zbMATH — the first resource for mathematics

A stable finite-difference time-domain scheme for local time-stepping on an adaptive mesh. (English) Zbl 1452.78033
Summary: Physical effects driven by strong electromagnetic fields often occur in regions of highly localized fields on a scattering object. Unfortunately, the most common numerical technique for simulating time-domain electromagnetics, known as Finite-Difference Time-Domain (FDTD), is ill-equipped to handle such problems. A common solution to capture physics across many spatial scales is to use an adaptive mesh, which resolves temporal or spatial features exactly when and where they are needed, avoiding extra computation in space-time regions where it is unnecessary. We present a minimal modification to the FDTD algorithm that allows for a stable late-time solution to Maxwell’s equations on an adaptive mesh with a Courant-Friedrichs-Levy limit of 5/6. An emphasis is placed on creating a simple, flexible, and easy to understand algorithm. The algorithm is implemented in 1D, 2D and 3D for geometries which are dynamic or possess large disparities in spatial or temporal scales. An example is presented which demonstrates the use of the algorithm in a resonant dielectric disc with a small slot.
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
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Kourtzanidis, K.; Rogier, F.; Boeuf, J.-P., ADI-FDTD modeling of microwave plasma discharges in air towards fully three-dimensional simulations, Comput. Phys. Commun., 195, 49-60 (2015) · Zbl 1344.76096
[2] Appel, A. W., An efficient program for many-body simulation, SIAM J. Sci. Stat. Comput., 6, 85-103 (1985)
[3] Grengard, L.; Rokhlin, V., The rapid evaluation of potential fields in three dimensions, ((1988), Springer: Springer Berlin, Heidelberg), 121-141
[4] Taflove, A.; Hagness, S. C., Computational Electrodynamics: The Finite-Difference Time-Domain Method (2005), Artech House
[5] Hamm, J.; Renn, F.; Hess, O., Dispersive media subcell averaging in the FDTD method using corrective surface currents, IEEE Trans. Antennas Propag., 62, 832-838 (2014)
[6] Farjadpour, A.; Roundy, D.; Rodriguez, A.; Ibanescu, M.; Bermel, P.; Joannopoulos, J. D.; Johnson, S. G.; Burr, G. W., Improving accuracy by subpixel smoothing in the finite-difference time domain, Opt. Lett., 31, 2972 (2006)
[7] Deinega, A.; Valuev, I., Subpixel smoothing for conductive and dispersive media in the finite-difference time-domain method, Opt. Lett., 32, 3429 (2007)
[8] Liu, J.; Brio, M.; Moloney, J. V., Subpixel smoothing finite-difference time-domain method for material interface between dielectric and dispersive media, Opt. Lett., 37, 4802 (2012)
[9] Horesh, L.; Haber, E., A second order discretization of Maxwell’s equations in the quasi-static regime on OcTree grids, SIAM J. Sci. Comput., 33, 2805-2822 (2011) · Zbl 1232.65019
[10] Coomar, A.; Arntsen, C.; Lopata, K. A.; Pistinner, S.; Neuhauser, D., Near-field: a finite-difference time-dependent method for simulation of electrodynamics on small scales, J. Chem. Phys., 135, Article 084121 pp. (2011)
[11] Kunz, K.; Simpson, L., A technique for increasing the resolution of finite-difference solutions of the Maxwell equation, IEEE Trans. Electromagn. Compat., EMC-23, 419-422 (1981)
[12] Zivanovic, S. S.; Yee, K. S.; Mei, K. K., A subgridding method for the time-domain finite-difference method to solve Maxwell’s equations, IEEE Trans. Microw. Theory Tech., 39, 471-479 (1991)
[13] Prescott, D. T.; Shuley, N. V., A method for incorporating different sized cells into the finite-difference time-domain analysis technique, IEEE Microw. Guided Wave Lett., 2, 434-436 (1992)
[14] Thoma, P.; Weiland, T., A consistent subgridding scheme for the finite difference time domain method, Int. J. Numer. Model. Electron. Netw., Devices Fields, 9, 359-374 (1996)
[15] Thoma, P.; Weiland, T., Numerical stability of finite difference time domain methods, IEEE Trans. Magn., 34, 2740-2743 (1998)
[16] Chevalier, M. W.; Luebbers, R. J., FDTD local grid with material traverse, (IEEE Antennas Propag. Soc. Int. Symp. (1996), IEEE), 116-119, Dig
[17] Okoniewski, M.; Okoniewska, E.; Stuchly, M. A., Three-dimensional subgridding algorithm for FDTD, IEEE Trans. Antennas Propag., 45, 422-429 (1997)
[18] White, M. J.; Iskander, M. F.; Huang, Z., Development of a multigrid FDTD code for three-dimensional applications, IEEE Trans. Antennas Propag., 45, 1512-1517 (1997)
[19] Yu, W.; Mittra, R., A new subgridding method for the finite-difference time-domain (FDTD) algorithm, Microw. Opt. Technol. Lett., 21, 330-333 (1999)
[20] Krishnaiah, K. M.; Railton, C. J., A stable subgridding algorithm and its application to eigenvalue problems, IEEE Trans. Microw. Theory Tech., 47, 620-628 (1999)
[21] Wang, Shumin; Teixeira, F. L.; Lee, R.; Lee, Jin-Fa, Optimization of subgridding schemes for FDTD, IEEE Microw. Wirel. Compon. Lett., 12, 223-225 (2002)
[22] Vaccari, A.; Cala’ Lesina, A.; Cristoforetti, L.; Pontalti, R., Parallel implementation of a 3D subgridding FDTD algorithm for large simulations, Prog. Electromagn. Res., 120, 263-292 (2011)
[23] Marrone, M.; Mittra, R.; Yu, W., A novel approach to deriving a stable hybridized FDTD algorithm using the cell method, (IEEE Antennas Propag. Soc. Int. Symp. Dig. Held Conjunction with Usn. North Am. Radio Sci. Meet. (Cat. No. 03CH37450) (2003), IEEE), 340-343
[24] Xiao, Kai; Pommerenke, D. J.; Drewniak, J. L., A three-dimensional FDTD subgridding method with separate spatial and temporal subgridding interfaces, (2005 Int. Symp. Electromagn. Compat. 2005. 2005 Int. Symp. Electromagn. Compat. 2005, EMC 2005 (2005), IEEE), 578-583
[25] Podebrad, O.; Clemens, M.; Weiland, T., New flexible subgridding scheme for the finite integration technique, IEEE Trans. Magn., 39, 1662-1665 (2003)
[26] Brio, M.; Moloney, J. V.; Zakharian, A. R., FDTD based second-order accurate local mesh refinement method for Maxwell’s equations in two space dimensions, Commun. Math. Sci., 2, 497-513 (2004) · Zbl 1083.65084
[27] Chang, C.; Sarris, C. D., A spatial filter-enabled high-resolution subgridding scheme for stable FDTD modeling of multiscale geometries, (2011 IEEE MTT-S Int. Microw. Symp. (2011), IEEE), 1-4
[28] Donderici, B.; Teixeira, F. L., Improved FDTD subgridding algorithms via digital filtering and domain overriding, IEEE Trans. Antennas Propag., 53, 2938-2951 (2005)
[29] Gaffar, M.; Jiao, D., An explicit and unconditionally stable FDTD method for electromagnetic analysis, IEEE Trans. Microw. Theory Tech., 62, 2538-2550 (2014)
[30] Collino, F.; Fouquet, T.; Joly, P., Conservative space-time mesh refinement methods for the FDTD solution of Maxwell’s equations, J. Comput. Phys., 211, 9-35 (2006) · Zbl 1107.78015
[31] Meglicki, Z.; Gray, S. K.; Norris, B., Multigrid FDTD with Chombo, Comput. Phys. Commun., 176, 109-120 (2007) · Zbl 1196.78030
[32] Alsunaidi, M. A.; Al-Jabr, A. A., A general ADE-FDTD algorithm for the simulation of dispersive structures, IEEE Photonics Technol. Lett., 21, 817-819 (2009)
[33] Brio, M.; Dineen, C.; Moloney, J. V.; Zakharian, A. R., Stability of 2D FDTD algorithms with local mesh refinement for Maxwell’s equations, Commun. Math. Sci., 4, 345-374 (2006) · Zbl 1146.78010
[34] Remis, R. F., On the stability of the finite-difference time-domain method, J. Comput. Phys., 163, 249-261 (2000) · Zbl 0994.78016
[35] Edelvik, F.; Schuhmann, R.; Weiland, T., A general stability analysis of FIT/FDTD applied to lossy dielectrics and lumped elements, Int. J. Numer. Model. Electron. Netw., Devices Fields, 17, 407-419 (2004) · Zbl 1054.78029
[36] Zakharian, A. R.; Brio, M.; Dineen, C.; Moloney, J. V., Second-order accurate FDTD space and time grid refinement method in three space dimensions, IEEE Photonics Technol. Lett., 18, 1237-1239 (2006)
[37] Ghrist, M.; Fornberg, B.; Driscoll, T. A., Staggered time integrators for wave equations, SIAM J. Numer. Anal., 38, 718-741 (2000) · Zbl 0973.65070
[38] Fornberg, B., Fast generation of weights in finite difference formulas, (Recent Dev. Numer. Methods Softw. ODEs/DAEs/PDEs (1992), WORLD SCIENTIFIC), 97-123
[39] Achilleos, G., Errors within the Inverse Distance Weighted (IDW) interpolation procedure, Geocarto Int., 23, 429-449 (2008)
[40] Boeuf, J.-P.; Chaudhury, B.; Zhu, G. Q., Theory and modeling of self-organization and propagation of filamentary plasma arrays in microwave breakdown at atmospheric pressure, Phys. Rev. Lett., 104, Article 015002 pp. (2010)
[41] Kourtzanidis, K.; Raja, L. L., Analysis and characterization of microwave plasma generated with rectangular all-dielectric resonators, Plasma Sources Sci. Technol., 26, Article 045007 pp. (2017)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.