计算数学
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计算数学  2019, Vol. 41 Issue (4): 419-439    DOI:
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矩阵形式二次修正Maxwell-Dirac系统的多尺度算法
付姚姚1,2, 曹礼群1,3
1. 中国科学院大学, 北京 100190;
2. 中国科学院数学与系统科学研究院计算数学与科学工程计算研究所, 北京 100190;
3. 中国科学院数学与系统科学研究院计算数学与科学工程计算研究所, 科学与工程计算国家重点实验室, 国家数学与交叉科学中心, 北京 100190
THE MULTISCALE ALGORITHMS FOR THE MAXWELL-DIRAC SYSTEM IN MATRIX FORM WITH QUADRATIC CORRECTION
Fu Yaoyao1,2, Cao Liqun1,3
1. University of Chinese Academy of Sciences, Beijing 100190, China;
2. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
3. LSEC, NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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摘要 带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有:一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.
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关键词Maxwell-Dirac系统   二次修正   矩阵形式   多尺度渐近方法   Crank-Nicolson有限差分方法   自适应棱单元方法     
Abstract: The Maxwell-Dirac system with quadratic correction has a wide applications in materials science such as topological insulators, graphene, superconductors and so on. In this paper, we first present the Dirac equation in matrix form with quadratic correction. Combining the Maxwell's equations, we present the homogenization method and the multiscale asymptotic method for the modified Maxwell-Dirac system in matrix form with rapidly oscillating discontinuous coefficients in a bounded Lipschitz convex domain under the Weyl gauge. Based on the multiscale asymptotic expansions of the solution of the Maxwell-Dirac system, by using the Crank-Nicolson finite difference method and the adaptive edge element method, we developed the multiscale algorithms for solving the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients. Numerical examples are then carried out to validate the method presented in this paper.
Key wordsMaxwell-Dirac system   quadratic correction   matrix form   the multiscale asymptotic expansion   the Crank-Nicolson finite difference method   the adaptive edge element method   
收稿日期: 2019-01-23; 出版日期: 2019-11-16
基金资助:

国家自然科学基金重点项目(91330202)、面上项目(11571353)资助.

通讯作者: 曹礼群,Email:clq@lsec.cc.ac.cn.     E-mail: clq@lsec.cc.ac.cn
引用本文:   
. 矩阵形式二次修正Maxwell-Dirac系统的多尺度算法[J]. 计算数学, 2019, 41(4): 419-439.
. THE MULTISCALE ALGORITHMS FOR THE MAXWELL-DIRAC SYSTEM IN MATRIX FORM WITH QUADRATIC CORRECTION[J]. Mathematica Numerica Sinica, 2019, 41(4): 419-439.
 
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[1] 付姚姚, 曹礼群, 马楚鹏. 有界区域Weyl规范下具有周期间断系数Maxwell-Dirac系统多尺度算法[J]. 计算数学, 2019, 40(2): 111-129.

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