Two-Potential Formalism for Numerical Solution of the Maxwell Equations

TL;DR

This paper introduces a two-potential formulation of Maxwell's equations that results in a hyperbolic system, simplifying numerical simulations of electromagnetic phenomena using shock-capturing methods.

Contribution

The paper presents a novel two-potential formalism transforming Maxwell's equations into a hyperbolic system without differential constraints, enhancing numerical simulation efficiency.

Findings

Successfully modeled electromagnetic wave propagation using high-order shock-capturing schemes.

Demonstrated the formulation's effectiveness in vacuum and inhomogeneous media.

Provided examples showing improved numerical stability and accuracy.

Abstract

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.