Globally constraint-preserving FR/DG scheme for Maxwell's equations at all orders
Globally constraint-preserving FR/DG scheme for Maxwell's equations at all orders
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DOI：
10.1016/j.jcp.2019.06.003
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发表时间：
2018-09
期刊：
影响因子：
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通讯作者：
Arijit Hazra;P. Chandrashekar;D. Balsara
中科院分区：
文献类型：
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作者：
Arijit Hazra;P. Chandrashekar;D. Balsara
计算电动力学（CED）是麦克斯韦方程组的数值解，在科学和工程中的一些问题中起着不可思议的重要作用。高精度的解是需要的，而不连续伽辽金（DG）法是实现数值CED高精度的较好方法之一。麦克斯韦方程组有一对对合约束，在离散水平上全局满足约束的模拟方案是非常理想的。Balsara和Käppeli（2019）给出了四阶CED全局约束保持DG格式的von Neumann稳定性分析。本文的重点是发展理论，并记录了DGTD方案在恒定介电常数和磁导率介质中的优越耗散和色散。在本文中，我们提出了一种精度达到五阶的差分差分差分方法，并分析了它们在空间中介电常数和磁导率变化较大时的性能。我们的DGTD方案通过在网格面上配置电位移和磁感应以及它们的高阶模式来实现约束保持。我们的第一个发现是，在四阶及更高的精度下，除了以脸为中心的模式外，还必须发展一些以区域为中心的模式。众所周知，DG格式中的极限步长会导致格式最优精度的降低；尽管这些方案在使用weno型限制器时仍然保留了它们的正式精度顺序。在本文中，我们记录了在不需要任何DG方案限制的情况下介电常数和磁导率变化几乎一个数量级的模拟。这一非常有利的第二个发现确保了即使在介电常数和渗透率存在较大空间变化的情况下，DGTD方案也能保持最佳精度。我们还研究了这些问题中的电磁能守恒。我们的第三个发现表明，即使介电常数和磁导率在空间中变化很大，电磁能量也能很好地守恒；只要电导率为零。
Computational electrodynamics (CED), the numerical solution of Maxwell's equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG) method is one of the better ways of delivering high accuracy in numerical CED. Maxwell's equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable. Balsara and Käppeli (2019) presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to fourth order. That paper was focused on developing the theory and documenting the superior dissipation and dispersion of DGTD schemes in media with constant permittivity and permeability. In this paper we present working DGTD schemes for CED that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space.Our DGTD schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh. Our first finding is that at fourth and higher orders of accuracy, one has to evolve some zone-centered modes in addition to the face-centered modes. It is well-known that the limiting step in DG schemes causes a reduction of the optimal accuracy of the scheme; though the schemes still retain their formal order of accuracy with WENO-type limiters. In this paper, we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the DG scheme. This very favorable second finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability. We also study the conservation of electromagnetic energy in these problems. Our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space; as long as the conductivity is zero.