temporal waveforms or the sinusoidal steady-state response at any frequency within the excitation spectrum. FDTD treats nonlinear behavior naturally. Being a time-domain 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 semi-classical standpoint. 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. 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 structure-dependent Green functions. 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. 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. 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). Implementations There are hundreds of simulation tools (e.g. OmniSim, XFdtd, Lumerical, CST Studio Suite, OptiFDTD etc.) that implement FDTD algorithms, many optimized to run on parallel-processing clusters. See also Computational electromagnetics Eigenmode expansion Beam propagation method Finite-difference frequency-domain Finite element method Scattering-matrix method Discrete dipole approximation References Further reading 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. This interview touches on how the development of FDTD ties into the century and one-half history of Maxwell's theory of electrodynamics: Nature Photonics interview The following university-level textbooks provide a good general introduction to the FDTD method: External links Free software/Open-source software FDTD projects: FDTD++: advanced, fully featured FDTD software, along with sophisticated material models and predefined fits as well as discussion/support forums and email support openEMS (Fully 3D Cartesian & Cylindrical graded mesh EC-FDTD Solver, written in C++, using a Matlab/Octave-Interface) pFDTD (3D C++ FDTD codes developed