Skip to main content
SpringerLink
Log in

Numerical Analysis of AVF Methods for Three-Dimensional Time-Domain Maxwell’s Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose two schemes [AVF(2) and AVF(4)] for Maxwell’s equations, by discretizing the Hamiltonian formulation with Fourier pseudospectral method for spatial discretization and average vector field method for time integration. Both AVF(2) and AVF(4) hold the two Hamiltonian energies automatically, while being energy-, momentum- and divergence-preserving, unconditionally stable, non-dissipative and spectral accurate. Rigorous error estimates are obtained for the proposed schemes. The numerical dispersion relations are also investigated. Numerical experiments support well the theoretical analysis results. The proposed schemes are valid for the regular domain, but invalid for the domain with complex geometries.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

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

    Article  MATH  Google Scholar 

  2. Liu, Y.: Fourier analysis of numerical algorithms for the Maxwell equations. J. Comput. Phys. 124, 396–416 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Namiki, T.: A new FDTD algorithm based on alternating direction implicit method. IEEE Trans. Microw. Theory Tech. 47, 2003–2007 (1999)

    Article  Google Scholar 

  4. Zheng, F., Chen, Z., Zhang, J.: Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method. IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000)

    Article  Google Scholar 

  5. Zhao, A.: Analysis of the numerical dispersion of the 2-Dalternating-direction implicit FDTD method. IEEE Trans. Microw. Theory Tech. 50, 1156–1164 (2002)

    Article  Google Scholar 

  6. Diamanti, N., Giannopoulos, A.: Implementation of ADI-FDTD subgrids in ground penetrating radar FDTD models. J. Appl. Geophys. 67, 309–317 (2009)

    Article  Google Scholar 

  7. Chen, W.B., Li, X.J., Liang, D.: Energy-conserved splitting finite-difference time-domain methods for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal. 48, 1530–1554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, W.B., Li, X.J., Liang, D.: Energy-conserved splitting FDTD methods for Maxwell’s equations. Numer. Math. 108, 445–485 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liang, D., Yuan, Q.: The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations. J. Comput. Phys. 243, 344–364 (2013)

    Article  MathSciNet  Google Scholar 

  10. Anderson, N., Arthurs, A.M.: Helicity and variational principles for Maxwell’s equations. Int. J. Electron. 54, 861–864 (1983)

    Article  Google Scholar 

  11. Marsden, J.E., Weinstein, A.: The Hamiltonian structure of the Maxwell–Vlasov equations. Physica D 4, 394–406 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sha, W., Huang, Z.X., Wu, X.L., Chen, M.S.: Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation. J. Comput. Phys. 225, 33–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cai, J.X., Wang, Y.S., Wang, B., Jiang, B.: New multisymplectic self-adjoint scheme and its composition scheme for the time-domain Maxwell’s equations. J. Math. Phys. 47(123508), 1–16 (2006)

    Google Scholar 

  14. Sun, Y., Tse, P.S.P.: Symplectic and multisymplectic numerical methods for Maxwell’s equations. J. Comput. Phys. 230, 2076–2094 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhu, H.J., Song, S.H., Chen, Y.M.: Multi-symplectic wavelet collocation method for Maxwell’s equations. Adv. Appl. Math. Mech. 3, 663–688 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41(045206), 1–7 (2008)

    MathSciNet  Google Scholar 

  17. McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. A 357, 1021–1046 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. McLachlan, R.I., Quispel, G.R.W., Turner, G.S.: Numerical integrators that preserve symmetries and reversing symmetries. SIAM J. Numer. Anal. 35, 586–599 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)

    MathSciNet  Google Scholar 

  20. Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Energy-preserving integrators and the structure of B-series. Found Comput. Math. 10, 673–693 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cieśliński, J.L.: Improving the accuracy of the AVF method. J. Comput. Appl. Math. 259, 233–243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Celledoni, E., Grimm, V., McLaren, R.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Perserving energy resp. dissipation in numerical PDEs using the ”Average Vector Field” method. J. Comput. Phys. 231, 6770–6789 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dahlby, M., Owren, B.: A general framework for deriving integral preserving numerical methods for PDEs. SIAM J. Sci. Comput. 33, 2318–2340 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhong, Z.Y., Jin, S., Markowich, P.A., Sparber, C., Zheng, C.X.: A time-splitting spectral scheme for the Maxwell–Dirac system. J. Comput. Phys. 208, 761–789 (2005)

    Article  MathSciNet  Google Scholar 

  25. Liu, Q.H.: The PSTD algorithm: a time-domain method requiring only two cells per wavelength. Microw. Opt. Technol. Lett. 15, 158–165 (1997)

    Article  Google Scholar 

  26. Liu, Q.H.: Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm. IEEE Trans. Geosci. Remote Sens. 37, 917–926 (1999)

    Article  Google Scholar 

  27. Hesthaven, J., Warburton, T.: High-order nodal methods on unstructured grids, Part I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 1–34 (2002)

    Article  Google Scholar 

  28. Kopriva, D., Woodruff, S., Hussaini, M.: Discontinuous spectral element approximation of Maxwell’s equations, in Discontinuous Galerkin Methods. In: Cockburn, B., Karniadakis, G., Shu, C.W. (eds.) Theory, computation and applications. Lecture notes computer science engineering, vol. 11, pp. 355–361. Springer, Berlin (2000)

  29. Perugia, I., Schötzau, D.: The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72, 1179–1214 (2002)

    Article  Google Scholar 

  30. Guo, B.Y., Ma, H.P.: The Fourier pseudospectral method for three-dimensional vorticity equations. Acta Math. Appl. Sin. 4, 55–68 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lee, J., Fornberg, B.: A split step approach for the 3-D Maxwell’s equations. J. Comput. Appl. Math. 158, 485–505 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)

    Article  MathSciNet  Google Scholar 

  33. Gao, L.P., Zhang, B.: Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations. Sci. Chin. Math. 56, 1705–1726 (2013)

    Article  MATH  Google Scholar 

  34. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods: evolution to complex geometries and applications to fluid dynamics. Springer-Verlag, Berlin, Heidelberg (2007)

Download references

Acknowledgments

The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions.

Author information

Authors and Affiliations

  1. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Jiaxiang Cai, Yushun Wang & Yuezheng Gong

  2. School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, Jiangsu, People’s Republic of China

    Jiaxiang Cai

Authors
  1. Jiaxiang Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yushun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuezheng Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiaxiang Cai.

Additional information

The work was supported by the National Natural Science Foundation of China (11201169, 41231173 and 11271195) and the Graduate Education Innovation Project of Jiangsu Province of China (CXLX13_366).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cai, J., Wang, Y. & Gong, Y. Numerical Analysis of AVF Methods for Three-Dimensional Time-Domain Maxwell’s Equations. J Sci Comput 66, 141–176 (2016). https://doi.org/10.1007/s10915-015-0016-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0016-5

Keywords

Mathematics Subject Classification

Search

Navigation