Finite-difference time-___domain method: Difference between revisions

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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]] (finding approximate solutions to the associated system of [[differential equation]]s). Since it is a [[time ___domain|time-___domain]] method, FDTD solutions can cover a wide [[frequency]] range with a single [[computer simulation|simulation]] run, and treat nonlinear material properties in a natural way.

The FDTD method belongs in the general class of [[Discretization|grid]]-based differential numerical modeling methods ([[finite difference methods]]). The time-dependent [[Maxwell's equations]] (in [[Partial differential equation|partial differential]] form) are discretized using [[central difference|central-difference]] approximations to the space and time [[partial derivative]]s. The resulting [[finite difference method|finite-difference]] equations are solved in either software or hardware in a [[leapfrog integration|leapfrog]] manner: the [[electric field]] [[vector component]]s in a volume of space are solved at a given instant in time; then the [[magnetic field]] vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved.

== History ==

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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) -->

{{Copypaste|section|date=January 2021}}<!-- 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.