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

Leon Herles, Mario Mally, Jörg Ostrowski, Sebastian Schöps, Melina Merkel

TL;DR

This work tackles the low-frequency instability of full Maxwell simulations by extending a stabilized two-step Maxwell formulation to the time domain. It combines a generalized tree-cotree gauge to eliminate the curl-curl kernel with a decoupled, two-step time integration: an electroquasistatic step using the trapezoidal rule and a magnetic inductive step using Newmark-$\beta$, enabling stable integration as $\Delta t$ grows large. The approach yields a kernel-free, well-conditioned system that can be solved in parallel across steps and supports nonlinear, temperature-dependent material behavior through a selective linearization in the first step. Numerical tests in 3D academic and planar-coil configurations confirm stability, accuracy, and improved conditioning, highlighting the method’s potential for broadband and multiphysics electromagnetic simulations.

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.

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

TL;DR

This work tackles the low-frequency instability of full Maxwell simulations by extending a stabilized two-step Maxwell formulation to the time domain. It combines a generalized tree-cotree gauge to eliminate the curl-curl kernel with a decoupled, two-step time integration: an electroquasistatic step using the trapezoidal rule and a magnetic inductive step using Newmark-, enabling stable integration as grows large. The approach yields a kernel-free, well-conditioned system that can be solved in parallel across steps and supports nonlinear, temperature-dependent material behavior through a selective linearization in the first step. Numerical tests in 3D academic and planar-coil configurations confirm stability, accuracy, and improved conditioning, highlighting the method’s potential for broadband and multiphysics electromagnetic simulations.

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.

Paper Structure

This paper contains 12 sections, 23 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Formulation
  3. Discretization
  4. Spatial Discretization
  5. Temporal Discretization
  6. Instability of the Full Maxwell model
  7. Tree-Cotree Decomposition
  8. Stabilization
  9. Numerical Tests
  10. Academic Example
  11. Simulation of Setup with Planar Coil
  12. Conclusion

Figures (8)

  • Figure 1: Setup of the academic test example; adapted from Herles_2025aa.
  • Figure 2: Electric flux density $\left\lVert\boldsymbol{D}\right\rVert_{2}$ for different times.
  • Figure 3: Condition number of different system matrices of academic test example from \ref{['fig:academicTestEx']} over time step size.
  • Figure 4: Error \ref{['eq:error']} of time domain w.r.t. frequency-domain.
  • Figure 5: Problem setup of the planar coil. Taken from Herles_2025aa.
  • ...and 3 more figures