Mathematical models for dispersive electromagnetic waves: an overview <s> Numerical simulations <s> Part 1 Reinventing electromagnetics: background history of space-grid time-domain techniques for Maxwell's equations scaling to very large problem sizes defense applications dual-use electromagnetics technology. Part 2 The one-dimensional scalar wave equation: propagating wave solutions finite-difference approximation of the scalar wave equation dispersion relations for the one-dimensional wave equation numerical group velocity numerical stability. Part 3 Introduction to Maxwell's equations and the Yee algorithm: Maxwell's equations in three dimensions reduction to two dimensions equivalence to the wave equation in one dimension. Part 4 Numerical stability: TM mode time eigenvalue problem space eigenvalue problem extension to the full three-dimensional Yee algorithm. Part 5 Numerical dispersion: comparison with the ideal dispersion case reduction to the ideal dispersion case for special grid conditions dispersion-optimized basic Yee algorithm dispersion-optimized Yee algorithm with fourth-order accurate spatial differences. Part 6 Incident wave source conditions for free space and waveguides: requirements for the plane wave source condition the hard source total-field/scattered field formulation pure scattered field formulation choice of incident plane wave formulation. Part 7 Absorbing boundary conditions for free space and waveguides: Bayliss-Turkel scattered-wave annihilating operators Engquist-Majda one-way wave equations Higdon operator Liao extrapolation Mei-Fang superabsorption Berenger perfectly-matched layer (PML) absorbing boundary conditions for waveguides. Part 8 Near-to-far field transformation: obtaining phasor quantities via discrete fourier transformation surface equivalence theorem extension to three dimensions phasor domain. Part 9 Dispersive, nonlinear, and gain materials: linear isotropic case recursive convolution method linear gyrontropic case linear isotropic case auxiliary differential equation method, Lorentz gain media. Part 10 Local subcell models of the fine geometrical features: basis of contour-path FD-TD modelling the simplest contour-path subcell models the thin wire conformal modelling of curved surfaces the thin material sheet relativistic motion of PEC boundaries. Part 11 Explicit time-domain solution of Maxwell's equations using non-orthogonal and unstructured grids, Stephen Gedney and Faiza Lansing: nonuniform, orthogonal grids globally orthogonal global curvilinear co-ordinates irregular non-orthogonal unstructured grids analysis of printed circuit devices using the planar generalized Yee algorithm. Part 12 The body of revolution FD-TD algorithm, Thomas Jurgens and Gregory Saewert: field expansion difference equations for on-axis cells numerical stability PML absorbing boundary condition. Part 13 Modelling of electromagnetic fields in high-speed electronic circuits, Piket-May and Taflove. (part contents). <s> BIB001 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> Numerical simulations <s> We develop an effective medium description of a two-dimensional photonic band-gap medium composed of dielectric cylinders of large dielectric constant. Using the transfer matrix method we have calculated reflection coefficients for a slab of the composite and plane-wave incidence, as well as the (complex) wavevector for the infinite system. From these quantities we derive an effective permittivity and permeability for the composite. In the case of p-polarized incidence the composite displays a negative magnetic permeability at microwave frequencies due to single-scatterer resonances in the medium. <s> BIB002
We have performed 2D simulations in the (x, y)-plane, considering the 2D Maxwell system for the electromagnetic field (E x , E y , H z ). We use the scaling ε 0 = µ 0 = 1 and consider Ω e = Ω m = 1 and two sets of values for the absorption parameters: α e = α m = 0.1 and α e = α m = 1. The domain of computation is the square [−0.5, 0.5] × [−0.5, 0.5]. On the boundary of the domain, a perfectly conducting boundary condition E × n = 0 is used. The system is initialized with E x (x, 0) = E y (x, 0) = e −300(x 2 +y 2 ) and 2 ) , while for the rest of the unknowns zero initial conditions are used. We compared results obtained separately with the two systems (107) ('exact solution') and (82), with measures (110) ('approximate solution'). The equations were discretized as follows: • For the solution of (107) we use the Yee scheme (see for instance BIB001 ), where dispersive terms are discretized with the help of the trapezoid rule. We take the mesh size ∆x = ∆y and ∆t = ∆x 2 . • The discretization of (82) requires an additional step consisting in the approximation of the ξ-integrals. In order to do so, we first perform the change of variables ξ = α tan τ in (111) and discretize the above integral, using the Gauss-Legendre quadrature rule [47, pp. 177-178] on the interval 0, π 2 , with quadrature weights {w , 1 ≤ ≤ N q } and quadrature nodes {τ , 1 ≤ ≤ N q }: The model (82) is thus approximated by a local Lorentz system (53) with N e = N m = N q and ω e, = α e tan τ , ω m, = α m tan τ , Ω We first tested the approximation of (107) with the discrete augmented system. In figures 11 and 12, we represent the evolution of H z at the origin as a function of time in the time window BIB002  , for different values of N q . In both cases we observe the convergence of the approximate solution (red curves) towards the exact solution (blue curve) when N q increases. In the case of small absorption, figure 11 , the convergence is attained quite quickly, but for large absorption, figure 12 , fairly accurate solution is obtained only with 40 quadrature points (which is not surprising since the augmented models are non-dissipative). In both cases, the approximation of the exact dissipative model with the discrete non-dissipative model does not provide an efficient numerical method. However, our exact dissipative model is already itself local, which is not always the case. In figure 13 , we plot the variations of the energy E loc (t) (108) as a function of time for the two cases. The curves confirm the theoretical decay of this energy, which, as expected, is much stronger decay in the case α e = α m = 1. In figure 14 , we study the variations of the energy E(t) for the exact solution (red curve), compared with the variations of the energy E(t) for the approximate solution for different values of N q (blue curve). We see that, contrary to E loc (t), E(t) is not a decreasing function of time, even though it tends to 0 when t → +∞, as it will be proven in the next section.