Nonreflecting Boundary Conditions for Time-Domain
             Acoustic and Electromagnetic Wave Propagation

                         Bradley K. Alpert


The exact nonreflecting boundary conditions for the wave equation and
Maxwell's equations---well-known to be nonlocal in both space and
time---can be expressed as a convolution of the solution at the
boundary from the time of quiescence to the present.  These boundary
conditions, derived by separation of variables in Cartesian,
cylindrical, and spherical coordinate systems, appear to require
extensive history of the solution on the boundary.  We show, however,
that by appropriate representation of the convolution kernels, this
history can be reduced to order $O(n\log n)$, where $n$ is the number
of points in the discretization of the boundary, and is independent of
the integration period.  This observation leads to effective
implementations of nonreflecting boundary conditions, examples of
which are presented.

This is joint work with Leslie Greengard and Thomas Hagstrom.