Energy analysis of the ADI-FDTD method with fourth-order accuracy in time for Maxwell's equations

Li Zhang; Maohua Ran; Hanyue Zhang; Li Zhang; Maohua Ran; Hanyue Zhang

doi:10.3934/math.2023012

AIMS Mathematics

2023, Volume 8, Issue 1: 264-284. doi: 10.3934/math.2023012

Previous Article Next Article

Research article

Energy analysis of the ADI-FDTD method with fourth-order accuracy in time for Maxwell's equations

1.
V. C. & V. R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu 610068, China
2.
School of Mathematical Science, Sichuan Normal University, Chengdu 610068, China
3.
School of Mathematics, Aba Teachers University, Aba 623002, China

Received: 28 June 2022 Revised: 31 August 2022 Accepted: 01 September 2022 Published: 28 September 2022
MSC : 65M06, 65N15

In this work, the ADI-FDTD method with fourth-order accuracy in time for the 2-D Maxwell's equations without sources and charges is proposed. We mainly focus on energy analysis of the proposed ADI-FDTD method. By using the energy method, we derive the numerical energy identity of the ADI-FDTD method and show that the ADI-FDTD method is approximately energy-preserving. In comparison with the energy in theory, the numerical one has two perturbation terms and can be used in computation in order to keep it approximately energy-preserving. Numerical experiments are given to show the performance of the proposed ADI-FDTD method which confirm the theoretical results.
- Maxwell's equations,
- ADI-FDTD,
- energy method,
- approximately energy-preserving
Citation: Li Zhang, Maohua Ran, Hanyue Zhang. Energy analysis of the ADI-FDTD method with fourth-order accuracy in time for Maxwell's equations[J]. AIMS Mathematics, 2023, 8(1): 264-284. doi: 10.3934/math.2023012

Related Papers:

Abstract

In this work, the ADI-FDTD method with fourth-order accuracy in time for the 2-D Maxwell's equations without sources and charges is proposed. We mainly focus on energy analysis of the proposed ADI-FDTD method. By using the energy method, we derive the numerical energy identity of the ADI-FDTD method and show that the ADI-FDTD method is approximately energy-preserving. In comparison with the energy in theory, the numerical one has two perturbation terms and can be used in computation in order to keep it approximately energy-preserving. Numerical experiments are given to show the performance of the proposed ADI-FDTD method which confirm the theoretical results.

References

[1]	K. S. Yee, Numerical solution of inition of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE T. Antenn. Propag., 14 (1966), 302–307. http://dx.doi.org/10.1109/TAP.1966.1138693 doi: 10.1109/TAP.1966.1138693
[2]	T. Namiki, A new FDTD algorithm based on alternating direction implicit method, IEEE T. Microw. Theory, 47 (1999), 2003–2007. http://dx.doi.org/10.1109/22.795075 doi: 10.1109/22.795075
[3]	P. Ciarlet, J. Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numer. Math., 82 (1999), 193–219. http://dx.doi.org/10.1007/s002110050417 doi: 10.1007/s002110050417
[4]	L. Mu, J. Wang, X. Ye, S. Zhang, A weak Galerkin finite element method for the Maxwell's equations, J. Sci. Comput., 65 (2016), 363–386. http://dx.doi.org/10.1007/s10915-014-9964-4 doi: 10.1007/s10915-014-9964-4
[5]	A. Taflove, M. E. Brodwin, Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations, IEEE T. Microw. Theory, 23 (1975), 623–630. http://dx.doi.org/10.1109/tmtt.1975.1128640 doi: 10.1109/tmtt.1975.1128640
[6]	J. E. Dendy, G. Fairweather, Alternating-direction Galerkin methods for parabolic and hyperbolic problems on rectangular polygons, SIAM J. Numer. Anal., 12 (1975), 144–163. http://dx.doi.org/10.2307/2156287 doi: 10.2307/2156287
[7]	R. Holland, Implicit three-dimensinal finite differencing of Maxwell's equations, IEEE T. Nucl. Sci., 31 (1984), 1322–1326. http://dx.doi.org/10.1109/tns.1984.4333504 doi: 10.1109/tns.1984.4333504
[8]	R. Leis, Initial boundary value problems in mathematical physics, New York, Wiley, 1986. http://dx.doi.org/10.1007/978-3-663-10649-4
[9]	H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262–268. http://dx.doi.org/10.1016/0375-9601(90)90092-3 doi: 10.1016/0375-9601(90)90092-3
[10]	S. Li, L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 32 (1995), 1839–1875. http://dx.doi.org/10.2307/2158531 doi: 10.2307/2158531
[11]	A. Taflove, S. Hagness, Computational electrodynamics: The finite-difference time-domain method, 2Eds., Boston, Artech House, 2000. http://dx.doi.org/10.1016/0021-9169(96)80449-1
[12]	F. Zheng, Z. Chen, J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method, IEEE T. Microw. Theory, 48 (2000), 1550–1558. http://dx.doi.org/10.1109/22.869007 doi: 10.1109/22.869007
[13]	T. Namiki, K. Ito, Investigation of numerical errors of the two-dimensional ADI-FDTD method, IEEE T. Microw. Theory, 48 (2000), 1950–1956. http://dx.doi.org/10.1109/22.883876 doi: 10.1109/22.883876
[14]	F. Zheng, Z. Chen, Numerical dispersion analysis of the unconditionally stable 3-D ADI-FDTD method, IEEE T. Microw. Theory, 49 (2001), 1006–1009. http://dx.doi.org/10.1109/22.920165 doi: 10.1109/22.920165
[15]	S. D. Gedney, G. Liu, J. A. Roden, A. Zhu, Perfectly matched layer media with CFS for an unconditional stable ADI-FDTD method, IEEE T. Antenn. Propag., 49 (2001), 1554–1559. http://dx.doi.org/10.1109/8.964091 doi: 10.1109/8.964091
[16]	J. Douglas, S. Kim, Improved accuracy for locally one-dimensional methods for parabolic equations, Math. Mod. Method. Appl. S., 11 (2001), 1563–1579. http://dx.doi.org/10.1142/s0218202501001471 doi: 10.1142/s0218202501001471
[17]	A. P. Zhao, Analysis of the numerical dispersion of the 2-D alternating-direction implicit FDTD method, IEEE T. Microw. Theory, 50 (2002), 1156–1164. http://dx.doi.org/10.1109/22.993419 doi: 10.1109/22.993419
[18]	Z. Q. Xie, C. H. Chan, B. Zhang, An explict four-order staggered finite-difference time-domain method for Maxwell's equations, J. Comput. Appl. Math., 147 (2002), 75–98. http://dx.doi.org/10.1016/S0377-0427(02)00394-1 doi: 10.1016/S0377-0427(02)00394-1
[19]	L. Gao, B. Zhang, D. Liang, The splitting finite-difference time-domain methods for Maxwell's equations in two dimensions, J. Comput. Appl. Math., 205 (2007), 207–230. http://dx.doi.org/10.1016/j.cam.2006.04.051 doi: 10.1016/j.cam.2006.04.051
[20]	B. Li, W. Sun, Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media, SIAM J. Numer. Anal., 51 (2013), 1959–1977. http://dx.doi.org/10.1137/120871821 doi: 10.1137/120871821
[21]	J. Wang, A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation, J. Sci. Comput., 60 (2014), 390–407. http://dx.doi.org/10.1007/s10915-013-9799-4 doi: 10.1007/s10915-013-9799-4
[22]	D. Li, J. Wang, Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system, J. Sci. Comput., 72 (2017), 892–915. http://dx.doi.org/10.1007/s10915-017-0381-3 doi: 10.1007/s10915-017-0381-3
[23]	E. L. Tan, Y. H. Ding, ADI-FDTD Method with fourth order accuracy in time, IEEE T. Microw. Theory, 18 (2008), 296–298. http://dx.doi.org/10.1109/lmwc.2008.922099 doi: 10.1109/lmwc.2008.922099
[24]	W. Chen, X. Li, D. Liang, Energy-conserved splitting FDTD methods for Maxwell's equations, Numer. Math., 108 (2008), 445–485. http://dx.doi.org/10.1007/s00211-007-0123-9 doi: 10.1007/s00211-007-0123-9
[25]	W. Chen, X. Li, D. Liang, Symmetric energy-conserved splitting FDTD scheme for the Maxwell's equations, Commun. Comput. Phys., 6 (2009), 804–825. http://dx.doi.org/10.4208/cicp.2009.v6.p804 doi: 10.4208/cicp.2009.v6.p804
[26]	W. Li, D. Liang, Symmetric energy-conserved S-FDTD scheme for two-dimensional Maxwell's equations in negative index metamaterials, J. Sci. Comput., 69 (2016), 696–735. http://dx.doi.org/10.1007/s10915-016-0214-9 doi: 10.1007/s10915-016-0214-9
[27]	M. H. Carpenter, D. Gottlieb, S. Abarbanel, Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes, J. Comput. Phys., 111 (1994), 220–236. http://dx.doi.org/10.1006/jcph.1994.1057 doi: 10.1006/jcph.1994.1057
[28]	D. Appel$\ddot{o}$, V. A. Bokil, Y. Cheng, F. Li, Energy stable SBP-FDTD methods for Maxwell-Duffing models in nonlinear photonics, IEEE J. Multiscale Mu., 4 (2019), 329–336. http://dx.doi.org/10.1109/jmmct.2019.2959587 doi: 10.1109/jmmct.2019.2959587
[29]	N. Sharan, P. T. Brady, D. Livescu, Time stability of strong boundary conditions in finite-difference schemes for hyperbolic systems, SIAM J. Numer. Anal., 60 (2022), 1331–1362. http://dx.doi.org/10.1137/21M1419957 doi: 10.1137/21M1419957
[30]	D. Liang, Q. Yuan, The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell's equations, J. Comput. Phys., 243 (2013), 344–364. http://dx.doi.org/10.1016/j.jcp.2013.02.040 doi: 10.1016/j.jcp.2013.02.040
[31]	L. Gao, X. Li, W. Chen, New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations, Math. Meth. Appl. Sci., 36 (2013), 440–455. http://dx.doi.org/10.1002/mma.2605 doi: 10.1002/mma.2605
[32]	L. Gao, M. Cao, R. Shi, H. Guo, Energy conservation and super convergence analysis of the EC-S-FDTD schemes for Maxwell's equations with periodic boundaries, Numer. Meth. Part. D. E., 35 (2019), 1562–1587. http://dx.doi.org/10.1002/num.22364 doi: 10.1002/num.22364
[33]	J. Xie, D. Liang, Z. Zhang, Energy-preserving local mesh-refined splitting FDTD schemes for two dimensional Maxwell's equations, J. Comput. Phys., 425 (2021), 109896. http://dx.doi.org/10.1016/j.jcp.2020.109896 doi: 10.1016/j.jcp.2020.109896
[34]	D. Wang, A. Xiao, W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys., 272 (2014), 644–655. http://dx.doi.org/10.1016/j.jcp.2014.04.047 doi: 10.1016/j.jcp.2014.04.047
[35]	P. Wang, C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238–251. http://dx.doi.org/10.1016/j.jcp.2014.03.037 doi: 10.1016/j.jcp.2014.03.037
[36]	M. Ran, C. Zhang, A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations, Commun. Nonlinear Sci., 41 (2016), 64–83. http://dx.doi.org/10.1016/j.cnsns.2016.04.026 doi: 10.1016/j.cnsns.2016.04.026
[37]	M. Ran, C. Zhang, A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator, Int. J. Comput. Math., 93 (2016), 1103–1118. http://dx.doi.org/10.1080/00207160.2015.1016924 doi: 10.1080/00207160.2015.1016924
[38]	H. Li, Y. Wang, Q. Sheng, An energy-preserving Crank-Nicolson Galerkin method for Hamiltonian partial differential equations, Numer. Meth. Part. D. E., 32 (2016), 1485–1504. http://dx.doi.org/10.1002/num.22062 doi: 10.1002/num.22062

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)