发表机构
Institute of Mathematics, University of Zürich; University of Trento; University of Science and Technology, Shenzhen; INRIA, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（苏黎世大学数学研究所; 特伦托大学; 深圳理工大学; 法国国家信息与自动化研究所、波尔多大学、法国国家科学研究中心、波尔多理工学院、IMB）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单纯形网格上Active Flux方法同时满足守恒形式与离散对合的性质，推导出声学旋度与麦克斯韦散度算子的显式公式，并通过三阶数值实验验证。
AI 中文摘要
我们研究了非结构化三角网格上Active Flux方法的对合保持性质。考虑了两个模型问题：满足速度场旋度对合的二维线性声学系统，以及满足散度对合的具有面内磁场的二维麦克斯韦方程组。两个系统都可以写成守恒形式。我们要解决的问题是，单一离散化如何同时满足守恒形式和对合。在单纯形上的离散Active Flux框架内，我们推导了离散旋度算子的显式公式，该算子被声学格式精确保持，以及被麦克斯韦系统保持的离散散度算子。该分析适用于一般单纯形网格和任意多项式次数，包括该方法在三阶以上的扩展。三阶格式的数值实验证实了理论结果。要点：1-Active Flux在非结构化三角网格上保持离散对合，2-线性声学格式保持的显式离散旋度算子，3-二维麦克斯韦方程组保持的显式离散散度算子，4-结果适用于一般单纯形网格和任意多项式次数，5-三阶数值实验证实了理论结果。
英文摘要
We study the involution-preserving properties of the Active Flux method on unstructured triangular meshes. Two model problems are considered: the two-dimensional linear acoustic system, which satisfies a curl involution on the velocity field, and the two-dimensional Maxwell equations with an in-plane magnetic field, which satisfy a divergence involution. Both systems can be written in conservative form. The question we address is how a single discretization can satisfy the conservative form and the involution at the same time. Within a discrete Active Flux framework on simplices, we derive explicit formulas for a discrete curl operator that is exactly preserved by the scheme for acoustics, and for a discrete divergence operator preserved for the Maxwell system. The analysis applies to general simplex meshes and to any polynomial degree, including extensions of the method beyond third order. Numerical experiments with the third-order scheme confirm the theoretical results. Highlights: 1-Active Flux on unstructured triangular meshes preserves discrete involutions, 2-Explicit discrete curl operator preserved by the scheme for linear acoustics, 3- Explicit discrete divergence operator preserved for 2D Maxwell equations, 4- Results hold on general simplex meshes,% and for any polynomial degree 5- Third-order numerical experiments confirm the theoretical results. \end{itemize}
