On Some Strategies for Computer Simulation of the Wave Propagation Using Finite Differences I. One–Dimensional FDTD Method

Abstract

Some strategies used in the computer simulation of wave phenomena by means of finite differences in time-domain (FDTD) method are reviewed and discussed here. It is shown that the wave equation in its discretized form possesses different properties in comparison with the true differential formulation. In this part the issues of stability and numerical dispersion are thoroughly investigated for the one-dimensional case represented here by waves on transmission lines and transversal electromagnetic plane wave.

On Some Aspects of the Complex–Envelope Finite–Differences Simulation of Wave Propagation in One–Dimensional Case

“…Let us consider the functions f (z, t) and g(z, t), representing similarly as in [4] the normalised wave amplitudes of the voltage u(z, t) and current i(z, t) on the transmission line, ie f (z, t) = u(z, t), g(z, t) = Z 0 i(z, t), Z 0 = L 0 /C 0 , or the normalised transversal components of the electric and magnetic field vectors E x (z, t), H y (z, t) pertaining to the homogeneous plane wave, ie…”

On Some Strategies for Computer Simulation of Thewave Propagation Using Finite Differences II. Multi–Dimensional FDTD Method

“…In addition the Ecomponents and the H -components are staggered along the time axis too, analogously to Fig. 1a in [1], ie the calculations are performed intermittently in discrete time instants E ⇒ n + For the sake of simplicity, in what follows the normalised quantities h x,y,z = H x,y,z √ Z 0 , and e x,y,z = E x,y,z / √ Z 0 are introduced, where Z 0 = µ/ε is the wave impedance of homogeneous media. The discretised equations (2) and (3) then read…”

“…Due to the sampling theorem k∆| x,z = π holds for the maximum representable values of the wavenumbers k x , k z with the sampling intervals ∆ x , ∆ z , and therefore A x,z | max = c∆ t /∆ x,z , leading to the ultimate power conservation condition (CFL condition (59) in [1] in two dimensions)…”

“…The von Neumann stability analysis as elucitaded in [1] leads in the two dimensional case for eg time-harmonic plane TE-wave to…”

“…As discussed in part one of this tutorial review [1] the computer simulation of electromagnetic wave propagation is a subject of standing interest. The approximation of derivatives by finite differences in Maxwell equations is one of the most transparent and straightforward methods used.…”