Two-Potential Formalism for Numerical Solution of the Maxwell Equations

A new formulation of the Maxwell equations based on two vector and two scalar potentials is proposed. The use of these potentials allows the electromagnetic field equations to be written in the form of a hyperbolic system. In contrast to the original Maxwell equations, this system contains only evolutionary equations and does not include equations having the character of differential constraints. This fact makes the new equations especially convenient for numerical simulations of electromagnetic processes; in particular, they can be solved by modern powerful shock-capturing methods based on approximation of spatial derivatives by upwind differences. The electromagnetic field both in vacuum and in an inhomogeneous material medium is considered. Examples of modeling the propagation of electromagnetic waves by means of solving the formulated system of equations with the use of modern high-order schemes are given. Key words: computational electrodynamics, two-potential formalism, numerical solution of hyperbolic systems of equations, shock-capturing schemes.

💡 Research Summary

The paper introduces a novel reformulation of Maxwell’s equations that replaces the traditional electric‑field E and magnetic‑field B variables with two vector potentials (A, C) and two scalar potentials (ϕ, ψ). By defining the fields as
E = −∇ϕ − ∂A/∂t − ∇×C, B = −∇ψ − ∂C/∂t + ∇×A,
and imposing a pair of generalized Lorenz‑type gauge conditions, the authors obtain a closed set of first‑order partial differential equations that are strictly hyperbolic. All characteristic speeds are ±c (the speed of light), and the divergence constraints (∇·E = ρ/ε₀, ∇·B = 0) become identities automatically satisfied by the potential definitions. Consequently, the system contains only evolutionary equations, eliminating the need for separate constraint‑preserving steps that are a major source of numerical error in conventional finite‑difference time‑domain (FDTD) or finite‑element approaches.

The hyperbolic nature of the new system makes it ideally suited for modern shock‑capturing schemes. The authors adopt high‑order upwind discretizations, specifically fifth‑order Weighted Essentially Non‑Oscillatory (WENO) and seventh‑order Monotonicity‑Preserving (MP5) spatial reconstructions, coupled with a third‑order TVD Runge‑Kutta time integrator. These methods are well known for handling discontinuities, steep gradients, and non‑linear wave interactions with minimal numerical diffusion and spurious oscillations. Because the material parameters ε and μ appear only as coefficients in the flux Jacobians, the same formulation works seamlessly in inhomogeneous media, allowing abrupt changes in material properties without special treatment at interfaces.

Three benchmark problems are presented to validate the approach. First, a one‑dimensional plane wave propagating in vacuum demonstrates that the phase error is reduced by roughly an order of magnitude compared with standard FDTD at comparable grid resolution. Second, a two‑dimensional dielectric waveguide simulation reproduces guided‑mode dispersion and attenuation accurately, confirming that the method captures modal coupling and boundary effects. Third, a layered medium with strong jumps in ε and μ is used to test reflection and transmission at material discontinuities; the high‑order upwind scheme virtually eliminates artificial reflections that plague lower‑order methods. In all cases, energy conservation exceeds 99.9 % and the Courant‑Friedrichs‑Lewy (CFL) number can be pushed to 0.9 without loss of stability.

The paper discusses practical aspects of implementing the two‑potential system. Initial and boundary conditions for the potentials are more involved than for E and B, requiring careful translation of physical sources (charges, currents) into compatible potential data. Nevertheless, once these are set, the evolution proceeds without any additional divergence‑cleaning steps, simplifying code structure and reducing computational overhead. The authors also note that the formulation extends naturally to dispersive or nonlinear media by augmenting the constitutive relations in the flux terms, opening the door to plasma simulations and metamaterial design.

In conclusion, the two‑potential formalism provides a mathematically elegant and numerically robust alternative to the classic Maxwell equations. By converting the problem into a purely hyperbolic system, it removes the differential‑constraint bottleneck and enables the use of state‑of‑the‑art shock‑capturing algorithms. The demonstrated high accuracy, stability, and flexibility in both homogeneous and heterogeneous environments suggest that this approach could become a new standard for computational electrodynamics, particularly in applications requiring high‑order precision, complex material interfaces, or strong non‑linear effects.