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Multi-Symplectic Wavelet Collocation Method for Maxwell’s Equations

Published online by Cambridge University Press:  03 June 2015

Huajun Zhu ,
Songhe Song  and
Yaming Chen
Huajun Zhu*
Affiliation:
Department of Mathematics and System Science, School of Science, National University of Defense Technology, Changsha 410073, China
Songhe Song*
Affiliation:
Department of Mathematics and System Science, School of Science, National University of Defense Technology, Changsha 410073, China
Yaming Chen*
Affiliation:
Department of Mathematics and System Science, School of Science, National University of Defense Technology, Changsha 410073, China
*
Corresponding author. Email: girl-zhu@163.com
Email: shsong@nudt.edu.cn
Email: chenym-08@163.com
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Abstract

In this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell’s equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.

Keywords

37M1565P1065M7065T60Multi-symplecticwavelet collocation methodMaxwell’s equationssymplecticconservation laws

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 6 , December 2011 , pp. 663 - 688
Copyright
Copyright © Global-Science Press 2011

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References

[1] Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE T. Antenn. Propag., 14 (1966), pp. 302–307.Google Scholar
[2] Krumpholz, M. and Katehi, L. P. B., MRTD: new time-domain schemes based on multiresolution analysis, IEEE T. Microw. Theory, 44 (1996), pp. 555–571.CrossRefGoogle Scholar
[3] Zheng, F., Chen, Z. and Zhang, J., Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method, IEEE T. Microw. Theory, 48 (2000), pp. 1550–1558.Google Scholar
[4] Gao, L., Zhang, B. and Liang, D., The splitting finite-difference time-domain methods for Maxwell’s equations in two dimensions, J. Comput. Appl. Math., 205 (2007), pp. 207–230.CrossRefGoogle Scholar
[5] Sha, W., Huang, Z. X., Wu, X. L. and Chen, M. S., Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation, J. Comput. Phys., 225 (2007), pp. 33–50.CrossRefGoogle Scholar
[6] Chen, W. B., Xi, X. J. and Liang, D., Energy-conserved splitting FDTD methods for Maxwell’s equations, Numer. Math., 108 (2008), pp. 445–485.CrossRefGoogle Scholar
[7] Xu, Y., van der Vegt, Jaap J. W. and Bokhove, Onno, Discontinuous Hamiltonian finite element method for linear hyperbolic systems, J. Sci. Comput., 3 (2008), pp. 241–265.CrossRefGoogle Scholar
[8] Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), pp. 473–499.CrossRefGoogle Scholar
[9] Bridges, T. J. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184–193.CrossRefGoogle Scholar
[10] Bridges, T. J. and Reich, S., Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Phys. D, 152–153 (2001), pp. 491–504.Google Scholar
[11] Hairer, E., Lubich, CH. and Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, 2002.CrossRefGoogle Scholar
[12] Islas, A. L. and Schober, C. M., On the preservation of phase structure under multisym-plectic discretization, J. Comput. Phys., 197 (2004), pp. 585–609.CrossRefGoogle Scholar
[13] Moore, B. E. and Reich, S., Multi-symplectic integrators for Hamiltonian PDEs, Future Gener. Comp. Sy., 19 (2003), pp. 395–402.CrossRefGoogle Scholar
[14] Asher, U. M. and Mclachlan, R. I., Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), pp. 255–269.CrossRefGoogle Scholar
[15] Hong, J. L. and Li, C., Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations, J. Comput. Phys., 211 (2006), pp. 448–472.CrossRefGoogle Scholar
[16] Hong, J. L., Jiang, S. S., Li, C. and Liu, H. Y., Explicit multisymplectic methods for Hamiltonian wave equations, Commun. Comput. Phys., 2 (2007), pp. 662–683.Google Scholar
[17] Hong, J. L. and Kong, L. H., Novel multi-symplectic integrators for nonlinear fourth-order Schrödinger equation with trapped term, Commun. Comput. Phys., 7 (2010), pp. 613–630.Google Scholar
[18] Chen, J. B., Symplectic and multisymplectic Fourier pseudospectral discretizations for the KleinGordon equation, Lett. Math. Phys., 75 (2006), pp. 293–305.CrossRefGoogle Scholar
[19] Chen, Y. M., Zhu, H. J. and Song, S. H., Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 181 (2010), pp. 1231–1241.CrossRefGoogle Scholar
[20] Cai, J. X., Wang, Y. S., Wang, B. and Jiang, B., New multisymplectic self-adjoint scheme and its composition scheme for the time-domain Maxwell’s equations, J. Math. Phys., 47 (2006), 123508.CrossRefGoogle Scholar
[21] Lee, J. and Fornberg, B., A split step approach for the 3-D Maxwell’s equations, J. Comput. Appl. Math., 158 (2003), pp. 485–505.CrossRefGoogle Scholar
[22] Kong, L. H., Hong, J. L. and Zhang, J. J., Splitting multisymplectic integrators for Maxwell’s equations, J. Comput. Phys., 229 (2010), pp. 4259–4278.CrossRefGoogle Scholar
[23] Zhu, H. J., Tang, L. Y., Song, S. H., Tang, Y. F. and Wang, D. S., Symplectic wavelet collocation method for Hamiltonian wave equations, J. Comput. Phys., 229 (2010), pp. 2550–2572.CrossRefGoogle Scholar
[24] Zhu, H. J., Song, S. H. and Tang, Y. F., Multi-symplectic wavelet collocation method for the nonlinear Schrödinger and Camassa-Holm equations, Comput. Phys. Commun., 182 (2010), pp. 616–627.CrossRefGoogle Scholar
[25] Zhu, H. J., Chen, Y. M., Song, S. H. and Hu, H. Y., Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations, Appl. Numer. Math., 61 (2010), pp. 308–321.CrossRefGoogle Scholar
[26] Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal., 6 (1992), pp. 1716–1740.CrossRefGoogle Scholar
[27] Bertoluzza, S. and Naldi, G., A wavelet collocation method for the numerical solution of partial differential equations, Appl. Comput. Harm. Anal., 3 (1996), pp. 1–9.CrossRefGoogle Scholar
[28] Vasilyev, O. V. and Paolucci, S., A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comput. Phys., 138 (1997), pp. 16–56.CrossRefGoogle Scholar