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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]]).
=== Development of FDTD and Maxwell's equations===<!-- Contents of the chronology, despite being referenced with the original articles, appears to be largely taken in verbatim from Taflove and Hagness's book. (Chapter 1) -->
An appreciation of the basis, technical development, and possible future of FDTD numerical techniques for Maxwell's equations can be developed by first considering their history. The following lists some of the key publications in this area.
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| 1994 || Thomas ''et al'' introduced a Norton's equivalent circuit for the FDTD space lattice, which permits the SPICE circuit analysis tool to implement accurate subgrid models of nonlinear electronic components or complete circuits embedded within the lattice.<ref name="thomas94" />
|-
| 1994 || Berenger introduced the highly effective, perfectly matched layer (PML) ABC for two-dimensional FDTD grids,<ref name="berenger94" /> which was extended to non-orthogonal meshes by Navarro ''et al''
|-
| 1994 || Chew and Weedon introduced the coordinate stretching PML that is easily extended to three dimensions, other coordinate systems and other physical equations.<ref name="chewweedon94" />
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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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=== Weaknesses of FDTD modeling===
[[File:Numerical dispersion of a pulse signal in 1D FDTD.ogg|thumb|right|290px|[[Numerical dispersion]] of a square pulse signal in a simple one-dimensional FDTD scheme. [[Ringing artifacts]] around the edges of the pulse are heavily accentuated ([[Gibbs phenomenon]]) and the signal distorts as it propagates, even in the absence of a [[Dispersion (optics)|dispersive medium]]. This artifact is a direct result of the discretization scheme.<ref name="taflove05"/>]]
* Since FDTD requires that the entire computational ___domain be gridded, and the grid spatial discretization must be sufficiently fine to resolve both the smallest electromagnetic wavelength and the smallest geometrical feature in the model, very large computational domains can be developed, which results in very long solution times. Models with long, thin features, (like wires) are difficult to model in FDTD because of the excessively large computational ___domain required. Methods such as [[eigenmode expansion]] can offer a more efficient alternative as they do not require a fine grid along the z-direction.<ref name="phot_cad" />
* There is no way to determine unique values for permittivity and permeability at a material interface.
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== Popularity ==
{{Original research|section|date=August 2013}}<!-- Contents of the section, despite being referenced with the original source, the content of the section appears to be largely taken in verbatim from Taflove and Hagness's book. (Chapter 1) -->
<!-- The following text is from Computational Electrodynamics: The
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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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| pages = 32–74
| year = 1928
| language =
| url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN002272636
| doi = 10.1007/BF01448839
| jfm = 54.0486.01
| mr = 1512478
|bibcode = 1928MatAn.100...32C
}}</ref>
<ref name="obrien1950">
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| doi = 10.1002/sapm1950291223
}}</ref>
<ref name="vonneumann49">
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|issue=3 | pages = 232–237
|date=March 1950
| doi=10.1063/1.1699639
|bibcode = 1950JAP....21..232V }}</ref>
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| journal=LEOS Newsletter
| year=2008
}}</ref>
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{{cite journal
|author1=A. Deinega |author2=I. Valuev |title=Long-time behavior of PML absorbing boundaries for layered periodic structures
|journal=
|volume= 182
|issue=1 |pages= 149–151
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|doi=10.1016/j.cpc.2010.06.006
|bibcode = 2011CoPhC.182..149D }}</ref>
<ref name="navarro94">
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|year=1994
|doi=10.1049/el:19941139
|bibcode=1994ElL....30.1654N }}</ref>
<ref name="SorianoNavarro2004">
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|year=2004
|doi= 10.1063/1.1753661
|bibcode=2004JAP....95.8011S |hdl=10550/12837
|hdl-access=free
}}</ref>
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| pages= 3155–3160
| doi=10.1109/TAP.2003.818809|arxiv = physics/0208060 |bibcode = 2003ITAP...51.3155D
| issue= 11 |s2cid=119095479 }}
</ref>
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| doi= 10.1109/10.730440
| pmid= 9835195
| issue= 12|s2cid=6169784 }}
</ref>
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| volume= 17
| pages= 239–244
| url=http://handle.dtic.mil/100.2/ADP013476| archive-url=https://web.archive.org/web/20120527162538/http://handle.dtic.mil/100.2/ADP013476| url-status=dead| archive-date=May 27, 2012}}
</ref>
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|bibcode = 2002IAWPL...1...53S
|url-status = dead
|
|
|citeseerx = 10.1.1.694.4837
|s2cid = 368964
}}
</ref>
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| pages= 2416–2421
| doi= 10.1109/TNS.1977.4329229|bibcode = 1977ITNS...24.2416H
| issue= 6
}}
</ref>
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|issue=10 | pages= 1251–1254
| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper54.pdf
| doi=10.1109/68.329654|bibcode = 1994IPTL....6.1251J |s2cid=46710331 }}
</ref>
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<ref name="katz91">{{cite journal
|author1=D. S. Katz |author2=A. Taflove|author2-link=Allen Taflove |author3=M. J. Piket-May|authorlink3=Melinda Piket-May |author4=K. R. Umashankar | title= FDTD analysis of electromagnetic wave radiation from systems containing horn antennas
| journal= IEEE Transactions on Antennas and Propagation
| year= 1991
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| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper51.pdf
| doi= 10.1109/75.311494
| issue= 8|s2cid=10156811 }}
</ref>
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| pages= 333–341
| doi=10.1109/TEMC.1978.303727
| issue= 2|s2cid=31666283 }}
</ref>
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<ref name="liu97">{{cite book
| author= Q. H. Liu
| title= IEEE Antennas and Propagation Society International Symposium 1997. Digest
| chapter= The pseudospectral time-___domain (PSTD) method: A new algorithm for solutions of Maxwell's equations
| year= 1997
| volume= 1
| pages= 122–125
| doi=10.1109/APS.1997.630102
| isbn= 978-0-7803-4178-4
}}
</ref>
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| pages= 1059–1068
| doi=10.1109/8.55618|bibcode = 1990ITAP...38.1059M
| issue= 7 |s2cid=31583883 }}
</ref>
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| pages= 377–382
| doi= 10.1109/TEMC.1981.303970
| issue= 4
}}
</ref>
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|year=2006
|doi=10.1109/tap.2006.875484
|bibcode = 2006ITAP...54.1818A |s2cid=25120679 }}</ref>
<ref name="ramahi97">{{cite journal
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| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper53.pdf
| doi= 10.1109/75.324711
| issue= 10|s2cid=24572883 }}
</ref>
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| volume= 7
| pages= 599–604
| doi= 10.1002/mop.4650071304
| issue= 13|bibcode=1994MiOTL...7..599C }}
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| pages= 334–339
| url= http://www3.interscience.wiley.com/journal/73504513/abstract
| archive-url= https://archive.today/20130105125334/http://www3.interscience.wiley.com/journal/73504513/abstract
| url-status= dead
| archive-date= 2013-01-05
| doi=10.1002/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A
| issue= 5
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| doi= 10.1109/10.1360
| pmid= 3350546
| issue= 3|s2cid=20350396 }}
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| doi= 10.1109/TEMC.1980.303879
| issue= 3 | bibcode= 1980ITElC..22..191T
| s2cid= 39236486
}}
</ref>
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| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper10.pdf
| doi=10.1109/TEMC.1983.304133
| issue= 4 |s2cid=40419955 }}
</ref>
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<ref name="taflove05">{{cite book
| author=[[Allen Taflove]] and [[Susan Hagness|Susan C. Hagness]]
| title=Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed.
| publisher=Artech House Publishers
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<ref name="thomas94">{{cite journal
|author1=V. A. Thomas |author2=M. E. Jones |author3=M. J. Piket-May |authorlink3=Melinda Piket-May|author4=A. Taflove |author5=E. Harrigan | title= The use of SPICE lumped circuits as sub-grid models for FDTD high-speed electronic circuit design
| journal= IEEE Microwave and Guided Wave Letters
| year= 1994
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|issue=5 | pages= 141–143
| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper49.pdf
| doi=10.1109/75.289516|s2cid=32905331 }}
</ref>
<ref name="tirkas91">{{cite book
|author1=P. A. Tirkas |author2=C. A. Balanis |
| volume= 3
| pages= 1750–1753
| doi=10.1109/APS.1991.175196
| isbn= 978-0-7803-0144-3|s2cid=122038624 }}
</ref>
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| pages= 333–335
| doi=10.1109/75.244870
| issue= 9|s2cid=27549555 }}
</ref>
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| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper9.pdf
| doi= 10.1109/TEMC.1982.304054
| issue= 4 |bibcode=1982ITElC..24..397U |s2cid=37962500 }}
</ref>
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| pages= 302–307
| doi= 10.1109/TAP.1966.1138693
| issue= 3
}}
</ref>
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| journal= Mississippi State University, Interaction Notes
| volume= 44
| year= 1969
| url= http://ece-research.unm.edu/summa/notes/In/0044.pdf }}
</ref>
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| doi=10.1109/22.869007|bibcode = 2000ITMTT..48.1550Z
| issue= 9 }}
</ref>
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| pages= 3596–3600
| doi= 10.1109/TAP.2008.2005544|bibcode = 2008ITAP...56.3596A
| issue= 11 |s2cid=31351974 }}
</ref>
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| pages=3248–3252
| year=2008
| doi = 10.1109/TAP.2008.929447 |bibcode = 2008ITAP...56.3248T |s2cid= 29617214 }}
</ref>
<ref name="oliveira2021">{{cite journal
| title=Least Squares Finite-Difference Time-Domain
| author=R. M. S. de Oliveira | author2=R. R. Paiva
| journal=IEEE Transactions on Antennas and Propagation
| year=2021
| volume=69 | issue=9 | pages=6111–6115 | doi = 10.1109/TAP.2021.3069576 | bibcode=2021ITAP...69.6111D | s2cid=234307029 }}
</ref>
}}
{{Refbegin}}
The following article in ''Nature Milestones: Photons'' illustrates the historical significance of the FDTD method as related to Maxwell's equations:
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|date=May 2010
| doi=10.1038/nmat2639
|
| doi-access=free
| url-access=subscription
}}
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| publisher=available online
| year=2010
| url=http://
{{Refend}}
*
{{commons category}}
[[Free software]]/[[Open-source software]] FDTD projects:
* [http://www.fdtdxx.com FDTD++]: advanced, fully featured FDTD software, along with sophisticated material models and predefined fits as well as discussion/support forums and email support
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* [https://web.archive.org/web/20110517102321/http://www.its.caltech.edu/~seheon/FDTD.html pFDTD] (3D C++ FDTD codes developed by Se-Heon Kim)
* [https://web.archive.org/web/20090626051810/http://www.thecomputationalphysicist.com/ JFDTD] (2D/3D C++ FDTD codes developed for nanophotonics by Jeffrey M. McMahon)
* [http://www.ece.ncsu.edu/oleg/wiki/WOLFSIM WOLFSIM] {{Webarchive|url=https://web.archive.org/web/20080702190617/http://www.ece.ncsu.edu/oleg/wiki/WOLFSIM |date=2008-07-02 }} (NCSU) (2-D)
* [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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