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 ==

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 Kane Yee's FDTD scheme, presented in his seminal 1966 paper,<ref name="yee66" /> was to apply centered finite difference operators on staggered grids in space and time for each electric and magnetic vector field component in Maxwell's curl equations.

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) -->

{{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.

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| 2009 || Oliveira and Sobrinho applied the FDTD method for simulating lightning strokes in a power substation<ref name="oliveira09" />

|-

| 2012 || Moxley ''et al'' developed a generalized finite-difference time-___domain quantum method for the N-body interacting Hamiltonian.<ref name="Moxley2012" />

|-

| 2013 || Moxley ''et al'' developed a generalized finite-difference time-___domain scheme for solving nonlinear Schrödinger equations.<ref name="Moxley2013" />

|-

| 2014 || Moxley ''et al'' developed an implicit generalized finite-difference time-___domain scheme for solving nonlinear Schrödinger equations.<ref name="Moxley2014" />

|-

| 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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== Popularity ==

{{CopypasteOriginal research|section|date=JanuaryAugust 20212013}}<!-- Contents of the chronologysection, despite being referenced with the original articlessource, the content of the section appears to be largely taken in verbatim from Taflove and Hagness's book. (Chapter 1) -->

{{Original research|section|date=August 2013}}

{{Copypaste|section|date=January 2021}}<!-- 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" />

the dominant computational electrodynamics techniques (as well as potentially other [[multi-physics|multiphysics]] problems).<ref name="taflove05" />

=== Implementations ===

There are hundreds of simulation tools (e.g. Tidy3D, OmniSim, XFdtd, Lumerical, CST Studio Suite, [[Optiwave Systems|OptiFDTD]] etc.) that implement FDTD algorithms, many optimized to run on parallel-processing clusters.

==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="Moxley2012">

{{cite journal

|author1=F. I. Moxley III |author2=T. Byrnes |author3=F. Fujiwara |author4=W. Dai | title = A generalized finite-difference time-___domain quantum method for the N-body interacting Hamiltonian

| journal = Computer Physics Communications

| volume = 183

| issue = 11

| pages = 2434–2440

| year = 2012

| doi=10.1016/j.cpc.2012.06.012

|bibcode = 2012CoPhC.183.2434M }}</ref>

<ref name="Moxley2014">{{cite book

| author=Frederick Moxley

| display-authors=etal

| title=Contemporary Mathematics: Mathematics of Continuous and Discrete Dynamical Systems

| publisher=American Mathematical Society

| year=2014

| isbn=978-0-8218-9862-8

| url=https://www.ams.org/bookstore-getitem?item=CONM-618| author-link=Frederick Moxley

}}

</ref>

<ref name="Moxley2013">

{{cite journal

|author1=F. I. Moxley III |author2=D. T. Chuss |author3=W. Dai | title = A generalized finite-difference time-___domain scheme for solving nonlinear Schrödinger equations

| journal = Computer Physics Communications

| volume = 184

| issue = 8

| pages = 1834–1841

| year = 2013

| doi=10.1016/j.cpc.2013.03.006

|bibcode = 2013CoPhC.184.1834M }}</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

| doi-access=free

| url-access=subscription

}}

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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]

* [httphttps://sourceforge.net/projects/bigboy bigboy] (unmaintained, no release files. must get source from cvs)

* [httphttps://sourceforge.net/projects/pfdtd/files/ Parallel (MPI&OpenMP) FDTD codes in C++] (developed by Zs. Szabó)

* [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.)