Wideband matching of FDTD-PIC using a multi-phase velocity operator

The basis for a multi-phase velocity non-reflecting boundary condition is found in the use of the Higdon operator. A high order implementation of the method provides a boundary method equally suited to the injection of multiple signals as well as providing a high quality absorbing boundary across a broad frequency spectrum. The result is a near perfect absorption of scattered outgoing waves originating from the simulation interior. The method has been successfully applied in many wave equation environments for a variety of media from acoustic (shallow water equation) to electromagnetic waves in dispersive media. The method is easily motivated and formulated in Cartesian coordinates. The Higdon operator of order J describing the propagation of the transverse electric, ET, along the z-axis is written as follows: [Π
_j=1
^J
(∂
_t
+v
_j
∂
_z
)]E
_T
=0. The parameters represent phase velocities of the wave. This reduces to the familiar 1 dimensional wave equation for J=1, in which the sign of the phase velocity indicates either a forward or backward traveling wave. Use of J=2 leads to the 2nd order phase velocity model previously reported by the author. Indeed, we have investigated the case for J=3 as well, but found the standard implementation when converted to a finite difference operator form extremely cumbersome. Increasing the order of J requires the method capture information more remote spatially and temporally from the boundary. This increases the potential effect of geometry variation as the information capture is projected deeper into the simulation. Fortunately, there is an alternative approach suggested by Givoli and Neta. They suggest recasting the Higdon operator in terms of auxiliary functions of arbitrarily high order. Converted to a finite difference representation, each order remains a first order difference method but couples adjacent orders of the auxiliary functions.