Two-Potential Formalism in Maxwell's Equations

• Two-potential formalism is a representation of electromagnetic fields using two vector and two scalar potentials, reformulating Maxwell's equations into a strictly hyperbolic system without divergence constraints.
• It employs Lorentz-type gauge conditions to eliminate non-physical zero-speed modes, making the approach optimal for high-order numerical schemes and shock-capturing in varied media.
• The formalism removes elliptic subproblems, enabling efficient field reconstruction and effective parallel computation in modern high-performance electrodynamics simulations.

The two-potential formalism is a formulation of Maxwell’s equations in which electromagnetic fields are represented through two vector potentials and two scalar potentials, yielding a symmetric, strictly hyperbolic system. By eliminating differential constraints present in the standard approach, this formalism produces a pure evolution system advantageous for high-order numerical schemes, particularly in computational electrodynamics for media with variable dielectric and magnetic properties (Kudryavtsev et al., 2012).

1. Definition of Potentials and Gauges

The formalism works in units with $c=1$ and uses SI-like notation for both vacuum and linear isotropic media with possibly inhomogeneous permittivity $\varepsilon(x)$ and permeability $\mu(x)$.

Vector Potentials:

• $\mathbf{A}(\mathbf{x},t)$: “Magnetic” vector potential with $\mathbf{B} = \nabla \times \mathbf{A}$,
• $\mathbf{C}(\mathbf{x},t)$: “Electric” vector potential with $\mathbf{E} = -\nabla \times \mathbf{C}$.

Scalar Potentials:

• $\varphi(\mathbf{x},t)$: Standard electric scalar potential, $$\mathbf{E} = -\nabla \varphi - \partial_t\mathbf{A}$$,
• $\psi(\mathbf{x},t)$: Magnetic scalar potential, $\varepsilon(x)$0.

Gauge Conditions:

To produce a strictly hyperbolic system free of non-physical zero-speed modes, Lorentz-type gauge conditions are imposed.

• In vacuum ($\varepsilon(x)$1): $\varepsilon(x)$2, $\varepsilon(x)$3.
• In a medium: $\varepsilon(x)$4, $\varepsilon(x)$5.

These conditions ensure all physical modes propagate at finite speed and complete the closure of the system.

2. Reformulation of Maxwell’s Equations

Starting from the Maxwell equations (allowing for electric and magnetic sources $\varepsilon(x)$6):

$\varepsilon(x)$7

For the homogeneous case (vacuum, no sources), two dual representations of $\varepsilon(x)$8 and $\varepsilon(x)$9 follow:

• $\mu(x)$0
• $\mu(x)$1

Equating these yields two first-order vector equations:

• $\mu(x)$2
• $\mu(x)$3

When sources or polarization/magnetization (auxiliary vectors $\mu(x)$4, $\mu(x)$5, $\mu(x)$6, $\mu(x)$7) are present, these appear as additive terms; their specific forms depend on source modeling.

For media with $\mu(x)$8 and $\mu(x)$9, the system generalizes to: $\mathbf{A}(\mathbf{x},t)$0 (Kudryavtsev et al., 2012)

3. Hyperbolic Structure and Elimination of Constraints

Unlike the standard Maxwell curl equations, which impose divergence constraints ($\mathbf{A}(\mathbf{x},t)$1, $\mathbf{A}(\mathbf{x},t)$2) handled by cleaning procedures or elliptic solves, the two-potential formalism encodes these constraints into the gauge equations. The resulting eight-component system is strictly hyperbolic: it consists entirely of first-order time evolution equations, without hidden constraints or subproblems requiring elliptic solvers.

For any propagation direction, the principal symbol $\mathbf{A}(\mathbf{x},t)$3 (from the system's block structure) yields eigenvalues $\mathbf{A}(\mathbf{x},t)$4 (where $\mathbf{A}(\mathbf{x},t)$5), each of multiplicity four, and there are no zero-speed (stationary) modes. This ensures robust propagation of all physical and gauge information at the appropriate finite characteristic speed.

4. Practical Numerical Discretization

High-order discretization is central to the formalism's computational appeal. Spatial derivatives—curl and divergence/gauge terms—are discretized using fifth-order Weighted Essentially Non-Oscillatory (WENO) schemes in each coordinate. Temporally, a fourth-order Runge–Kutta–Gill method provides efficient and memory-saving integration.

The timestep is subject to a CFL-type condition: $\mathbf{A}(\mathbf{x},t)$6 with $\mathbf{A}(\mathbf{x},t)$7 for stability. In inhomogeneous media, material parameters $\mathbf{A}(\mathbf{x},t)$8 and $\mathbf{A}(\mathbf{x},t)$9 are located at cell centers and used pointwise—no special averaging is necessary.

Boundary conditions—such as perfect conductors, periodic, or open domains—are enforced directly on the potentials using symmetry/antisymmetry or characteristic-based inflow/outflow for $\mathbf{B} = \nabla \times \mathbf{A}$0 modes. The update for a single stage in the $\mathbf{B} = \nabla \times \mathbf{A}$1-direction is exemplified by: $\mathbf{B} = \nabla \times \mathbf{A}$2 where $\mathbf{B} = \nabla \times \mathbf{A}$3 are WENO5 numerical fluxes, and other derivatives are computed with five-point WENO stencils (Kudryavtsev et al., 2012).

5. Field Reconstruction and Implementation Considerations

The electromagnetic fields can be reconstructed from the four potentials post hoc using high-order centered differences:

• $\mathbf{B} = \nabla \times \mathbf{A}$4
• $\mathbf{B} = \nabla \times \mathbf{A}$5

A key feature is the relative smoothness of the potentials compared to the fields, increasing the efficiency and accuracy of post-processing field calculations.

The absence of elliptic subproblems allows near-linear computational complexity and high parallelizability. Since no divergence cleaning is required, the algorithm is well-suited for high-performance computing environments.

6. Applications and Computational Impact

This formalism is particularly advantageous for computational electrodynamics in both vacuum and materials with space-dependent $\mathbf{B} = \nabla \times \mathbf{A}$6. It accommodates sharp interfaces and discontinuities through shock-capturing, non-oscillatory schemes. Modern high-order methods applied within this framework enable accurate simulation of electromagnetic wave propagation, including situations with inhomogeneous or discontinuous material properties (Kudryavtsev et al., 2012).

A plausible implication is that, because the equations are strictly hyperbolic and all modes propagate causally, the approach is naturally resilient against numerical artifacts commonly associated with standard Maxwell solvers, such as spurious non-propagating (zero-speed) modes and poor handling of divergence constraints.

7. Summary Table: Comparison of Maxwell and Two-Potential Systems

Feature	Standard Maxwell (Fields)	Two-Potential Formalism
Differential constraints	Present ($\mathbf{B} = \nabla \times \mathbf{A}$7, $\mathbf{B} = \nabla \times \mathbf{A}$8)	None (constraints into gauges)
System type	Mixed (evolutionary + elliptic)	Strictly hyperbolic (evolutionary)
Zero-speed modes	Yes	No
Suitability for shock-capturing	Indirect (cleaning required)	Direct (evolutionary update)
High-order parallelizability	Conditional	Natural

In summary, the two-potential formalism recasts Maxwell’s equations as a first-order, eight-component hyperbolic evolution system in the potentials $\mathbf{B} = \nabla \times \mathbf{A}$9, free of divergence constraints and compatible with high-order shock-capturing numerical schemes, in both vacuum and general inhomogeneous media (Kudryavtsev et al., 2012).