A stabilized Two-Step Formulation of Maxwell's Equations in the time-domain

TL;DR

This paper introduces a stabilized two-step time-domain formulation for Maxwell's equations that ensures numerical stability and robustness across frequency ranges, including static limits, using a generalized gauge and tailored discretization schemes.

Contribution

It extends a two-step Maxwell's equations formulation to the time domain with a generalized gauge, improving stability and applicability to nonlinear materials.

Findings

Method is stable and accurate in 3D simulations.

Effective in static and low-frequency regimes.

Applicable to nonlinear, temperature-dependent materials.

Abstract

Simulating electromagnetic fields across broad frequency ranges is challenging due to numerical instabilities at low frequencies. This work extends a stabilized two-step formulation of Maxwell's equations to the time-domain. Using a Galerkin discretization in space, we apply two different time-discretization schemes that are tailored to the first- and second-order in time partial differential equations of the two-step solution procedure used here. To address the low-frequency instability, we incorporate a generalized tree-cotree gauge that removes the singularity of the curl-curl operator, ensuring robustness even in the static limit. Numerical results on academic and application-oriented 3D problems confirm stability, accuracy, and the method's applicability to nonlinear, temperature-dependent materials.