UNCONDITIONALLY STABLE TIME STEPPING METHOD FOR MIXED FINITE ELEMENT MAXWELL SOLVERS
UNCONDITIONALLY STABLE TIME STEPPING METHOD FOR MIXED FINITE ELEMENT MAXWELL SOLVERS
复制标题
混合有限元麦克斯韦求解器的无条件稳定时间步进方法
DOI：
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发表时间：
2020
影响因子：
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通讯作者：
B. Shanker
中科院分区：
文献类型：
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作者：
Z. Crawford;Jie Li;A. Christlieb;B. Shanker
计算电磁场的时域有限元法（TD-FEM）已得到广泛研究。TD-FEM求解通常使用Newmark-Beta方法。TD-FEM的挑战之一是存在随时间增长的DC零空间。这可以通过直接求解麦克斯韦方程来克服。一种方法，称为时域混合有限元法（TDMFEM），离散麦克斯韦方程使用适当的空间基组和蛙跳时间步进。通常，用于离散场量的基函数是低阶的。它是条件稳定的，并且时间步长和网格依赖特征值之间有很强的联系，很像Courant-Friedrichs-Lewy（CFL）条件。这意味着时间步长可以非常小。为了克服这一挑战，我们使用Newmark-Beta方法。这项工作的主要贡献是发展，严格证明，无条件稳定的高阶TD-MFEM不同的边界条件。此外，我们分析零空间的系统，并证明稳定性和收敛性。所有结果均与条件稳定的蛙跳方法进行比较。
Time domain finite element methods (TD-FEM) for computing electromagnetic fields are well studied. TD-FEM solution is typically effected using Newmark-Beta methods. One of the challenges of TD-FEM is the presence of a DC null-space that grows with time. This can be overcome by solving Maxwell equations directly. One approach, called time domain mixed finite element method (TDMFEM), discretizes Maxwell’s equations using appropriate spatial basis sets and leapfrog time stepping. Typically, the basis functions used to discretize field quantities have been low order. It is conditionally stable, and there is a strong link between time step size and mesh dependent eigenvalues, much like the Courant-Friedrichs-Lewy (CFL) condition. This implies that the time step sizes can be very small. To overcome this challenge, we use the Newmark-Beta approach. The principal contribution of this work is the development of, and rigorous proof of, unconditional stability for higher order TD-MFEM for different boundary conditions. Further, we analyze nullspaces of the resulting system, and demonstrate stability and convergence. All results are compared against the conditionally stable leapfrog approach.