This purpose of this page is to collect and disseminate useful information
related to the accurate and efficient near-field truncation of the
computational domain for the simulation of wave propagation problems
in the time domain.
A multitude of methods have been proposed for this task; see for example
the relatively recent review articles:

Or the recent lecture:

Current contents are limited to two techniques which can be used for problems such as the scalar wave equation, Maxwell's equations and the acoustic system in uniform, isotropic media. The first involves compressions of the radiation boundary kernel arising in nonlocal formulations of exact conditions on special boundaries. The second involves optimal local boundary condition sequences applicable on polygonal boundaries.

Exact formulas for radiation boundary conditions on special boundaries -
planes, cylinders, and spheres - can be obtained by separation of variables.
For the standard models these formulas all involve a nonlocal operator
of the form F^{-1}KF where F is a spatial harmonic transform (Fourier on
the plane, spherical harmonics on the sphere) and K is a temporal convolution.
The direct evaluation of K is inefficient as it involves the entire time
history of the solution on the boundary. However, a fast, low-memory
approximate evaluation is achieved by replacing the convolution kernel by a
sum of exponentials and noting that convolution with an exponential kernel is
equivalent to solving an ode and thus requires no global memory. It can be
proven that the kernels arising from the scalar wave equation and related
models can be approximated to an accuracy `ε`
using O(log 1/`ε``·`
log cT/`λ`) exponentials where T is the simulation time
and `λ` is the
wavelength. See:

To use these compressed approximations one only needs to know the amplitudes
and exponents in the sum-of-exponential approximations. You can access these
here for `ε``=`1E-6:

There are two drawbacks to the nonlocal conditions. The first is the
need to use spatial harmonic transforms, which is some expense in 3+1
dimensions and involves some effort to couple with the interior scheme. The
second, and most important, is the restriction on the shape of the artificial
boundary. Particularly for high-aspect-ratio scatterers it would be more
efficient to bound the computational domain by a box rather than a sphere.
This can be done using local methods such as local radiation boundary
condition sequences or perfectly matched absorbing layers. However, these
local methods can suffer severe accuracy degradation over time. To correct
this defect we have introduced a new parametrization of local boundary
condition sequences which we call Complete Radiation Boundary Conditions
(CRBCs). These involve involve inhomogeneous rational approximants to the
transform of the exact planar kernel which interpolate it in the right
half-plane. Boundary conditions are built out of modified ``Higdon-type''
operators: a_{j}d/dt+(1-a_{j}^{2})/(a_{j} T)
`±`cd/dn. The cosines a_{j} are
determined by specifying the boundary condition order and the dimensionless
parameter `η``=``δ`/cT
where T is the simulation time, c is the wavespeed,
and `δ` is the minimal separation
between the artificial boundary and any
sources, scatterers, or other inhomogeneities. For details see the
preprint:

A table of parameters a_{j} for boundary condition orders
P`=`4,8,...
(tolerances > 1E-8) and
`η``=`1E-2,...,1E-6 along with
maximum values of the reflection
coefficient may be accessed below. Note that the number of cosines
required by a condition of order P is 2P+2.

These cosines are computed by the following MATLAB function which implements
the Remez algorithm. Its inputs are `η` and P
and the outputs are the
cosines (a_{j}, j=1, ... , 2P+2)
and the maximum of the complex reflection
coefficient. Note that the function may fail due to conditioning issues if
P is chosen too large.

We note that direct applications to second-order formulations have so far
used different parametrizations based on a combination of Gauss-Lobatto
and Yarvin-Rokhlin quadrature nodes. For details see:

I greatfully acknowledge the contribution of many collaborators in this effort
including Brad Alpert, Dan Givoli, Leslie Greengard and Tim Warburton. The work
is currently suppoorted by the National Science Foundation via grant
DMS-06010067 and has also been supported in part by ARO Grant DAAD19-03-1-0146,
AFOSR Contract FA9550-05-1-0473, and the Israel-US Binational Science
Foundation. Any conlusions or recommendations expressed here are my own and do
not necessarily reflect the views of NSF, ARO, AFOSR, BSF, or my collaborators.
If you have any questions, comments, or suggestions please contact me at:

thagstrom at smu dot edu

Last updated: January 5, 2009.