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

2025-07-24

AI Summary

Low-frequency numerical instability severely limits time-domain simulations of broadband electromagnetic fields. Method: This paper proposes a stabilized two-step time-domain finite element method, combining Galerkin spatial discretization with a stepwise temporal integration scheme. Crucially, it introduces the generalized tree–cotree gauge into the time-domain framework for the first time, effectively eliminating the singularity of the curl operator and ensuring convergence in the static limit. Contribution/Results: The method overcomes the low-frequency instability inherent in conventional time-domain approaches and natively supports nonlinear and temperature-dependent material constitutive relations. Validated on multiple academic and industrial-grade 3D benchmark problems, it demonstrates robust stability across the entire frequency spectrum, high accuracy, and strong computational reliability. Thus, it establishes a scalable, physically consistent time-domain paradigm for complex multiphysics electromagnetic simulations.

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• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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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.

Problem

• Extends stabilized two-step Maxwell's equations to time-domain
• Addresses low-frequency instability via generalized tree-cotree gauge
• Validates stability and accuracy for nonlinear materials

Innovation

• Stabilized two-step Maxwell's equations in time-domain
• Galerkin discretization with tailored time schemes
• Generalized tree-cotree gauge for low-frequency stability