二维含时麦克斯韦方程组的块对角质量矩阵任意阶元胞方法
An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations
摘要：
本文介绍了一种新的数值方法，在二维非结构网格上求解含时麦克斯韦方程组。这依赖于引入一个新的网格，这是重心对偶细胞复杂的开始单纯网格，并近似两个未知领域的几何实体的两个对偶复杂的积分量。仔细选择的基函数产生廉价可逆的块对角系统矩阵的离散时间步进计划。本贡献的主要新颖之处在于将任意多项式度的近似函数空间，定义通过一个新的参考细胞。所提出的方法，虽然是一种不连续Galerkin方法，既不需要引入用户调整的惩罚参数的切向跳跃的领域，也没有数值耗散，以实现稳定性。事实上，一个精确的电磁能量守恒定律的半离散计划证明，它是在几个数值试验表明，由此产生的算法提供了虚假的解决方案与预期的收敛阶。
英文摘要：
We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis-functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.
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