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We extend the nonstandard (NS) finite difference time domain (FDTD) methodology, originally developed to solve Maxwell’s equations in linear materials, to nonlinear ones. We validate it by computing harmonics generation in a nonlinear dielectric and comparing with theory. The methodology also applies to the quantum electrodynamics that describes the interaction of charged particles with electromagnetic fields, and also to the Ginzburg-Landau model of superconductivity.
James B. Cole (Sun,) studied this question.