Finitedifference timedomain method
Finitedifference timedomain 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 equations). Since it is a timedomain method, FDTD solutions can cover a wide frequency range with a single simulation run, and treat nonlinear material properties in a natural way.
The FDTD method belongs in the general class of gridbased differential numerical modeling methods (finite difference methods). The timedependent Maxwell's equations (in partial differential form) are discretized using centraldifference approximations to the space and time partial derivatives. The resulting finitedifference equations are solved in either software or hardware in a leapfrog manner: the electric field vector components 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 steadystate electromagnetic field behavior is fully evolved.
History
Finite difference schemes for timedependent partial differential equations (PDEs) have been employed for many years in computational fluid dynamics problems,[1] including the idea of using centered finite difference operators on staggered grids in space and time to achieve secondorder accuracy.[1] The novelty of Kane Yee's FDTD scheme, presented in his seminal 1966 paper,[2] 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 "Finitedifference timedomain" and its corresponding "FDTD" acronym were originated by Allen Taflove in 1980.[3] 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 nearDC (ultralowfrequency geophysics involving the entire Earthionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics, solitons, and biophotonics).[4] In 2006, an estimated 2,000 FDTDrelated publications appeared in the science and engineering literature (see Popularity). As of 2013, there are at least 25 commercial/proprietary FDTD software vendors; 13 freesoftware/opensourcesoftware FDTD projects; and 2 freeware/closedsource FDTD projects, some not for commercial use (see External links).
Development of FDTD and Maxwell's equations
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.
Partial chronology of FDTD techniques and applications for Maxwell's equations.[5]  

year  event 
1928  Courant, Friedrichs, and Lewy (CFL) publish seminal paper with the discovery of conditional stability of explicit timedependent finite difference schemes, as well as the classic FD scheme for solving secondorder wave equation in 1D and 2D.[6] 
1950  First appearance of von Neumann's method of stability analysis for implicit/explicit timedependent finite difference methods.[7] 
1966  Yee described the FDTD numerical technique for solving Maxwell's curl equations on grids staggered in space and time.[2] 
1969  Lam reported the correct numerical CFL stability condition for Yee's algorithm by employing von Neumann stability analysis.[8] 
1975  Taflove and Brodwin reported the first sinusoidal steadystate FDTD solutions of two and threedimensional electromagnetic wave interactions with material structures;[9] and the first bioelectromagnetics models.[10] 
1977  Holland and Kunz & Lee applied Yee's algorithm to EMP problems.[11][12] 
1980  Taflove coined the FDTD acronym and published the first validated FDTD models of sinusoidal steadystate electromagnetic wave penetration into a threedimensional metal cavity.[3] 
1981  Mur published the first numerically stable, secondorder accurate, absorbing boundary condition (ABC) for Yee's grid.[13] 
1982–83  Taflove and Umashankar developed the first FDTD electromagnetic wave scattering models computing sinusoidal steadystate nearfields, farfields, and radar crosssection for two and threedimensional structures.[14][15] 
1984  Liao et al reported an improved ABC based upon spacetime extrapolation of the field adjacent to the outer grid boundary.[16] 
1985  Gwarek introduced the lumped equivalent circuit formulation of FDTD.[17] 
1986  Choi and Hoefer published the first FDTD simulation of waveguide structures.[18] 
1987–88  Kriegsmann et al and Moore et al published the first articles on ABC theory in IEEE Transactions on Antennas and Propagation.[19][20] 
1987–88, 1992  Contourpath subcell techniques were introduced by Umashankar et al to permit FDTD modeling of thin wires and wire bundles,[21] by Taflove et al to model penetration through cracks in conducting screens,[22] and by Jurgens et al to conformally model the surface of a smoothly curved scatterer.[23] 
1988  Sullivan et al published the first 3D FDTD model of sinusoidal steadystate electromagnetic wave absorption by a complete human body.[24] 
1988  FDTD modeling of microstrips was introduced by Zhang et al.[25] 
1990–91  FDTD modeling of frequencydependent dielectric permittivity was introduced by Kashiwa and Fukai,[26] Luebbers et al,[27] and Joseph et al.[28] 
1990–91  FDTD modeling of antennas was introduced by Maloney et al,[29] Katz et al,[30] and Tirkas and Balanis.[31] 
1990  FDTD modeling of picosecond optoelectronic switches was introduced by Sano and Shibata,[32] and ElGhazaly et al.[33] 
1992–94  FDTD modeling of the propagation of optical pulses in nonlinear dispersive media was introduced, including the first temporal solitons in one dimension by Goorjian and Taflove;[34] beam selffocusing by Ziolkowski and Judkins;[35] the first temporal solitons in two dimensions by Joseph et al;[36] and the first spatial solitons in two dimensions by Joseph and Taflove.[37] 
1992  FDTD modeling of lumped electronic circuit elements was introduced by Sui et al.[38] 
1993  Toland et al published the first FDTD models of gain devices (tunnel diodes and Gunn diodes) exciting cavities and antennas.[39] 
1993  Aoyagi et al present a hybrid Yee algorithm/scalarwave equation and demonstrate equivalence of Yee scheme to finite difference scheme for electromagnetic wave equation.[40] 
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.[41] 
1994  Berenger introduced the highly effective, perfectly matched layer (PML) ABC for twodimensional FDTD grids,[42] which was extended to three dimensions by Katz et al,[43] and to dispersive waveguide terminations by Reuter et al.[44] 
1994  Chew and Weedon introduced the coordinate stretching PML that is easily extended to three dimensions, other coordinate systems and other physical equations.[45] 
1995–96  Sacks et al and Gedney introduced a physically realizable, uniaxial perfectly matched layer (UPML) ABC.[46][47] 
1997  Liu introduced the pseudospectral timedomain (PSTD) method, which permits extremely coarse spatial sampling of the electromagnetic field at the Nyquist limit.[48] 
1997  Ramahi introduced the complementary operators method (COM) to implement highly effective analytical ABCs.[49] 
1998  Maloney and Kesler introduced several novel means to analyze periodic structures in the FDTD space lattice.[50] 
1998  Nagra and York introduced a hybrid FDTDquantum mechanics model of electromagnetic wave interactions with materials having electrons transitioning between multiple energy levels.[51] 
1998  Hagness et al introduced FDTD modeling of the detection of breast cancer using ultrawideband radar techniques.[52] 
1999  Schneider and Wagner introduced a comprehensive analysis of FDTD grid dispersion based upon complex wavenumbers.[53] 
2000–01  Zheng, Chen, and Zhang introduced the first threedimensional alternatingdirection implicit (ADI) FDTD algorithm with provable unconditional numerical stability.[54][55] 
2000  Roden and Gedney introduced the advanced convolutional PML (CPML) ABC.[56] 
2000  Rylander and Bondeson introduced a provably stable FDTD  finiteelement timedomain hybrid technique.[57] 
2002  Hayakawa et al and Simpson and Taflove independently introduced FDTD modeling of the global Earthionosphere waveguide for extremely lowfrequency geophysical phenomena.[58][59] 
2003  DeRaedt introduced the unconditionally stable, “onestep” FDTD technique.[60] 
2008  Ahmed, Chua, Li and Chen introduced the threedimensional locally onedimensional (LOD)FDTD method and proved unconditional numerical stability.[61] 
2008  Taniguchi, Baba, Nagaoka and Ametani introduced a Thin Wire Representation for FDTD Computations for conductive media[62] 
2009  Oliveira and Sobrinho applied the FDTD method for simulating lightning strokes in a power substation[63] 
2010  Chaudhury and Boeuf demonstrated the numerical procedure to couple FDTD and Plasma fluid model for studying Microwave Plasma Interaction.[64] 
2012  Moxley et al developed a generalized finitedifference timedomain quantum method for the Nbody interacting Hamiltonian.[65] 
2013  Moxley et al developed a generalized finitedifference timedomain scheme for solving nonlinear Schrödinger equations.[66] 
2014  Moxley et al developed an implicit generalized finitedifference timedomain scheme for solving nonlinear Schrödinger equations.[67] 
FDTD models and methods
When Maxwell's differential equations are examined, it can be seen that the change in the Efield in time (the time derivative) is dependent on the change in the Hfield across space (the curl). This results in the basic FDTD timestepping relation that, at any point in space, the updated value of the Efield in time is dependent on the stored value of the Efield and the numerical curl of the local distribution of the Hfield in space.[2]
The Hfield is timestepped in a similar manner. At any point in space, the updated value of the Hfield in time is dependent on the stored value of the Hfield and the numerical curl of the local distribution of the Efield in space. Iterating the Efield and Hfield updates results in a marchingintime process wherein sampleddata analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory.
This description holds true for 1D, 2D, and 3D FDTD techniques. When multiple dimensions are considered, calculating the numerical curl can become complicated. Kane Yee's seminal 1966 paper proposed spatially staggering the vector components of the Efield and Hfield about rectangular unit cells of a Cartesian computational grid so that each Efield vector component is located midway between a pair of Hfield vector components, and conversely.[2] This scheme, now known as a Yee lattice, has proven to be very robust, and remains at the core of many current FDTD software constructs.
Furthermore, Yee proposed a leapfrog scheme for marching in time wherein the Efield and Hfield updates are staggered so that Efield updates are conducted midway during each timestep between successive Hfield updates, and conversely.[2] On the plus side, this explicit timestepping scheme avoids the need to solve simultaneous equations, and furthermore yields dissipationfree numerical wave propagation. On the minus side, this scheme mandates an upper bound on the timestep to ensure numerical stability.[9] As a result, certain classes of simulations can require many thousands of timesteps for completion.
Using the FDTD method
To implement an FDTD solution of Maxwell's equations, a computational domain must first be established. The computational domain is simply the physical region over which the simulation will be performed. The E and H fields are determined at every point in space within that computational domain. The material of each cell within the computational domain must be specified. Typically, the material is either freespace (air), metal, or dielectric. Any material can be used as long as the permeability, permittivity, and conductivity are specified.
The permittivity of dispersive materials in tabular form cannot be directly substituted into the FDTD scheme. Instead, it can be approximated using multiple Debye, Drude, Lorentz or critical point terms. This approximation can be obtained using open fitting programs[68] and does not necessarily have physical meaning.
Once the computational domain and the grid materials are established, a source is specified. The source can be current on a wire, applied electric field or impinging plane wave. In the last case FDTD can be used to simulate light scattering from arbitrary shaped objects, planar periodic structures at various incident angles,[69][70] and photonic band structure of infinite periodic structures.[71][72]
Since the E and H fields are determined directly, the output of the simulation is usually the E or H field at a point or a series of points within the computational domain. The simulation evolves the E and H fields forward in time.
Processing may be done on the E and H fields returned by the simulation. Data processing may also occur while the simulation is ongoing.
While the FDTD technique computes electromagnetic fields within a compact spatial region, scattered and/or radiated far fields can be obtained via neartofarfield transformations.[14]
Strengths of FDTD modeling
Every modeling technique has strengths and weaknesses, and the FDTD method is no different.
 FDTD is a versatile modeling technique used to solve Maxwell's equations. It is intuitive, so users can easily understand how to use it and know what to expect from a given model.
 FDTD is a timedomain technique, and when a broadband pulse (such as a Gaussian pulse) is used as the source, then the response of the system over a wide range of frequencies can be obtained with a single simulation. This is useful in applications where resonant frequencies are not exactly known, or anytime that a broadband result is desired.
 Since FDTD calculates the E and H fields everywhere in the computational domain as they evolve in time, it lends itself to providing animated displays of the electromagnetic field movement through the model. This type of display is useful in understanding what is going on in the model, and to help ensure that the model is working correctly.
 The FDTD technique allows the user to specify the material at all points within the computational domain. A wide variety of linear and nonlinear dielectric and magnetic materials can be naturally and easily modeled.
 FDTD allows the effects of apertures to be determined directly. Shielding effects can be found, and the fields both inside and outside a structure can be found directly or indirectly.
 FDTD uses the E and H fields directly. Since most EMI/EMC modeling applications are interested in the E and H fields, it is convenient that no conversions must be made after the simulation has run to get these values.
Weaknesses of FDTD modeling
 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 zdirection.[73]
 There is no way to determine unique values for permittivity and permeability at a material interface.
 Space and time steps must satisfy the CFL condition, or the leapfrog integration used to solve the partial differential equation is likely to become unstable.
 FDTD finds the E/H fields directly everywhere in the computational domain. If the field values at some distance are desired, it is likely that this distance will force the computational domain to be excessively large. Farfield extensions are available for FDTD, but require some amount of postprocessing.[4]
 Since FDTD simulations calculate the E and H fields at all points within the computational domain, the computational domain must be finite to permit its residence in the computer memory. In many cases this is achieved by inserting artificial boundaries into the simulation space. Care must be taken to minimize errors introduced by such boundaries. There are a number of available highly effective absorbing boundary conditions (ABCs) to simulate an infinite unbounded computational domain.[4] Most modern FDTD implementations instead use a special absorbing "material", called a perfectly matched layer (PML) to implement absorbing boundaries.[42][46]
 Because FDTD is solved by propagating the fields forward in the time domain, the electromagnetic time response of the medium must be modeled explicitly. For an arbitrary response, this involves a computationally expensive time convolution, although in most cases the time response of the medium (or Dispersion (optics)) can be adequately and simply modeled using either the recursive convolution (RC) technique, the auxiliary differential equation (ADE) technique, or the Ztransform technique. An alternative way of solving Maxwell's equations that can treat arbitrary dispersion easily is the Pseudospectral SpatialDomain method (PSSD), which instead propagates the fields forward in space.
Grid truncation techniques
The most commonly used grid truncation techniques for openregion FDTD modeling problems are the Mur absorbing boundary condition (ABC),[13] the Liao ABC,[16] and various perfectly matched layer (PML) formulations.[4][42][46] The Mur and Liao techniques are simpler than PML. However, PML (which is technically an absorbing region rather than a boundary condition per se) can provide ordersofmagnitude lower reflections. The PML concept was introduced by J.P. Berenger in a seminal 1994 paper in the Journal of Computational Physics.[42] Since 1994, Berenger's original splitfield implementation has been modified and extended to the uniaxial PML (UPML), the convolutional PML (CPML), and the higherorder PML. The latter two PML formulations have increased ability to absorb evanescent waves, and therefore can in principle be placed closer to a simulated scattering or radiating structure than Berenger's original formulation.
To reduce undesired numerical reflection from the PML additional back absorbing layers technique can be used.[74]
Popularity
Notwithstanding both the general increase in academic publication
throughput during the same period and the overall expansion of interest
in all Computational electromagnetics (CEM) techniques, there are
seven primary reasons for the tremendous expansion of interest in FDTD
computational solution approaches for Maxwell's equations:
 FDTD does not require a matrix inversion. Being a fully explicit computation, FDTD avoids the difficulties with matrix inversions that limit the size of frequencydomain integralequation and finiteelement electromagnetics models to generally fewer than 10^{9} electromagnetic field unknowns.[4] FDTD models with as many as 10^{9} field unknowns have been run; there is no intrinsic upper bound to this number.[4]
 FDTD is accurate and robust. The sources of error in FDTD calculations are well understood, and can be bounded to permit accurate models for a very large variety of electromagnetic wave interaction problems.[4]
 FDTD treats impulsive behavior naturally. Being a timedomain technique, FDTD directly calculates the impulse response of an electromagnetic system. Therefore, a single FDTD simulation can provide either ultrawideband temporal waveforms or the sinusoidal steadystate response at any frequency within the excitation spectrum.[4]
 FDTD treats nonlinear behavior naturally. Being a timedomain technique, FDTD directly calculates the nonlinear response of an electromagnetic system. This allows natural hybriding of FDTD with sets of auxiliary differential equations that describe nonlinearities from either the classical or semiclassical standpoint.[4] One research frontier is the development of hybrid algorithms which join FDTD classical electrodynamics models with phenomena arising from quantum electrodynamics, especially vacuum fluctuations, such as the Casimir effect.[4][75]
 FDTD is a systematic approach. With FDTD, specifying a new structure to be modeled is reduced to a problem of mesh generation rather than the potentially complex reformulation of an integral equation. For example, FDTD requires no calculation of structuredependent Green functions.[4]
 Parallelprocessing computer architectures have come to dominate supercomputing. FDTD scales with high efficiency on parallelprocessing CPUbased computers, and extremely well on recently developed GPUbased accelerator technology.[4]
 Computer visualization capabilities are increasing rapidly. While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods, which generate timemarched arrays of field quantities suitable for use in color videos to illustrate the field dynamics.[4]
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 multiphysics problems).[4]
Implementations
There are hundreds of simulation tools (e.g. XFdtd, Lumerical, CST studio suite, OptiFDTD etc.) that implement FDTD algorithms, many optimized to run on parallelprocessing clusters.
Frederick Moxley suggests further applications with computational quantum mechanics and simulations.[76]
See also
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Hartmut Ruhl; Nils Moscḧuring; Nina Elkina (2012). "Computational Physics Course 17104 Lecture 9" (PDF). Cite journal requires
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Further reading
The following article in Nature Milestones: Photons illustrates the historical significance of the FDTD method as related to Maxwell's equations:
 David Pile (May 2010). "Milestone 2 (1861) Maxwell's equations". Nature Milestones: Photons. doi:10.1038/nmat2639. Retrieved 17 June 2010.
Allen Taflove's interview, "Numerical Solution," in the January 2015 focus issue of Nature Photonics honoring the 150th anniversary of the publication of Maxwell's equations. This interview touches on how the development of FDTD ties into the century and onehalf history of Maxwell's theory of electrodynamics:
The following universitylevel textbooks provide a good general introduction to the FDTD method:
 Karl S. Kunz; Raymond J. Luebbers (1993). The Finite Difference Time Domain Method for Electromagnetics. CRC Press. ISBN 9780849386572. Archived from the original on 20071210. Retrieved 20060805.
 Allen Taflove; Susan C. Hagness (2005). Computational Electrodynamics: The FiniteDifference TimeDomain Method, 3rd ed. Artech House Publishers. ISBN 9781580538329.
 Wenhua Yu; Raj Mittra; Tao Su; Yongjun Liu; Xiaoling Yang (2006). Parallel FiniteDifference TimeDomain Method. Artech House Publishers. ISBN 9781596930858.
 John B. Schneider (2010). Understanding the FDTD Method. available online.
External links
Free software/Opensource software FDTD projects:
 FDTD++: advanced, fully featured FDTD software, with included C++ source code, along with sophisticated material models and predefined fits as well as discussion/support forums and email support
 openEMS (Fully 3D Cartesian & Cylindrical graded mesh ECFDTD Solver, written in C++, using a Matlab/OctaveInterface)
 pFDTD (3D C++ FDTD codes developed by SeHeon Kim)
 JFDTD (2D/3D C++ FDTD codes developed for nanophotonics by Jeffrey M. McMahon)
 WOLFSIM (NCSU) (2D)
 Meep (MIT, 2D/3D/cylindrical parallel FDTD)
 (Geo) Radar FDTD
 bigboy (unmaintained, no release files. must get source from cvs)
 toyFDTD
 Parallel (MPI&OpenMP) FDTD codes in C++ (developed by Zs. Szabó)
 FDTD code in Fortran 90
 FDTD code in C for 2D EM Wave simulation
 Angora (3D parallel FDTD software package, maintained by Ilker R. Capoglu)
 GSvit (3D FDTD solver with graphics card computing support, written in C, graphical user interface XSvit available)
 gprMax (Open Source (GPLv3), 3D/2D FDTD modelling code in Python/Cython developed for GPR but can be used for general EM modelling.)
Freeware/Closed source FDTD projects (some not for commercial use):
 EMTL (Electromagnetic Template Library) (Free С++ library for electromagnetic simulations. The current version implements mainly the FDTD).