volume | PIER Journals

Abstract

The connections between Maxwell's equations and symplectic matrix are studied. First, we analyze the continuous-time Maxwell's differential equations in free space and verify its time evolution matrix (TEMA) is symplectic-unitary matrix for complex space or symplectic-orthogonal matrix for real space. Second, the spatial differential operators are discretized by pseudo-spectral (PS) approach with collocated grid and by finite-difference (FD) method with staggered grid. For the PS approach, the TEMA conserves the symplectic-unitary property. For the FD method, the TEMA conserves the symplectic-orthogonal property. Finally, symplectic integration scheme is used in the time direction. In particular, we find the symplectiness of the TEMA also can be conserved. The mathematical proofs presented are helpful for further numerical study of symplectic schemes.

1. Feng, K. and M. Z. Qin, Symplectic Geometric Algorithm for Hamiltonian Systems, Zhejiang Science&Technology Press, 2003.

2. Sanz-Serna, J. M. and M. P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, 1994.

3. Hirono, T., W. Lui, S. Seki, and Y. Yoshikuni, "A threedimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator," IEEE Transactions on Microwave Theory andT echniques, Vol. 49, 1640-1648, Sept. 2001.
doi:10.1109/22.942578

4. Sha, W., X. L. Wu, and M. S. Chen, "A diagonal split-cell model for the high-order symplectic FDTD scheme," PIERS Online, Vol. 2, 715-719, Jun. 2006.
doi:10.2529/PIERS060903035033

5. Sha, W., Z. X. Huang, M. S. Chen, and X. L. Wu, "Survey on symplectic finite-difference time-domain schemes for Maxwell's equations," IEEE Transactions on Antennas and Propagation, Vol. 56, 493-500, Feb. 2008.
doi:10.1109/TAP.2007.915444

6. Shi, Y. and C. H. Liang, "Multidomain pseudospectral time domain algorithm using a symplectic integrator," IEEE Transactions on Antennas and Propagation, Vol. 55, 433-439, Feb. 2007.
doi:10.1109/TAP.2006.889906

7. Reich, S., "Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations ," Journal of Computational Physics, Vol. 157, 473-499, Jan. 2000.
doi:10.1006/jcph.1999.6372

8. Teixeira, F. L., "Geometric aspects of the simplicial discretization of Maxwell's equations," Progress In Electromagnetics Research, Vol. 32, 171-188, 2001.
doi:10.2528/PIER00080107

9. Zhou, X. L., "On independence completeness of Maxwell's equations and uniqueness theorems in electromagnetics," Progress In Electromagnetics Research, Vol. 64, 117-134, 2006.
doi:10.2528/PIER06061302

10. Everitt, W. N. and L. Markus, "Complex symplectic geometry with applications to ordinary differential operators ," Transactions of the American Mathematical Society, Vol. 351, 4905-4945, Dec. 1999.
doi:10.1090/S0002-9947-99-02418-6

11. Anderson, N. and A. M. Arthurs, "Helicity and variational principles for Maxwell's equations," International Journal of Electronics, Vol. 54, 861-864, Jun. 1983.
doi:10.1080/00207218308938781

12. Farago, I., R. Horvath, and W. H. A. Schilders, "Investigation of numerical time-integrations of Maxwell's equations using the staggered grid spatial discretization," International Journal of Numerical Modelling-electronic Networks Devices and Fields, Vol. 18, 149-169, Mar.-Apr. 2005.
doi:10.1002/jnm.570

13. Kole, J. S., M. T. Figge, and H. De Raedt, "Higher-order unconditionally stable algorithms to solve the time-dependent Maxwell equations," Physical Review E, Vol. 65, Jun. 2002.
doi:10.1103/PhysRevE.65.066705

14. Yoshida, H., "Construction of higher order symplectic integrators," Physica D: Nonlinear Phenomena, Vol. 46, 262-268, Nov. 1990.

15. Dopico, F. M. and C. R. Johnson, "Complementary bases in symplectic matrices and a proof that their determinant is one," Linear Algebra andIts Applications, Vol. 419, 772-778, Dec. 2006.
doi:10.1016/j.laa.2006.06.014

Dr. Wei Sha

Dr. Xian-Liang Wu

Dr. Zhixiang Huang

Dr. Ming-Sheng Chen