Note that the time step size obeys the Courant condition for the numerical solution to be stable. We use the numpy library for the required scientific functions, matplotlib for plotting and tqdm for timing the FDTD loop. Now we need to set the simulation up as follows. 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. We finally have the foundation of the 1D FDTD algorithm. Finally, we enumerate two numerical examples to confirm that the POD reduced-order FDTD extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium. The Finite-Difference Time-Domain (FDTD) method is a rigorous and powerful tool for modeling nano-scale optical devices. Finite-difference time-domain (FDTD) is one of the primary computational electrodynamics modeling techniques available.
And then, we formulate three sets of POD bases by means of the solutions obtained from the classical FDTD scheme on the very short time interval, establish the POD reduced-order FDTD extrapolating scheme, and furnish the error estimates among the POD reduced-order FDTD solutions and the classical FDTD solutions as well as the accuracy solution for the 2D Maxwell equations in a lossy medium. For this purpose, we first review the classical FDTD scheme and associated results for the 2D Maxwell equations in a lossy medium. This paper is concerned with establishing a reduced-order finite difference time-domain (FDTD) extrapolating scheme based on proper orthogonal decomposition (POD) method for the two-dimensional (2D) Maxwell equations in a lossy medium.