On the relation between FDTD and Fibonacci polynomials

In this paper we show that the Finite-Difference Time-Domain method (FDTD method) follows the recurrence relation for Fibonacci polynomials. More precisely, we show that FDTD approximates the electromagnetic field by Fibonacci polynomials in ΔtA, where Δt is the time step and A is the first-order Maxwell system matrix. By exploiting the connection between Fibonacci polynomials and Chebyshev polynomials of the second kind, we easily obtain the Courant-Friedrichs-Lewy (CFL) stability condition and we show that to match the spectral width of the system matrix, the time step should be chosen as large as possible, that is, as close to the CFL upper bound as possible.