Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part II, higher order FVTD schemes
Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part II, higher order FVTD schemes
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具有约束保持、多维黎曼求解器和子单元分辨率的材料介质中的计算电动力学 - 第二部分，高阶 FVTD 方案
DOI：
10.1016/j.jcp.2017.10.013
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发表时间：
2018
影响因子：
4.1
通讯作者：
Montecinos, Gino
中科院分区：
文献类型：
--
作者：
Balsara, Dinshaw S.;Garain, Sudip;Taflove, Allen;Montecinos, Gino
时域有限差分（FDTD）方法在计算电动力学领域有很好的应用，其成功的部分原因在于它能够满足麦克斯韦方程组的约束条件。尽管如此，在本系列的前一篇文章中，我们能够提出一个二阶精度的计算电动力学（CED）的Goddom格式，它满足所有相同的约束条件，同时保留了Goddom格式的所有传统优点。本文将时域有限体积（FVTD）方法推广到二阶精度以上，并在FDTD方法的基础上保留了一种对原始变量错开的策略，使电位移和磁感应矢量场的约束保持非常有利。这是完成与约束保持重建方法，在本文中扩展到第三和第四阶的精度。一维迎风的想法，从Goddom计划有显着的修改，使用多维迎风黎曼求解器开发的第一作者。在本文中，我们将展示如何使用它们的背景下，一个更高的顺序计划CED。我们还报告的进展时间步进。我们展示了如何龙格-库塔IMEX计划可以适应CED，即使在存在刚性源项所带来的大电导率，以及强大的空间变化的介电常数和磁导率。我们还制定了非常有效的ADER时间步进策略，赋予我们的方法与子细胞解决能力。因此，我们的方法可以是刚性稳定的，并解决了显着的子细胞内的材料属性的变化在一个区域。此外，我们提出的ADER计划，适用于所有双曲型偏微分方程的刚性源项，并在所有订单的精度。我们新的ADER公式提供了一个治疗的刚性源项，比以前的ADER计划更有效。本文的电子增刊中给出了生成任意刚性源项偏微分方程ADER时间更新格式的计算机代数系统脚本，并给出了数值求解物质介质中麦克斯韦方程的二阶、三阶和四阶精度格式。还提出了几个严格的测试表明，该方法的工作原理，并满足其设计目标，即使材料的介电常数和磁导率变化的数量级在几个区域。此外，由于该方法是无条件稳定和子细胞解决的刚性源项的存在下（即对于涉及巨大的变化，电导率在几个区域的问题），它可以准确地处理这样的问题，而不会减少任何时间步长。我们还表明，提高精度的顺序提供了独特的优势，解决子细胞的材料特性的变化。最重要的是，我们表明，当精度要求是严格的高阶计划提供最短的时间解决方案。这使得一个令人信服的情况下，使用高阶，子细胞解决方案在CED。
The Finite Difference Time Domain (FDTD) scheme has served the computational electrodynamics community very well and part of its success stems from its ability to satisfy the constraints in Maxwell's equations. Even so, in the previous paper of this series we were able to present a second order accurate Godunov scheme for computational electrodynamics (CED) which satisfied all the same constraints and simultaneously retained all the traditional advantages of Godunov schemes. In this paper we extend the Finite Volume Time Domain (FVTD) schemes for CED in material media to better than second order of accuracy.From the FDTD method, we retain a somewhat modified staggering strategy of primal variables which enables a very beneficial constraint-preservation for the electric displacement and magnetic induction vector fields. This is accomplished with constraint-preserving reconstruction methods which are extended in this paper to third and fourth orders of accuracy. The idea of one-dimensional upwinding from Godunov schemes has to be significantly modified to use the multidimensionally upwinded Riemann solvers developed by the first author. In this paper, we show how they can be used within the context of a higher order scheme for CED.We also report on advances in timestepping. We show how Runge–Kutta IMEX schemes can be adapted to CED even in the presence of stiff source terms brought on by large conductivities as well as strong spatial variations in permittivity and permeability. We also formulate very efficient ADER timestepping strategies to endow our method with sub-cell resolving capabilities. As a result, our method can be stiffly-stable and resolve significant sub-cell variation in the material properties within a zone. Moreover, we present ADER schemes that are applicable to all hyperbolic PDEs with stiff source terms and at all orders of accuracy. Our new ADER formulation offers a treatment of stiff source terms that is much more efficient than previous ADER schemes. The computer algebra system scripts for generating ADER time update schemes for any general PDE with stiff source terms are also given in the electronic supplements to this paper.Second, third and fourth order accurate schemes for numerically solving Maxwell's equations in material media are presented in this paper. Several stringent tests are also presented to show that the method works and meets its design goals even when material permittivity and permeability vary by an order of magnitude over just a few zones. Furthermore, since the method is unconditionally stable and sub-cell-resolving in the presence of stiff source terms (i.e. for problems involving giant variations in conductivity over just a few zones), it can accurately handle such problems without any reduction in timestep. We also show that increasing the order of accuracy offers distinct advantages for resolving sub-cell variations in material properties. Most importantly, we show that when the accuracy requirements are stringent the higher order schemes offer the shortest time to solution. This makes a compelling case for the use of higher order, sub-cell resolving schemes in CED.