^c$, and applying (31), Equation (61) can be decomposed into the following system of the difference–differential equations

$$\begin{array}{cc}\hfill & -{\left({\Delta }_k\right)}^2{A}_{\sigma k,s,m}^1-{\left({\Delta }_s\right)}^2{A}_{k,\sigma s,m}^1-{\left({\Delta }_m\right)}^2{A}_{k,s,\sigma m}^1+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{A}_{k,s,m}^1}{d{t}^2}}={\mu }_0{J}_{\tau k,s,m}^{23},\hfill \\ \hfill & -{\left({\Delta }_k\right)}^2{A}_{\sigma k,s,m}^2-{\left({\Delta }_s\right)}^2{A}_{k,\sigma s,m}^2-{\left({\Delta }_m\right)}^2{A}_{k,s,\sigma m}^2+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{A}_{k,s,m}^2}{d{t}^2}}=-{\mu }_0{J}_{k,\tau s,m}^{13},\hfill \\ \hfill & -{\left({\Delta }_k\right)}^2{A}_{\sigma k,s,m}^3-{\left({\Delta }_s\right)}^2{A}_{k,\sigma s,m}^3-{\left({\Delta }_m\right)}^2{A}_{k,s,\sigma m}^3+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{A}_{k,s,m}^3}{d{t}^2}}={\mu }_0{J}_{k,s,\tau m}^{12}.\hfill \end{array}$$

In the same way, we derive a semi-discrete analogue of the wave equation for the scalar potential $\Phi$. Substitution (47) and (57) into (44), we obtain

$$-{\epsilon }_0{d}^c\ast d^c\Phi -{\epsilon }_0{\displaystyle \frac{d}{dt}}\left(d^c\ast A\right)=Q.$$

Applying ${\ast }^{-1}$ to both sides and using (30) along with the gauge condition (60), we obtain

$${\delta }^c{d}^c\Phi +{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2\Phi }{d{t}^2}}={\displaystyle \frac{1}{{\epsilon }_0}}{\ast }^{-1}Q.$$

Since, by definition, ${\delta }^c\Phi =0$, it follows that ${\delta }^c{d}^c\Phi ={\Delta }^c\Phi$, and thus we obtain the semi-discrete analog of the wave equation in the form

$${\Delta }^c\Phi +{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2\Phi }{d{t}^2}}={\displaystyle \frac{1}{{\epsilon }_0}}{\ast }^{-1}Q.$$

Accordingly, for any components of the forms $\Phi$ and Q we have

$$-{\left({\Delta }_k\right)}^2{\Phi }_{\sigma k,s,m}-{\left({\Delta }_s\right)}^2{\Phi }_{k,\sigma s,m}-{\left({\Delta }_m\right)}^2{\Phi }_{k,s,\sigma m}+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{\Phi }_{k,s,m}}{d{t}^2}}={\displaystyle \frac{1}{{\epsilon }_0}}{Q}_{k,s,m}.$$

4. 2D Discrete Maxwell’s Equations on a Combinatorial Torus

In this section, we reduce our semi-discrete model of the three-dimensional Maxwell’s equations to the two-dimensional case. To this end, we adopt a combinatorial model of the two-dimensional Euclidean space ${\mathbb{R}}^2$, as detailed in [24] or [3]. As an illustrative example, we consider the semi-discrete Maxwell’s equations on a combinatorial torus and derive an explicit expression for the general solution in this setting.

On the two-dimensional chain complex $C\left(2\right)=C\otimes C$, representing a combinatorial plane, the semi-discrete Maxwell’s equations retain the same form as in (42)–(45). The discrete electric field intensity E remains a 1-form, expressed as

$$E=\sum _{k,s}\left(E_{k,s}^1{e}_1^{k,s}+E_{k,s}^2{e}_2^{k,s}\right).$$

In this two-dimensional setting, the discrete magnetic field intensity H becomes a 0-form

$$H=\sum _{k,s}{H}_{k,s}{x}^{k,s}.$$

Accordingly, the discrete magnetic flux density B and the discrete charge density Q are represented as 2-forms:

$$B=\sum _{k,s}{B}_{k,s}{V}^{k,s},Q=\sum _{k,s}{Q}_{k,s}{V}^{k,s}.$$

The discrete electric flux density field D and the discrete current density J become 1-forms:

$$D=\sum _{k,s}\left(D_{k,s}^1{e}_1^{k,s}+D_{k,s}^2{e}_2^{k,s}\right),J=\sum _{k,s}\left(J_{k,s}^1{e}_1^{k,s}+J_{k,s}^2{e}_2^{k,s}\right).$$

Following the notation in [24], we have

$$d^{c}E=\sum _{k,s}\left({\Delta }_k{E}_{k,s}^2-{\Delta }_s{E}_{k,s}^1\right)V^{k,s}.$$

(62)

Then, the two-dimensional version of Equation (42) can be written in component form as

$${\Delta }_k{E}_{k,s}^2-{\Delta }_s{E}_{k,s}^1=-{\displaystyle \frac{d{B}_{k,s}}{dt}}$$

(63)

for any $k,s\in \mathbb{Z}$. Similarly, we obtain the discrete analogue of Equation (44)

$${\Delta }_k{D}_{k,s}^2-{\Delta }_s{D}_{k,s}^1=Q_{k,s}.$$

(64)

Since for the 0-form H we have

$$d^{c}H=\sum _{k,s}\left(\left({\Delta }_k{H}_{k,s}\right)e_{k,s}^1+\left({\Delta }_s{H}_{k,s}\right)e_{k,s}^2\right),$$

(65)

Equation (43) is equivalent to the following system of difference–differential equations

$$\begin{array}{cc}\hfill {\Delta }_k{H}_{k,s}& ={\displaystyle \frac{d{D}_{k,s}^1}{dt}}+J_{k,s}^1,\hfill \\ \hfill {\Delta }_s{H}_{k,s}& ={\displaystyle \frac{d{D}_{k,s}^2}{dt}}+J_{k,s}^2.\hfill \end{array}$$

(66)

Finally, since in the 2-dimensional case $d^{c}B=0$ for any 2-form B, Equation (45) holds as an identity.

By the definition of the operation ∗ on the complex $K\left(2\right)$, as given in [24], we have

$$\ast E=\sum _{k,s}\left(-E_{k,\sigma s}^2{e}_1^{k,s}+E_{\sigma k,s}^1{e}_2^{k,s}\right),\ast H=\sum _{k,s}{H}_{k,s}{V}^{k,s}.$$

Then the two-dimensional discrete versions of the relations (47) and (46) can be written as

$$\begin{array}{cc}\hfill D_{k,s}^1& =-{\epsilon }_0{E}_{k,\sigma s}^2,\hfill \\ \hfill D_{k,s}^2& ={\epsilon }_0{E}_{\sigma k,s}^1,\hfill \end{array}$$

(67)

and

$$B_{k,s}={\mu }_0{H}_{k,s}.$$

(68)

It should be noted that in the two-dimensional model, we have

$$\ast \ast E=-\sum _{k,s}\left(E_{\sigma k,\sigma s}^1{e}_1^{k,s}+E_{\sigma k,\sigma s}^2{e}_2^{k,s}\right),\ast \ast H=\sum _{k,s}{H}_{\sigma k,\sigma s}{V}^{k,s}.$$

It follows immediately that for any r-form A the following identity holds

$$d^c(\ast \ast A)=-\ast \ast d^{c}A.$$

(69)

Compared to the three-dimensional case (see Formula (26)), the only difference is the sign on the right-hand side.

Similarly to the previous section, we now derive a semi-discrete wave equation for the discrete electric 1-form E in the two dimensional case. From the semi-discrete Maxwell’s equations, using (69), we have

$${\ast }^{-1}{d}^c\ast d^{c}E={\mu }_0\ast {\displaystyle \frac{d^{2}D}{d{t}^2}}+{\mu }_0\ast {\displaystyle \frac{dJ}{dt}}.$$

Using the definition of ${\delta }^c$ given by (30) and applying (67), this equation can be rewritten as

$${\delta }^c{d}^{c}E={\epsilon }_0{\mu }_0{\displaystyle \frac{d^2(\ast \ast E)}{d{t}^2}}+{\mu }_0\ast {\displaystyle \frac{dJ}{dt}}.$$

Assuming that the 2-form Q is equal to zero and using (35), we then obtain the semi-discrete wave equation in the form

$${\Delta }^{c}E+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2(\ast \ast E)}{d{t}^2}}={\mu }_0\ast {\displaystyle \frac{dJ}{dt}}.$$

(70)

Equation (70) is equivalent to the following system:

$$\begin{array}{cc}\hfill 4E_{k,s}^1-E_{\sigma k,s}^1-E_{k,\sigma s}^1-E_{\tau k,s}^1-E_{k,\tau s}^1-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{E}_{\sigma k,\sigma s}^1}{d{t}^2}}& =-{\mu }_0{\displaystyle \frac{d{J}_{k,\sigma s}^2}{dt}},\hfill \\ \hfill 4E_{k,s}^2-E_{\sigma k,s}^2-E_{k,\sigma s}^2-E_{\tau k,s}^2-E_{k,\tau s}^2-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{d^2{E}_{\sigma k,\sigma s}^2}{d{t}^2}}& ={\mu }_0{\displaystyle \frac{d{J}_{\sigma k,s}^1}{dt}}.\hfill \end{array}$$

Now, following [24], let us examine the two-dimensional semi-discrete Maxwell’s equations on a combinatorial torus in more detail. To begin, we associate the basis elements of the chain complex $C\left(2\right)$ with corresponding geometric objects in ${\mathbb{R}}^2$. As described in [24], consider a tiling of the plane ${\mathbb{R}}^2$ formed by the grid lines $x=k$ and $y=s$, where $k,s\in \mathbb{Z}$. Each open square defined by these lines is denoted by $V_{k,s}$, with its vertices labeled $x_{k,s},x_{\tau k,s},x_{k,\tau s}$, $x_{\tau k,\tau s}$, where $\tau k=k+1$. We define the edges $e_{k,s}^1$ and $e_{k,s}^2$ as the open intervals $(x_{k,s},x_{\tau k,s})$ and $(x_{k,s},x_{k,\tau s})$, respectively. These geometric elements correspond directly to the combinatorial objects - that is, the basis elements of the complex $C\left(2\right)$. Next, we introduce a combinatorial torus. Recall that the torus can be regarded as the topological space obtained by taking a rectangle and identifying each pair of opposite sides with the same orientation. Let V denote the open square that corresponds to the following 2-dimensional chain

$$V=V_{1,1}+V_{2,1}+V_{1,2}+V_{2,2}.$$

We then identify the points and intervals on the boundary of V as follows:

$$\begin{array}{c}\hfill x_{1,1}=x_{3,1}=x_{1,3}=x_{3,3},x_{1,2}=x_{3,2},x_{2,1}=x_{2,3},\\ \hfill e_{1,1}^1=e_{1,3}^1,e_{2,1}^1=e_{2,3}^1,e_{1,1}^2=e_{3,1}^2,e_{1,2}^2=e_{3,2}^2.\end{array}$$

(71)

The resulting geometric object is homeomorphic to the torus. For a visual representation, see Figure 1 in [24]. Let $C\left(T\right)$ denote the chain complex associated with this structure, referred to as a combinatorial model of the torus. Correspondingly, let $K\left(T\right)$ represent the cochain complex defined over $C\left(T\right)$. It is clear that the components of discrete forms defined on $C\left(T\right)$ satisfy the same conditions as in (71).

On the combinatorial torus $C\left(T\right)$, the 1-form E, the 0-form H, and the 2-form B can be expressed as

$$\begin{array}{cc}\hfill E& =E_{1,1}^1{e}_1^{1,1}+E_{2,1}^1{e}_1^{2,1}+E_{1,2}^2{e}_2^{1,2}+E_{1,1}^2{e}_2^{1,1}\hfill \\ \hfill & +E_{1,2}^1{e}_1^{1,2}+E_{2,2}^1{e}_1^{2,2}+E_{2,2}^2{e}_2^{2,2}+E_{2,1}^2{e}_2^{2,1},\hfill \end{array}$$

$$H=H_{1,1}{x}^{1,1}+H_{2,1}{x}^{2,1}+H_{1,2}{x}^{1,2}+H_{2,2}{x}^{2,2},$$

and

$$B=B_{1,1}{V}^{1,1}+B_{2,1}{V}^{2,1}+B_{1,2}{V}^{1,2}+B_{2,2}{V}^{2,2}.$$

Using this notation, the discrete exterior derivatives $d^{c}E$ and $d^{c}H$, given by (62) and (65), take the form

$$\begin{array}{c}\hfill d^{c}E=(E_{1,1}^1-E_{1,2}^1+E_{2,1}^2-E_{1,1}^2)V^{1,1}+(E_{2,1}^1-E_{2,2}^1-E_{2,1}^2+E_{1,1}^2)V^{2,1}\\ \hfill +(E_{1,2}^1-E_{1,1}^1+E_{2,2}^2-E_{1,2}^2)V^{1,2}+(E_{2,2}^1-E_{2,1}^1+E_{1,2}^2-E_{2,2}^2)V^{2,2}.\end{array}$$

$$\begin{array}{cc}\hfill d^{c}H& =(H_{2,1}-H_{1,1})e_1^{1,1}+(H_{1,1}-H_{2,1})e_1^{2,1}+(H_{1,1}-H_{1,2})e_2^{1,2}\hfill \\ \hfill & +(H_{1,2}-H_{1,1})e_2^{1,1}+(H_{2,2}-H_{1,2})e_1^{1,2}+(H_{1,2}-H_{2,2})e_1^{2,2}\hfill \\ \hfill & +(H_{2,1}-H_{2,2})e_2^{2,2}+(H_{2,2}-H_{2,1})e_2^{2,1}.\hfill \end{array}$$

Accordingly, Equation (63) on $C\left(T\right)$ becomes:

$$\begin{array}{c}\hfill E_{1,1}^1-E_{1,2}^1+E_{2,1}^2-E_{1,1}^2=-{\displaystyle \frac{d{B}_{1,1}}{dt}},\\ \hfill E_{2,1}^1-E_{2,2}^1-E_{2,1}^2+E_{1,1}^2=-{\displaystyle \frac{d{B}_{2,1}}{dt}},\\ \hfill E_{1,2}^1-E_{1,1}^1+E_{2,2}^2-E_{1,2}^2=-{\displaystyle \frac{d{B}_{1,2}}{dt}},\\ \hfill E_{2,2}^1-E_{2,1}^1+E_{1,2}^2-E_{2,2}^2=-{\displaystyle \frac{d{B}_{2,2}}{dt}}.\end{array}$$

(72)

Similarly, Equation (64) takes the form:

$$\begin{array}{c}\hfill D_{1,1}^1-D_{1,2}^1+D_{2,1}^2-D_{1,1}^2=Q_{1,1},\\ \hfill D_{2,1}^1-D_{2,2}^1-D_{2,1}^2+D_{1,1}^2=Q_{2,1},\\ \hfill D_{1,2}^1-D_{1,1}^1+D_{2,2}^2-D_{1,2}^2=Q_{1,2},\\ \hfill D_{2,2}^1-D_{2,1}^1+D_{1,2}^2-D_{2,2}^2=Q_{2,2}.\end{array}$$

(73)

Finally, the system (66) reads:

$$\begin{array}{cc}\hfill H_{2,1}-H_{1,1}& ={\displaystyle \frac{d{D}_{1,1}^1}{dt}}+J_{1,1}^1,\hfill \\ \hfill H_{1,1}-H_{2,1}& ={\displaystyle \frac{d{D}_{2,1}^1}{dt}}+J_{2,1}^1,\hfill \\ \hfill H_{1,1}-H_{1,2}& ={\displaystyle \frac{d{D}_{1,2}^2}{dt}}+J_{1,2}^2,\hfill \\ \hfill H_{1,2}-H_{1,1}& ={\displaystyle \frac{d{D}_{1,1}^2}{dt}}+J_{1,1}^2,\hfill \\ \hfill H_{2,2}-H_{1,2}& ={\displaystyle \frac{d{D}_{1,2}^1}{dt}}+J_{1,2}^1,\hfill \\ \hfill H_{1,2}-H_{2,2}& ={\displaystyle \frac{d{D}_{2,2}^1}{dt}}+J_{2,2}^1,\hfill \\ \hfill H_{2,1}-H_{2,2}& ={\displaystyle \frac{d{D}_{2,2}^2}{dt}}+J_{2,2}^2,\hfill \\ \hfill H_{2,2}-H_{2,1}& ={\displaystyle \frac{d{D}_{2,1}^2}{dt}}+J_{2,1}^2.\hfill \end{array}$$

(74)

Thus, Equations (72)–(74) represent a semi-discrete counterpart of Maxwell’s equations on the combinatorial torus. According to (71) the relations (67) and (68) become

$$\begin{array}{cc}\hfill D_{1,1}^1& =-{\epsilon }_0{E}_{1,2}^2,D_{2,1}^1=-{\epsilon }_0{E}_{2,2}^2,D_{1,2}^1=-{\epsilon }_0{E}_{1,1}^2,D_{2,2}^1=-{\epsilon }_0{E}_{2,1}^2,\hfill \\ \hfill D_{1,1}^2& ={\epsilon }_0{E}_{2,1}^1,D_{2,1}^2={\epsilon }_0{E}_{1,1}^1,D_{1,2}^2={\epsilon }_0{E}_{2,2}^1,D_{2,2}^2={\epsilon }_0{E}_{1,2}^1,\hfill \end{array}$$

(75)

and

$$B_{1,1}={\mu }_0{H}_{1,1},B_{2,1}={\mu }_0{H}_{2,1},B_{1,2}={\mu }_0{H}_{1,2},B_{2,2}={\mu }_0{H}_{2,2}.$$

(76)

A natural question in this framework is whether the system of semi-discrete Maxwell equations on the combinatorial torus is solvable. The following discussion addresses this question. For simplicity, we adopt natural units in which the fundamental constants ${\mu }_0$ and ${\epsilon }_0$ are set to 1. We also assume that $Q=0$ and $J=0$, meaning that we are considering a region free of charges and currents. Under these assumptions, and using relations (75) and (76), Equations (72) and (74) reduce to the following system of linear homogeneous ordinary differential equations:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{E}_{1,1}^1}{dt}}& =H_{2,2}-H_{2,1},\hfill \\ \hfill {\displaystyle \frac{d{E}_{2,1}^1}{dt}}& =H_{1,2}-H_{1,1},\hfill \\ \hfill {\displaystyle \frac{d{E}_{1,2}^2}{dt}}& =-H_{2,1}+H_{1,1},\hfill \\ \hfill {\displaystyle \frac{d{E}_{1,1}^2}{dt}}& =-H_{2,2}+H_{1,2},\hfill \\ \hfill {\displaystyle \frac{d{E}_{1,2}^1}{dt}}& =H_{2,1}-H_{2,2},\hfill \\ \hfill {\displaystyle \frac{d{E}_{2,2}^1}{dt}}& =H_{1,1}-H_{1,2},\hfill \\ \hfill {\displaystyle \frac{d{E}_{2,2}^2}{dt}}& =-H_{1,1}+H_{2,1},\hfill \\ \hfill {\displaystyle \frac{d{E}_{2,1}^2}{dt}}& =-H_{1,2}+H_{2,2},\hfill \\ \hfill {\displaystyle \frac{d{H}_{1,1}}{dt}}& =-E_{1,1}^1+E_{1,2}^1-E_{2,1}^2+E_{1,1}^2,\hfill \\ \hfill {\displaystyle \frac{d{H}_{2,1}}{dt}}& =-E_{2,1}^1+E_{2,2}^1+E_{2,1}^2-E_{1,1}^2,\hfill \\ \hfill {\displaystyle \frac{d{H}_{1,2}}{dt}}& =-E_{1,2}^1+E_{1,1}^1-E_{2,2}^2+E_{1,2}^2,\hfill \\ \hfill {\displaystyle \frac{d{H}_{2,2}}{dt}}& =-E_{2,2}^1+E_{2,1}^1-E_{1,2}^2+E_{2,2}^2.\hfill \end{array}$$

(77)

Similarly, the system (73) becomes:

$$\begin{array}{c}\hfill E_{1,1}^1-E_{2,1}^1-E_{1,2}^2+E_{1,1}^2=0,\\ \hfill E_{2,1}^1-E_{1,1}^1-E_{2,2}^2+E_{2,1}^2=0,\\ \hfill E_{1,2}^1-E_{2,2}^1-E_{1,1}^2+E_{1,2}^2=0,\\ \hfill E_{2,2}^1-E_{1,2}^1+E_{2,2}^2-E_{2,1}^2=0.\end{array}$$

(78)

We proceed as in [24] and present a matrix form of Equations (77) and (78). Let us introduce the following row vectors:

$$\left[H\right]=\left[H_{1,1}{H}_{2,1}{H}_{1,2}{H}_{2,2}\right],\left[E\right]=\left[E_{1,1}^1{E}_{2,1}^1{E}_{1,2}^2{E}_{1,1}^2{E}_{1,2}^1{E}_{2,2}^1{E}_{2,2}^2{E}_{2,1}^2\right],$$

$$\left[EH\right]=\left[\begin{array}{c}{E}_{1,1}^1{E}_{2,1}^1{E}_{1,2}^2{E}_{1,1}^2{E}_{1,2}^1{E}_{2,2}^1{E}_{2,2}^2{E}_{2,1}^2{H}_{1,1}{H}_{2,1}{H}_{1,2}{H}_{2,2}\end{array}\right].$$

Denote by ${[·]}^{\top }$ the corresponding column vector. Using this notation, Equations (77) and (78) can be rewritten as

$${\displaystyle \frac{d}{dt}}{\left[EH\right]}^{\top }=\mathcal{M}·{\left[EH\right]}^{\top },$$

(79)

and

$${\mathcal{M}}_1·{\left[EH\right]}^{\top }={\left[0\right]}^{\top },$$

(80)

respectively, where

$$\mathcal{M}=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& -1\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 1\\ -1& 0& 0& 1& 1& 0& 0& -1& 0& 0& 0& 0\\ 0& -1& 0& -1& 0& 1& 0& 1& 0& 0& 0& 0\\ 1& 0& 1& 0& -1& 0& -1& 0& 0& 0& 0& 0\\ 0& 1& -1& 0& 0& -1& 1& 0& 0& 0& 0& 0\end{array}\right]$$

and

$${\mathcal{M}}_1=\left[\begin{array}{cccccccc}1& -1& -1& 1& 0& 0& 0& 0\\ -1& 1& 0& 0& 0& 0& -1& 1\\ 0& 0& 1& -1& 1& -1& 0& 0\\ 0& 0& 0& 0& -1& 1& 1& -1\end{array}\right].$$

By applying row reduction, we obtain the row echelon form of ${\mathcal{M}}_1$:

$$\left[\begin{array}{cccccccc}1& -1& 0& 0& 0& 0& 1& -1\\ 0& 0& 1& -1& 0& 0& 1& -1\\ 0& 0& 0& 0& 1& -1& -1& 1\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right].$$

Hence, the matrix ${\mathcal{M}}_1$ has rank 3. It follows that a solution of Equation (80) (or, equivalently, the system (78)) can be expressed as

$$\begin{array}{c}\hfill E_{1,1}^1=E_{2,1}^1-E_{2,2}^2+E_{2,1}^2,\\ \hfill E_{1,2}^2=E_{1,1}^2-E_{2,2}^2+E_{2,1}^2,\\ \hfill E_{1,2}^1=E_{2,2}^1+E_{2,2}^2-E_{2,1}^2,\end{array}$$

(81)

where the variables $E_{2,1}^1$, $E_{1,1}^2$, $E_{2,2}^1$, $E_{2,2}^2$, and $E_{2,1}^2$ can be chosen arbitrarily. Under condition (81) the system (79) reduces to the following system of nine equations:

$${\displaystyle \frac{d}{dt}}{\left[\tilde{E}H\right]}^{\top }={\mathcal{M}}_2·{\left[\tilde{E}H\right]}^{\top },$$

(82)

where

$$\left[\tilde{E}H\right]=\left[\begin{array}{ccccccccc}{E}_{2,1}^1& E_{1,1}^2& E_{2,2}^1& E_{2,2}^2& E_{2,1}^2& H_{1,1}& H_{2,1}& H_{1,2}& H_{2,2}\end{array}\right]$$

and

$${\mathcal{M}}_2=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& -1& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& -1\\ 0& 0& 0& 0& 0& 1& 0& -1& 0\\ 0& 0& 0& 0& 0& -1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& -1& 1\\ -1& 1& 1& 2& -3& 0& 0& 0& 0\\ -1& -1& 1& 0& 1& 0& 0& 0& 0\\ 1& 1& -1& -4& 3& 0& 0& 0& 0\\ 1& -1& -1& 2& -1& 0& 0& 0& 0\end{array}\right].$$

By direct computation, the characteristic polynomial $\chi \left(\lambda \right)$ of the matrix ${\mathcal{M}}_3$ is found to be

$$\chi \left(\lambda \right)=-{\lambda }^3{(\lambda -2)}^2{(\lambda +2)}^2({\lambda }^2+8).$$

This factorization reveals the eigenvalues of ${\mathcal{M}}_3$ as follows: $\lambda =0$ with multiplicity 3; $\lambda =\pm 2$, each with multiplicity 2; and $\lambda =\pm 2\sqrt{2}i$, each with multiplicity 1. Accordingly, we can compute eigenvectors for each eigenvalues. The following three eigenvectors correspond to $\lambda =0$

$$\begin{array}{c}\hfill {\mathbf{h}}_1={\left[\begin{array}{ccccccccc}1& 0& 1& 0& 0& 0& 0& 0& 0\end{array}\right]}^{\top },\\ \hfill {\mathbf{h}}_2={\left[\begin{array}{ccccccccc}0& 1& 0& 1& 1& 0& 0& 0& 0\end{array}\right]}^{\top },\\ \hfill {\mathbf{h}}_3={\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 1& 1& 1& 1\end{array}\right]}^{\top }.\end{array}$$

For $\lambda =-2$ the corresponding eigenvectors are

$$\begin{array}{cc}\hfill {\mathbf{h}}_4& ={\left[\begin{array}{ccccccccc}-{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& 0& -1& 1& 0\end{array}\right]}^{\top },\hfill \\ \hfill {\mathbf{h}}_5& ={\left[\begin{array}{ccccccccc}-{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& -1& 0& 0& 1\end{array}\right]}^{\top },\hfill \end{array}$$

and for $\lambda =2$, the eigenvectors are

$$\begin{array}{cc}\hfill {\mathbf{h}}_6& ={\left[\begin{array}{ccccccccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& 0& -1& 1& 0\end{array}\right]}^{\top },\hfill \\ \hfill {\mathbf{h}}_7& ={\left[\begin{array}{ccccccccc}{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& -1& 0& 0& 1\end{array}\right]}^{\top }.\hfill \end{array}$$

respectively. Finally, for the complex eigenvalues $\lambda =\pm 2\sqrt{2}i$ the corresponding eigenvectors are given by ${\mathbf{h}}_8\pm i{\mathbf{h}}_9$, where

$$\begin{array}{cc}\hfill {\mathbf{h}}_8& ={\left[\begin{array}{ccccccccc}0& 0& 0& 0& 0& 1& -1& -1& 1\end{array}\right]}^{\top },\hfill \\ \hfill {\mathbf{h}}_9& ={\left[\begin{array}{ccccccccc}-{\displaystyle \frac{\sqrt{2}}{2}}& -{\displaystyle \frac{\sqrt{2}}{2}}& {\displaystyle \frac{\sqrt{2}}{2}}& -{\displaystyle \frac{\sqrt{2}}{2}}& {\displaystyle \frac{\sqrt{2}}{2}}& 0& 0& 0& 0\end{array}\right]}^{\top }.\hfill \end{array}$$

Thus, the general solution of Equation (82) can be written as

$$\begin{array}{cc}\hfill {\left[\tilde{E}H\right]}^{\top }& =c_1{\mathbf{h}}_1+c_2{\mathbf{h}}_2+c_3{\mathbf{h}}_3+c_4{\mathbf{h}}_4{e}^{-2t}+c_5{\mathbf{h}}_5{e}^{-2t}+c_6{\mathbf{h}}_6{e}^{2t}+c_7{\mathbf{h}}_7{e}^{2t}\hfill \\ \hfill & +c_8{\mathbf{h}}_{8}cos\left(2\sqrt{2}t\right)+c_9{\mathbf{h}}_{9}sin\left(2\sqrt{2}t\right),\hfill \end{array}$$

where $c_i\in \mathbb{R}$ are arbitrary constants. This expression, together with the representation (81), yields the general solution of the system of semi-discrete Maxwell’s Equation (79) on the combinatorial torus.

5. Conclusions

In this study, semi-discrete formulations of the three- and two-dimensional Maxwell’s equations that preserve the intrinsic geometric structure of the continuous model have been introduced and analyzed. The essential properties of the proposed semi-discrete framework were rigorously examined and compared with those of the classical Maxwell’s equations. The approach was further illustrated on a combinatorial two-dimensional torus, where the semi-discrete Maxwell’s equations reduce to a system of first-order linear ordinary differential equations, for which an explicit general solution was derived.

The discretization scheme presented in this work can be naturally extended to a four-dimensional setting, enabling the formulation of discrete Maxwell’s equations within the Minkowski spacetime framework, where the temporal coordinate is treated on an equal footing with the spatial dimensions. In such a setting, the Lorentzian structure of the metric must be incorporated into the definition of a discrete analogue of the Hodge star operator. A definition of such an operator has been proposed in [2]. These extensions are currently the focus of ongoing research.

Computational implementations, analysis of approximations to continuous solutions, and investigation of discrete dispersion effects represent important directions for future research. These will be pursued in subsequent work, including numerical validation of the semi-discrete scheme, convergence studies, and full time discretization with detailed dispersion analysis.