Finite-difference time-domain method

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 equations).

Since it is a time-domain method, FDTD solutions can cover a wide frequency range with a single simulation run, and treat nonlinear material properties in a natural way.

Finite difference schemes for time-dependent 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 second-order 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.

[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 near-DC (ultralow-frequency geophysics involving the entire Earth-ionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics, solitons, and biophotonics).

Iterating the E-field and H-field updates results in a marching-in-time process wherein sampled-data analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory.

[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.

[2] On the plus side, this explicit time-stepping scheme avoids the need to solve simultaneous equations, and furthermore yields dissipation-free numerical wave propagation.

However, PML (which is technically an absorbing region rather than a boundary condition per se) can provide orders-of-magnitude lower reflections.

[4] The following article in Nature Milestones: Photons illustrates the historical significance of the FDTD method as related to Maxwell's equations: 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.

In finite-difference time-domain method, "Yee lattice" is used to discretize Maxwell's equations in space. This scheme involves the placement of electric and magnetic fields on a staggered grid.
Illustration of a standard Cartesian Yee cell used for FDTD, about which electric and magnetic field vector components are distributed. [ 2 ] Visualized as a cubic voxel , the electric field components form the edges of the cube, and the magnetic field components form the normals to the faces of the cube. A three-dimensional space lattice consists of a multiplicity of such Yee cells. An electromagnetic wave interaction structure is mapped into the space lattice by assigning appropriate values of permittivity to each electric field component, and permeability to each magnetic field component.
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 dispersive medium . This artifact is a direct result of the discretization scheme. [ 4 ]