The system to discretize

is replaced by the extended system

where , , , and are four
free additional parameters which will be used in the discretized
approximation, either to model a physical effect (some dispersion with
for example), either to adjust some numerical corrections or to
model a specific boundary condition. The variables *E* and are
defined by where
and are the components propagating respectively forward and backward
along the axis, with identical definitions for *B*.

The discretized system takes the form

with

In practice, a numerical correction of the form . must be added to the first equation of the set (2)in order to satisfy the static limit in vacuum.

For , the system (2) reduces to (1) because, in vacuum, we have and :

- for 0, the discretized system reduces to the standard FDTD approximation in vacuum (or in a dispersive medium by tuning to a non-zero value);
- for , the wave equation in vacuum is modeled with some additional numerical control at the discretized level. A cut-off on the high frequencies is obtained and can be tuned by the value of (cf. Fig. 1), giving the possibility to reduce the numerical noise associated with the high frequencies as well as the numerical Cerenkov effect, which is caused by the high frequencies.

For at a boundary, the system reduces to the usual one-dimensional Sommerfeld outgoing-wave boundary condition and is extended to the second order approximation of the Engquist and Majda outgoing-wave boundary condition in higher dimension.

**Figure 1:** (when nonequal to zero) has an effect when using the
discretized form of the equations. On the left, it is shown that the short
wavelength waves ( is the mesh size) are damped when .
A benefit of this damping is a tunable reduction of numerical noise, has shown
on the right where the response of the system to a heavyside signal is
displayed for (top) and for (bottom).

Tue Jan 13 15:57:00 PST 1998