Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T04:58:20.412Z Has data issue: false hasContentIssue false
  • English
  • Français

Implicit DG Method for Time Domain Maxwell’s Equations Involving Metamaterials

Published online by Cambridge University Press:  09 September 2015

Jiangxing Wang ,
Ziqing Xie  and
Chuanmiao Chen
Jiangxing Wang
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China
Ziqing Xie*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China
Chuanmiao Chen
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing, College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China
*
*Corresponding author. Email: [email protected] (J. X. Wang), [email protected] (Z. Q. Xie), [email protected]
Get access

Abstract

An implicit discontinuous Galerkin method is introduced to solve the time-domain Maxwell’s equations in metamaterials. The Maxwell’s equations in metamaterials are represented by integral-differential equations. Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain. The fully discrete numerical scheme is proved to be unconditionally stable. When polynomial of degree at most p is used for spatial approximation, our scheme is verified to converge at a rate of O(τ2+hp+1/2). Numerical results in both 2D and 3D are provided to validate our theoretical prediction.

Keywords

65M1265M60Maxwell’s equationsmetamaterialsfully discteteDG method, L2-stabilityL2-error estimate
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 6 , December 2015 , pp. 796 - 817
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.Google Scholar
[2]Cockburn, B., Karniadakis, G. E. and Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications, Brown University, Providence, RI (US), 2000.CrossRefGoogle Scholar
[3]Cockburn, B., Li, F. and Shu, C. W.-, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194 (2004), pp. 488–610.Google Scholar
[4]Cockburn, B. and Shu, C. W.-, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), pp. 173–261.Google Scholar
[5]Cui, T. J., Smith, D. R. and Liu, R., Metamaterials: Theory, Design, and Applications, Springer, 2010.Google Scholar
[6]Di Pietro, D. A. and Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, Springer, 2011.Google Scholar
[7]Dolean, V., Fans, H., Fezoui, L. and Lanteri, S., Locally implicit discontinuous Galerkin method for time domain electromagnetics, J. Comput. Phys., 229 (2010), pp. 512–526.Google Scholar
[8]Fezoui, L., Lanteri, S., Lohrengel, S. and Piperno, S., Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Model. Math. Anal. Numer., 39 (2005), pp. 1149–1176.Google Scholar
[9]Hao, Y. and Mittra, R., FDTD Modeling of Metamaterials: Theory and Applications, Artech house, 2008.Google Scholar
[10]Hesthaven, J. S. and Warburton, T., Nodal high-order methods on unstructured grids I. Time-domain solution of Maxwell’s equations, J. Comput. Phys., 181 (2002), pp. 186–221.Google Scholar
[11]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008.Google Scholar
[12]Huang, Y., Li, J. and Yang, W., Interior penalty DG methods for Maxwells equations in dispersive media, J. Comput. Phys., 230 (2011), pp. 4559–4570.Google Scholar
[13]Huang, Y., Li, J. and Yang, W., Modeling backward wave propagation in metamaterials by the finite element time-domain method, SIAM J. Sci. Comput., 35 (2013), pp. B248–B274.Google Scholar
[14]Lanteri, S. and Scheid, C., Convergence of a discontinuous Galerkin scheme for the mixed time-domain Maxwell’s equations in dispersive media, IMA J. Numer. Anal., 33 (2013), pp. 432–459.Google Scholar
[15]Li, J., Posteriori error estimation for an interior penalty discontinuous Galerkin method for Maxwells equations in cold plasma, Adv. Appl. Math. Mech., 1 (2009), pp. 107–124.Google Scholar
[16]Li, J., Development of discontinuous Galerkin methods for Maxwells equations in metamaterials and perfectly matched layers, J. Comput. Appl. Math., 47 (2011), pp. 950–961.Google Scholar
[17]Li, J., Unified analysis of leap-frog methods for solving time-domain Maxwells equations in dispersive media, J. Sci. Comput., 47 (2011), pp. 1–26.Google Scholar
[18]Li, J. and Hesthaven, J. S., Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys,. 258 (2014), pp. 915–930.CrossRefGoogle Scholar
[19]Li, J. and Huang, Y., Time-domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer, 2013.Google Scholar
[20]Li, J., Waters, J. W. and Mchorro, E. A., An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwells equations in metamaterials, Comput. Methods Appl. Mech. Eng., 223 (2012), pp. 43–54.Google Scholar
[21]Lu, T., Zhang, P. and Cai, W., Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions, J. Comput. Phys., 200 (2004), pp. 549–580.Google Scholar
[22]Quarteroni, A., Quarteroni, A. M. and Valli, A., Numerical Approximation of Partial Differential Equations, Springer, 2008.Google Scholar
[23]Reed, W. H. and Hill, T., Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973).Google Scholar
[24]Tavlove, A. and Hagness, S. C., Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, 1995.Google Scholar
[25]Wang, B., Xie, Z. and Zhang, Z., Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, 229 (2010), pp. 8552–8563.Google Scholar
[26]Wang, B., Xie, Z. and Zhang, Z., Space-time discontinuous galerkin method for Maxwell equations in dispersive media, Acta Math. Sci., 34 (2014), pp. 1357–1376.Google Scholar
[27]Xie, Z., Wang, B. and Zhang, Z., Space-time discontinuous Galerkin method for Maxwells equations, Commun. Comput. Phys., 14 (2013), pp. 916–939.Google Scholar