A Second-Order Symplectic Partitioned Runge-Kutta Scheme for Maxwell’s Equations
Keywords:
A Second-Order Symplectic Partitioned Runge-Kutta Scheme for Maxwell’s EquationsAbstract
In this paper, we construct a new scheme
for approximating the solution to infinite dimensional
non-separable Hamiltonian systems of Maxwell’s
equations using the symplectic partitioned Runge-Kutta
(PRK) method. The scheme is obtained by discretizing
the Maxwell’s equations in the time direction based on
symplectic PRK method, and then evaluating the
equation in the spatial direction with a suitable finite
difference approximation. The scheme preserves the
symplectic structure in the time direction and shows
substantial benefits in numerical computation for
Hamiltonian system, especially in long-term
simulations. Also several numerical examples are
presented to verify the efficiency of the scheme.
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References
T. Hirono, W. W. Lui, and K. Yokoyama,
“Time-domain simulation of electromagnetic field
using a symplectic integrator,” IEEE Microwave
Guided Wave Lett., vol. 7, pp. 279–281, Sept.
T. Hirono, W. W. Lui, K. Yokoyama, and S. Seki,
“Stability and dispersion of the symplectic
fourth-order time-domain schemes for optical field
simulation,” J. Lightwave Technol., vol. 16, pp.
–1920, Oct. 1998.
T. Hirono, W. Lui, S Seki, and Y. Yoshikuni, “A
three-dimensional fourth-order finite-difference
time-domain scheme using a symplectic integrator
propagator,” IEEE Trans. Microwave Theory Tech.,
vol. 49, pp. 1640–1647, Sept. 2001.
I. Saitoh, Y. Suzuki, and N. Takahashi, “The
symplectic finite difference time domain method,”
IEEE Trans. Magn., vol. 37, pp. 3251–3254, Sept.
I. Saitoh and N. Takahashi, “Stability of symplectic
finite-difference time-domain methods,” IEEE
Trans. Magn., vol. 38, pp. 665–668, Mar. 2002.
R. Rieben, D. White, and G. Rodrigue, “High-order
symplectic integration methods for finite element
solutions to time dependent Maxwell equations,”
IEEE Trans. Antennas Propagat., vol. 52, no. 8, pp.
-2195, Aug. 2004.
ACES JOURNAL, VOL. 20, NO. 3, NOVEMBER 2005
S. Geng, “Construction of high order Symplectic
PRK methods,” J.Comput, Math, vol. 13, no.1, pp.
-50, 1995.
S. Geng, “Symplectic partitioned Runge-Kutta
method,” J.Comput, Math., vol. 11, no. 4, pp.
-372, 1993.
L.I. Schiff, Quantum Mechanics. London, U.K.,
McGraw-Hill, 1968,ch.14.
R.D. Ruth, “A canonical integration technique,”
IEEE Trans. Nucl. Science, vol. NS-30, pp.
-2671, Aug. 1983.
E. Forest and R.D.Ruth, “Four-order symplectic
integration,” Physica D., vol. 43, pp. 105-117,
G. Mur, “Absorbing boundary conditions for the
finite-difference approximation of the time-domain
electromagnetic field equations.” IEEE Trans.
Electromagn. Compt., EMC-23 (4), pp. 377-382,
Nov. 1981
