One approach in the construction of absorbing boundary conditions is based on suitable series or Fourier
expansions of the solution, and derives a hierarchy of local boundary conditions with increasing order of
accuracy [153, 46
, 240]. Typically, such higher-order local boundary conditions involve solving differential
equations at the boundary surface, where the order of the differential equation is increasing with the order
of the accuracy. This problem can be dealt with by introducing auxiliary variables at the boundary
surface [207, 208].
The starting point for a slightly different approach is an exact nonlocal boundary condition, which
involves the convolution with an appropriate integral kernel. A method based on an efficient approximation
of this integral kernel is then implemented; see, for instance, [16, 17] for the case of the 2D and 3D flat
wave equations and [271, 270
, 272
] for the Regge–Wheeler [347
] and Zerilli [453
] equations describing
linear gravitational waves on a Schwarzschild background. Although this method is robust, very accurate
and stable, it is based on detailed knowledge of the solutions, which might not always be available in more
general situations.
In the following, we illustrate some aspects of the problem of constructing absorbing boundary
conditions on some simple examples [372]. Specifically, we construct local absorbing boundary conditions
for the wave equation with a spherical outer boundary at radius
.
Consider first the one-dimensional case,
The general solution is a superposition of a left- and a right-moving solution, Therefore, the boundary conditions are perfectly absorbing according to our terminology. Indeed, the operator
Generalizing the previous example to higher dimensions is a nontrivial task. This is due to the fact that
there are infinitely many propagation directions for outgoing waves, and not just two as in the
one-dimensional case. Ideally, one would like to control all the propagation directions , which are
outgoing at the boundary (
, where
is the unit outward normal to the boundary), but this is
obviously difficult. Instead, one can try to control specific directions (starting with the one that is normal to
the outer boundary). Here, we illustrate the method of [46
] on the three-dimensional wave equation,
When the background is curved, it is not always possible to construct in- and outgoing solutions explicitly,
as in the previous example. Therefore, it is not even clear how a hierarchy of absorbing boundary conditions
should be formulated. However, in many applications the spacetime is asymptotically flat, and if the
boundary surface is placed sufficiently far from the strong field region, one can assume that the metric is a
small deformation of the flat, Minkowski metric. To first order in with
the ADM mass and
the areal radius of the outer boundary, these correction terms are given by those of the
Schwarzschild metric, and approximate in- and outgoing solutions for all
modes can again be
computed [372
].25
The
terms in the background metric induce two kind of corrections in the in- and outgoing solutions
. The first is a curvature correction term, which just adds
terms to the coefficients in the
sum of Eq. (5.144
). This term is local and still obeys Huygens’ principle. The second term is fast decaying
(it decays as
) and describes the backscatter off the curvature of the background. As a
consequence, it is nonlocal (it depends on the past history of the unperturbed solution) and violates
Huygens’ principle.
By construction, the boundary conditions are perfectly absorbing for outgoing waves with angular
momentum number
, including their curvature corrections to first order in
. If the first-order
correction terms responsible for the backscatter are taken into account, then
are not perfectly
absorbing anymore, but the spurious reflections arising from these correction terms have been estimated
in [372
] to decay at least as fast as
for monochromatic waves with wave number
satisfying
.
The well-posedness of higher-order absorbing boundary conditions for wave equations on a curved
background can be established by assuming the localization principle and the Laplace method [369]. Some
applications to general relativity are discussed in Sections 6 and 10.3.1.
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Living Rev. Relativity 15, (2012), 9
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