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{{Short description|Numerical analysis technique}}
[[File:Yee cell.png|thumb|250px|In finite-difference time-___domain method, "Yee lattice" is used to discretize [[Maxwell's equations]] in space. This scheme involves the placement of [[Electric field|electric]] and [[magnetic fields]] on a staggered grid.]]
'''Finite-difference time-___domain''' ('''FDTD''') or '''Yee's method''' (named after the Chinese American applied mathematician [[Kane S. Yee]], born 1934) is a [[numerical analysis]] technique used for modeling [[computational electrodynamics]].
== History ==
Finite difference schemes for time-dependent [[partial differential equation]]s (PDEs) have been employed for many years in [[computational fluid dynamics]] problems,<ref name="vonneumann49" /> including the idea of using centered finite difference operators on staggered grids in space and time to achieve second-order accuracy.<ref name="vonneumann49" />
The novelty of
The descriptor "Finite-difference time-___domain" and its corresponding "FDTD" acronym were originated by [[Allen Taflove]] in 1980.<ref name="taflove80" />
Since about 1990, FDTD techniques have emerged as primary means to computationally model many scientific and engineering problems dealing with [[electromagnetic wave]] interactions with material structures. Current FDTD modeling applications range from near-[[Direct current|DC]] (ultralow-frequency [[geophysics]] involving the entire Earth-[[ionosphere]] waveguide) through [[microwaves]] (radar signature technology, [[Antenna (radio)|antennas]], wireless communications devices, digital interconnects, biomedical imaging/treatment) to [[visible light]] ([[photonic crystal]]s, nano[[plasmon]]ics, [[soliton]]s, and [[biophotonics]]).<ref name="taflove05" /> In 2006, an estimated 2,000 FDTD-related publications appeared in the science and engineering literature (see [[#Popularity|Popularity]]). As of 2013, there are at least 25 commercial/proprietary FDTD software vendors; 13 free-software/[[Open source|open-source]]-software FDTD projects; and 2 freeware/closed-source FDTD projects, some not for commercial use (see [[#External links|External links]]).
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| 2009 || Oliveira and Sobrinho applied the FDTD method for simulating lightning strokes in a power substation<ref name="oliveira09" />
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| 2021 || Oliveira and Paiva developed the Least Squares Finite-Difference Time-Domain method (LS-FDTD) for using time steps beyond FDTD CFL limit.<ref name="oliveira2021" />
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While the FDTD technique computes electromagnetic fields within a compact spatial region, scattered and/or radiated far fields can be obtained via near-to-far-field transformations.<ref name="umashankar82" />
==== Stability ====
Due to the linearity of the FDTD method, the region of stability of the FDTD method may be determined by [[Von Neumann stability analysis]]. This method assumes that electric and magnetic fields are proportional to a monochromatic complex exponential. After a single time-step, the magnitude amplitude of the stable fields need to remain the same or less. This leads to the [[Courant–Friedrichs–Lewy condition]], which describes the relationship of the FDTD parameters to ensure stability.<ref name="taflove05"/>
=== Strengths of FDTD modeling ===
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# Parallel-processing computer architectures have come to dominate supercomputing. FDTD scales with high efficiency on parallel-processing CPU-based computers, and extremely well on recently developed GPU-based accelerator technology.<ref name="taflove05" />
# Computer visualization capabilities are increasing rapidly. While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods, which generate time-marched arrays of field quantities suitable for use in color videos to illustrate the field dynamics.<ref name="taflove05" />
# Anisotropy is treated naturally by the FDTD method. Yee cells, having components in each Cartesian direction, can be easily configured with anisotropic characteristics.<ref name="taflove05"/>
Taflove has argued that these factors combine to suggest that FDTD will remain one of the dominant computational electrodynamics techniques (as well as potentially other [[multi-physics|multiphysics]] problems).<ref name="taflove05" />
==See also==
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| jfm = 54.0486.01
| mr = 1512478
|bibcode = 1928MatAn.100...32C |s2cid=120760331 | url-access = subscription
}}</ref> <ref name="obrien1950">
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| doi = 10.1002/sapm1950291223
}}</ref>
<ref name="vonneumann49">
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| archive-date= 2013-01-05
| doi=10.1002/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A
| issue= 5| url-access= subscription}}
</ref>
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| access-date=17 June 2010
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* [http://ab-initio.mit.edu/meep/ Meep] ([[Massachusetts Institute of Technology|MIT]], 2D/3D/cylindrical parallel FDTD)
* [http://freshmeat.net/projects/radarfdtd/ (Geo-) Radar FDTD]
* [
* [
* [https://archive.today/20121217222254/http://cs.tu-berlin.de/~peutetre/sfdtd/ FDTD code in Fortran 90]
* [http://code.google.com/p/emwave2d/ FDTD code in C for 2D EM Wave simulation]
* {{usurped|1=[https://web.archive.org/web/20120911013524/http://angorafdtd.org/ Angora]}} (3D parallel FDTD software package, maintained by Ilker R. Capoglu)
* [http://gsvit.net/ GSvit] (3D FDTD solver with graphics card computing support, written in C, graphical user interface XSvit available)
*[http://www.gprmax.com gprMax] (Open Source (GPLv3), 3D/2D FDTD modelling code in Python/Cython developed for GPR but can be used for general EM modelling.)
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