An Active Flux Method for Maxwell's equations

Abstract

We present a novel Active Flux method and a divergence-free method using exact evolution operators for solving Maxwell's equations in three space dimensions on Cartesian grids. The proposed methods are fully discrete, third-order accurate and rely on a truly multidimensional evolution operator based on Kirchhoff's formula for the electromagnetic wave equation. The first method is a classical Active Flux method with globally continuous reconstruction while the second method is an adaptation of the Active Flux method to a locally divergence-free reconstruction by Balsara. The applied reconstruction is built from facial degrees of freedom and is no longer globally continuous. The classical method keeps the divergence error bounded. The divergence-free method with face averages initialized by a potential keeps the constraints at round-off. We compare the accuracy of both methods for smooth and discontinuous test problems.

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BibTeX RIS

BibTeX

@misc{kiechle2026active,
  title = {{An Active Flux Method for Maxwell's equations}},
  author = {Yanick Kiechle and Christiane Helzel},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03471},
  eprint = {2611.03471},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03471},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - An Active Flux Method for Maxwell's equations
AU  - Yanick Kiechle
AU  - Christiane Helzel
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03471
UR  - https://arxiv.org/abs/2611.03471
AB  - We present a novel Active Flux method and a divergence-free method using exact evolution operators for solving Maxwell's equations in three space dimensions on Cartesian grids. The proposed methods are fully discrete, third-order accurate and rely on a truly multidimensional evolution operator based on Kirchhoff's formula for the electromagnetic wave equation. The first method is a classical Active Flux method with globally continuous reconstruction while the second method is an adaptation of the Active Flux method to a locally divergence-free reconstruction by Balsara. The applied reconstruction is built from facial degrees of freedom and is no longer globally continuous. The classical method keeps the divergence error bounded. The divergence-free method with face averages initialized by a potential keeps the constraints at round-off. We compare the accuracy of both methods for smooth and discontinuous test problems.
ER  -

Abstract

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