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Developing and analyzing fourth-order difference methods for the metamaterial Maxwell’s equations

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Abstract

In this paper, we develop both a fourth order explicit scheme and a compact implicit scheme for solving the metamaterial Maxwell’s equations. A systematic technique is introduced to prove stability and error estimate for both schemes. Numerical results supporting our analysis are presented. To our best knowledge, our convergence theory and stability results are novel, and provide the first error estimate for the fourth order finite difference methods for Maxwell’s equations.

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References

  1. Bokil, V.A., Gibson, N.L.: Analysis of spatial high-order finite difference methods for Maxwell’s equations in dispersive media. IMA J. Numer. Anal. 32(3), 926–956 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenner, S.C., Gedicke, J., Sung, L.-Y.: An adaptive P 1 finite element method for two-dimensional transverse magnetic time harmonic Maxwell’s equations with general material properties and general boundary conditions. J. Sci. Comput. 68(2), 848–863 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Demkowicz, L., Li, J.: Numerical simulations of cloaking problems using a DPG method. Comput. Mech. 51(5), 661–672 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fang, J.: Time-Domain Finite Difference Computations for Maxwell’s Equations, Ph.D dissertation, EECS Dept., Univ. of California, Berkeley CA (1989)

  5. Fathy, A., Wang, C., Wilson, J., Yang, S.: A fourth order difference scheme for the Maxwell equations on Yee grid. J. Hyperbolic Diff. Equa. 5(3), 613–642 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hao, Y., Mittra, R.: FDTD Modeling of Metamaterials: Theory and Applications. Artech House Publishers, Boston (2008)

    MATH  Google Scholar 

  7. Hesthaven, J.S.: High-order accurate methods in time-domain computational electromagnetics: A review. In: Advances in Imaging and Electron Physics. Elsevier BV, pp. 59–123 (2003)

  8. Li, J., Hesthaven, J.S.: Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials. J. Comput. Phys. 258, 915–930 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, J., Huang, Y.: Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer Series in Computational Mathematics, vol. 43. Springer (2013)

  10. Li, J., Shi, C., Shu, C.-W.: Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials. Comput. Math. Appl. 73, 1760–1780 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, J., Shields, S.: Superconvergence analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes. Numer. Math. 134, 741–781 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, W., Liang, D., Lin, Y.: A new energy-conserved s-fdtd scheme for Maxwell’s equations in metamaterials. Int. J. Numer. Anal. Mod. 10, 775–794 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Mattsson, K., Nordström, J.: High order finite difference methods for wave propagation in discontinuous media. J. Comput. Phys. 220(1), 249–269 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Monk, P., Süli, E.: A convergence analysis of Yee’s scheme on nonuniform grid. SIAM J. Numer. Anal. 31, 393–412 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xie, Z., Wang, J., Wang, B., Chen, C.: Solving Maxwell’s equation in meta-materials by a CG-DG method. Commun. Comput. Phys. 19(5), 1242–1264 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, Z., Wang, L.-L., Rong, Z., Wang, B., Zhang, B.: Seamless integration of global Dirichlet-to-Neumann boundary condition and spectral elements for transformation electromagnetics. Comput. Methods Appl. Mech. Engrg. 301, 137–163 (2016)

    Article  MathSciNet  Google Scholar 

  17. Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propagat. 14, 302–307 (1966)

    Article  MATH  Google Scholar 

  18. Yefet, A., Turkel, E.: Fourth order compact implcit method for the Maxwell equations with discontinous coefficients. Appl. Numer. Math. 33, 125–134 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yefet, A., Petropoulos, P.G.: A staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equations. J. Comput. Phys. 168, 286–315 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao, S., Wei, G.W.: High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces. J. Comput. Phys. 200, 60–103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to Peter Monk for his insightful comments on an early version of the paper, which have been adopted to improve the paper. We also thank two referees for their helpful comments on improving our paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154-4020, USA

    Jichun Li & Min Chen

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, China

    Meng Chen

Authors
  1. Jichun Li
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  2. Meng Chen
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  3. Min Chen
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Corresponding author

Correspondence to Jichun Li.

Additional information

Communicated by: Aihui Zhou

Work supported by NSF grant DMS-1416742 and NSFC project 11671340.

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Li, J., Chen, M. & Chen, M. Developing and analyzing fourth-order difference methods for the metamaterial Maxwell’s equations. Adv Comput Math 45, 213–241 (2019). https://doi.org/10.1007/s10444-018-9614-8

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  • DOI: https://doi.org/10.1007/s10444-018-9614-8

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