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
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 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).