The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations. (English) Zbl 1349.78094
Summary: In this paper we develop a new spatial fourth-order energy-conserved splitting finite-difference time-domain method for Maxwell’s equations. Based on the staggered grids, the splitting technique is applied to lead to a three-stage energy-conserved splitting scheme. At each stage, using the spatial fourth-order difference operators on the strict interior nodes by a linear combination of two central differences, one with a spatial step and the other with three spatial steps, we first propose the spatial high-order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth-order accuracy in space step. The proposed scheme has the important properties: energy-conserved, unconditionally stable, non-dissipative, high-order accurate, and computationally efficient. We first prove that the scheme satisfies energy conversations and is in unconditional stability. We then prove the optimal error estimates of fourth-order in spatial step and second-order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence-free as well. Numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results.
MSC:
| 78M20 | Finite difference methods applied to problems in optics and electromagnetic theory |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q61 | Maxwell equations |
References:
| [1] | Ammari, H.; Kang, H.; Kim, E., Approximate boundary conditions for patch antennas mounted on thin dielectric layers, Commun. Comput. Phys., 1, 1076-1095 (2006) · Zbl 1114.78347 |
| [2] | Botchev, M. A.; Faragó, I.; Horváth, R., Application of the operator splitting to the Maxwell equations with the source term, Appl. Numer. Math., 59, 522-541 (2009) · Zbl 1159.78346 |
| [3] | Brekhovskikh, L. M., Waves in Layered Media (1980), Academic Press: Academic Press New York · Zbl 0558.73018 |
| [4] | Chen, W.; Li, X.; Liang, D., Energy-conserved splitting FDTD methods for Maxwell’s equations, Numer. Math., 108, 445-485 (2008) · Zbl 1185.78020 |
| [5] | Crandall, C. G.; Majda, A., The method of fractional steps for conservation laws, Numer. Math., 34, 285-314 (1980) · Zbl 0438.65076 |
| [6] | Davidson, D. B., Computational Electromagnetics for RF and Microwave Engineering (2005), Cambridge University Press |
| [7] | Diamanti, N.; Giannopoulos, A., Implementation of ADI-FDTD subgrids in ground penetrating radar FDTD models, J. Appl. Geophys., 67, 309-317 (2009) |
| [8] | Douglas, J.; Kim, S., Improved accuracy for locally one-dimensional methods for parabolic equations, Math. Models Methods Appl. Sci., 11, 1563-1579 (2001) · Zbl 1012.65095 |
| [9] | Douglas, J.; Rachford, H. H., On the numerical solutions of heat conduction problems in two and three space variables, Trans. Am. Math. Soc., 82, 421-439 (1956) · Zbl 0070.35401 |
| [10] | Du, C.; Liang, D., An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media, J. Comput. Phys., 229, 4501-4521 (2010) · Zbl 1305.76074 |
| [11] | Eyges, L., The Classical Electromagnetic Fields (1972), Addison-Wesley: Addison-Wesley Reading |
| [12] | Fang, J., Time domain finite difference computation for Maxwell’s equations, (Ph.D. Dissertation (1989), Univ. California: Univ. California Berkeley, CA) |
| [13] | Gao, L.; Zhang, B.; Liang, D., The splitting-difference time-domain methods for Maxwell’s equations in two dimensions, J. Comput. Appl. Math., 205, 207-230 (2007) · Zbl 1122.78021 |
| [14] | Gedney, S.; Liu, G.; Roden, J. A.; Zhu, A., Perfectly matched layer media with CFS for an unconditional stable ADI-FDTD method, IEEE Trans. Antennas Propag., 49, 1554-1559 (2001) · Zbl 1001.78026 |
| [15] | Holland, R., Implicit three-dimensional finite differencing of Maxwell’s equations, IEEE Trans. Nucl. Sci., 31, 1322-1326 (1984) |
| [16] | Huttunen, T.; Monk, P., The use of plane waves to approximate wave propagation in anisotropic media, J. Comput. Math., 25, 350-367 (2007) |
| [17] | Jurgens, H. M.; Zingg, W., Numerical solution of the time-domain Maxwell equations using high-accuracy finite difference methods, SIAM J. Sci. Comput., 22, 1675-1696 (2000) · Zbl 1049.78025 |
| [18] | Karlsen, K. H.; Lie, K. A., An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA J. Numer. Anal., 19, 1-28 (1999) |
| [19] | Leis, R., Initial Boundary Value Problems in Mathematical Physics (1986), Wiley: Wiley New York · Zbl 0599.35001 |
| [20] | Lu, Y., Some techniques for computing wave propagation in optical waveguides, Commun. Comput. Phys., 1, 1056-1075 (2006) · Zbl 1116.78030 |
| [21] | Monk, P.; Suli, E., A convergence analysis of Yee’s scheme on nonuniform grids, SIAM J. Numer. Anal., 31, 393-412 (1994) · Zbl 0805.65121 |
| [22] | Morton, K. W.; Mayers, D. F., Numerical Solution of Partial Differential Equations (2005), Cambridge University Press: Cambridge University Press New York · Zbl 1126.65077 |
| [23] | Nachman, A., A brief perspective on computational electromagnetics, J. Comput. Phys., 129, 237-239 (1996) |
| [24] | Namiki, T., A new FDTD algorithm based on alternating direction implicit method, IEEE Trans. Microw. Theory Tech., 47, 2003-2007 (1999) |
| [25] | Pani, A. K.; Fairweather, G.; Fernandes, R. I., Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term, SIAM J. Numer. Anal., 46, 344-364 (2008) · Zbl 1160.65068 |
| [26] | Peaceman, D. W.; Rachford, H. H., The numerical solution of parabolic and elliptic difference, J. Soc. Ind. Appl. Math., 3, 28-41 (1955) · Zbl 0067.35801 |
| [27] | Strong, G., On the construction and comparison of difference scheme, SIAM J. Numer. Anal., 5, 506-517 (1968) · Zbl 0184.38503 |
| [28] | Singh, G.; Tan, E. L.; Chen, Z. N., A split-step FDTD method for 3-D Maxwell’s equations in general anisotropic media, IEEE Trans. Antennas Propag., 58, 3647-3657 (2010) · Zbl 1369.78828 |
| [29] | Taflove, A.; Brodwin, M. E., Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations, IEEE Trans. Microw. Theory Tech., 23, 623-630 (1975) |
| [30] | Taflove, A.; Hagness, S., Computational Electrodynamics: The Finite-Difference Time-Domain Method (2000), Artech House: Artech House Boston, MA · Zbl 0963.78001 |
| [31] | Turkel, E.; Yefet, A., On the construction of a high order difference scheme for complex domains in a Cartesian grid, Appl. Numer. Math., 33, 113-124 (2000) · Zbl 0964.65098 |
| [32] | Wang, S.; Teixeira, F.; Chen, J., An iterative ADI-FDTD with reduced splitting error, IEEE Microw. Wirel. Comp. Lett., 15, 1531-1533 (2005) |
| [33] | Welfert, B. D., Analysis of iterated ADI-FDTD schemes for Maxwell curl equations, J. Comput. Phys., 222, 9-27 (2007) · Zbl 1129.78025 |
| [34] | Xie, Z.; Chan, C. H.; Zhang, B., An explicit fourth-order staggered finite-difference time-domain method for Maxwell’s equations, J. Comput. Appl. Math., 147, 75-98 (2002) · Zbl 1014.78015 |
| [35] | Yanenko, N. N., The method of fractional steps, (Holt, M., The solution of Problems of Mathematical Physics in Several Variables (1971), Springer-Verlag: Springer-Verlag New York) · Zbl 0209.47103 |
| [36] | 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) · Zbl 1155.78304 |
| [37] | Yefet, A.; Petropoulos, P. G., A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equation, J. Comput. Phys., 168, 286-315 (2001) · Zbl 0981.78012 |
| [38] | Zhang, F.; Chen, Z., Numerical dispersion analysis of the unconditionally stable ADI-FDTD method, IEEE Trans. Microw. Theory Tech., 49, 1006-1009 (2000) |
| [39] | Zhao, A., Analysis of the numerical dispersion of the 2-D alternating-direction implicit FDTD method, IEEE Trans. Microw. Theory Tech., 50, 1156-1164 (2002) |
| [40] | 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) |
| [41] | Zygiridis, T. T.; Tsiboukis, T. D., Optimized three-dimensional FDTD discretizations of Maxwell’s equations on Cartesian grids, J. Comput. Phys., 226, 2372-2388 (2007) · Zbl 1131.78014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
