We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems and which was originally introduced by Munz et al. for purely numerical purposes in order to treat the divergence constraints of the magnetic and the electric field in the vacuum Maxwell equations within general-purpose and non-structure-preserving numerical schemes for hyperbolic PDE. Among the many mathematically interesting features of the model are: i) its symmetric hyperbolicity, ii) the extra conservation law for the total energy density and, most importantly, iii) the very peculiar combination of the basic differential operators, since both, curl-curl and div-grad combinations are mixed within this kind of system. A similar mixture of Maxwell-type and acoustic-type subsystems has recently been also forwarded by Buchman et al. in the context of a reformulation of the Einstein field equations of general relativity in terms of tetrads. This motivates our interest in this class of PDE, since the system is by itself very interesting from a mathematical point of view and can therefore serve as useful prototype system for the development of new structure-preserving numerical methods. Up to now, to the best of our knowledge, there exists neither a rigorous variational derivation of this class of hyperbolic PDE systems, nor do exactly energy-conserving and asymptotic-preserving schemes exist for them. The objectives of this paper are to derive the Maxwell-GLM system from an underlying variational principle, show its consistency with Hamiltonian mechanics and special relativity, extend it to the general nonlinear case and to develop new exactly energy-conserving and asymptotic-preserving finite volume schemes for its discretization.

翻译：本文提出了一种新颖的Maxwell-GLM系统的变分推导方法。该系统通过广义拉格朗日乘子法（GLM）在原始真空Maxwell方程组基础上增加了两个辅助声学子系统，最初由Munz等人出于纯数值目的引入，旨在处理双曲型偏微分方程通用非结构保持数值格式中真空Maxwell方程组的磁场与电场散度约束条件。该模型具有多个数学上的有趣特征：i) 对称双曲性；ii) 总能量密度的额外守恒律；以及最重要的iii) 基本微分算子的特殊组合形式——旋度-旋度与散度-梯度组合在此类系统中同时存在。近期Buchman等人在用标架表述广义相对论爱因斯坦场方程的重构工作中，也提出了类似的Maxwell型与声学型子系统混合体系。这激发了我们对这类偏微分方程的研究兴趣，因为该系统本身具有重要的数学研究价值，可作为开发新型结构保持数值方法的原型系统。迄今为止，据我们所知，既不存在对此类双曲型偏微分方程系统的严格变分推导，也缺乏精确能量守恒且渐近保持的数值格式。本文的目标在于：从基本变分原理推导Maxwell-GLM系统，证明其与哈密顿力学及狭义相对论的一致性，将其推广至一般非线性情形，并为其离散化开发新型精确能量守恒且渐近保持的有限体积格式。