AI 中文总结
研究由 GLM 旋度清理与线性声学耦合产生的双曲型系统，通过变分原理推导方程并分析其性质。开发交错网格半隐式兼容数值格式，能保持关键性质，在傅里叶极限下渐近保持，守恒能量和不变量，通过测试案例验证结果。
AI 中文摘要
本文研究了由广义拉格朗日乘子（GLM）旋度清理与线性声学耦合产生的双曲型系统原型。首先表明，在无耗散情况下，方程可从潜在变分原理严格推导得出，所得系统是对称双曲型的。进一步分析表明它不仅守恒总能量，还存在一组由应用于状态变量的微分算子二次组合构成的不变量。添加刚性松弛源项可将系统扩展到耗散动力学，如描述在刚性松弛极限下与傅里叶定律渐近兼容的卡塔尼奥型热传递过程。开发了一种新的交错网格半隐式兼容数值格式来求解该系统，能在离散层面精确保持其关键性质。证明了该格式在傅里叶极限下是渐近保持的，收敛速率取决于初始数据。还表明该格式在无松弛时精确守恒总能量和所有不变量，有松弛时则一致耗散它们。这些结果在一组代表性测试案例中得到验证。
英文摘要
In this paper, we study a prototype hyperbolic system arising from the coupling of Generalized-Lagrangian-Multiplier (GLM) curl-cleaning with linear acoustics. We first show that, in the absence of dissipation, the equations can be rigorously derived from an underlying variational principle and that the resulting system is symmetric-hyperbolic. Further analysis shows that it not only conserves the total energy but also admits a set of invariants consisting of quadratic combinations of differential operators applied to the state variables. The addition of a stiff relaxation source term allows the system to be extended to dissipative dynamics, describing, for example, Cattaneo-type heat transfer processes that are asymptotically compatible with the Fourier law in the stiff relaxation limit; in this case, the total energy and invariants are dissipated accordingly. A new semi-implicit compatible numerical scheme on staggered grids is developed to solve this system while exactly preserving its key properties at the discrete level. In particular, we prove that the scheme is asymptotic-preserving in the Fourier limit, with a convergence rate depending on the initial data. We also show that the scheme conserves exactly the total energy as well as all the invariants in the absence of relaxation, and dissipates them consistently in its presence. These findings are demonstrated on a set of representative test cases.
