一种稳定性不受媒质损耗影响的CN-RK时域间断伽辽金方法

向汝; 马西奎; 马亮; 迟明珺; 王嘉玮

doi:10.12263/DZXB.20241004

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一种稳定性不受媒质损耗影响的CN-RK时域间断伽辽金方法

学术论文 | 更新时间：2025-12-27

- 一种稳定性不受媒质损耗影响的CN-RK时域间断伽辽金方法
- A CN-RK Discontinuous Galerkin Time-Domain Method Without the Stability Restriction Due to Material Loss
- 电子学报 2025年53卷第8期页码：2649-2657
- 作者机构：
  
  西安交通大学电气工程学院，陕西西安 710049
- 作者简介：
  
  [ "向汝女，1999年3月出生于湖北省利川市.现为西安交通大学电气工程学院博士研究生.主要研究方向为时域间断伽辽金方法及其在多尺度电磁问题中的应用等. E-mail: ruxiang@stu.xjtu.edu.cn" ]
  [ "马西奎男，1958年5月出生于陕西省渭南市.现为西安交通大学电气工程学院教授、博士生导师.主要研究方向为电气工程中的多物理场耦合理论及其数值分析软件技术、通信和电子系统中的电磁场与电磁波、电力电子系统非线性动力学理论等. E-mail: maxikui@mail.xjtu.edu.cn" ]
  [ "马亮男，1997年10月出生于安徽省无为市.现为西安交通大学电气工程学院博士研究生.主要研究方向为电磁场数值计算. E-mail: 3120104254@stu.xjtu.edu.cn" ]
  [ "迟明珺女，1998年1月出生于山东省青岛市.现为西安交通大学电气工程学院博士研究生.主要研究方向为电磁场数值计算. E-mail: 3120104253@stu.xjtu.edu.cn" ]
  [ "王嘉玮男，1990年12月出生于陕西省西安市.现为西安交通大学电气工程学院副教授.主要研究方向为电气设备的电磁兼容分析与优化设计、电磁场和多尺度多物理场建模仿真技术以及面向物理场分析的计算机辅助工程（CAE）软件开发等. E-mail: jwwang@xjtu.edu.cn" ]
- 基金信息：
  
  国家自然科学基金(52207014)
- DOI：10.12263/DZXB.20241004
  中图分类号： TM15;TN011
- 收稿：2024-11-06，
  
  录用：2025-07-15，
  
  纸质出版：2025-08-25
- 稿件说明：
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向汝, 马西奎, 马亮, 等. 一种稳定性不受媒质损耗影响的CN-RK时域间断伽辽金方法[J]. 电子学报, 2025, 53(08): 2649-2657.

XIANG Ru, MA Xi-kui, MA Liang, et al. A CN-RK Discontinuous Galerkin Time-Domain Method Without the Stability Restriction Due to Material Loss[J]. Acta Electronica Sinica, 2025, 53(08): 2649-2657.
向汝, 马西奎, 马亮, 等. 一种稳定性不受媒质损耗影响的CN-RK时域间断伽辽金方法[J]. 电子学报, 2025, 53(08): 2649-2657. DOI：10.12263/DZXB.20241004

XIANG Ru, MA Xi-kui, MA Liang, et al. A CN-RK Discontinuous Galerkin Time-Domain Method Without the Stability Restriction Due to Material Loss[J]. Acta Electronica Sinica, 2025, 53(08): 2649-2657. DOI：10.12263/DZXB.20241004

摘要

采用显式龙格-库塔（Runge-Kutta，RK）时间积分格式的时域间断伽辽金（Discontinuous Galerkin Time-Domain，DGTD）方法求解涉及有耗媒质的电磁问题时，时间步长会受到电导率或导磁率的严格限制.虽然已经有研究通过数值试验验证了有耗媒质对稳定性的影响，但缺乏严格的理论依据.因此，本文基于理论推导建立了有耗媒质中显式RK-DGTD方法的稳定性条件，得到时间步长上限的估计值，并对其准确性进行了数值验证.此外，为了提高计算效率，本文提出了一种适用于DGTD方法的隐式-显式时间离散格式，采用隐式Crank-Nicolson（CN）格式对损耗项进行离散，而对其余项采用显式RK格式，从理论上证明了该方法的时间稳定性不受媒质损耗的影响，并通过数值算例验证了其可靠性与效率.

Abstract

When discontinuous Galerkin time-domain (DGTD) methods with explicit runge-kutta (RK) schemes are applied to the analysis of electromagnetic (EM) problems involving lossy media

the time-step size is strictly limited by electric and magnetic conductivity. Although the influence of lossy media on stability has been verified by numerical experiments

it lacks strict theoretical support. Therefore

the stability condition of the explicit RK-DGTD methods in lossy media is established based on theoretical derivation. The upper limit of time-step size is estimated

and numerical results demonstrate its accuracy. In addition

to improve computational efficiency

an implicit-explicit time integration scheme is proposed for DGTD methods. The lossy terms are discretized using an implicit crank-nicolson (CN) scheme

while the remaining terms are discretized using an explicit RK scheme. The time stability of the proposed method is theoretically proven to be unaffected by material loss

and numerical experiments validate its reliability and efficiency.

关键词

Keywords

references

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