### 5.3 Absorbing boundary conditions

When modeling isolated systems, the boundary conditions have to be chosen such that they minimize spurious reflections from the boundary surface. This means that inside the computational domain, the solution of the IBVP should lie as close as possible to the true solution of the Cauchy problem on the unbounded domain. In this sense, the dynamics outside the computational domain is replaced by appropriate conditions on a finite, artificial boundary. Clearly, this can only work in particular situations, where the solutions outside the domain are sufficiently simple so that they can be computed and used to construct boundary conditions, which are, at least, approximately compatible with them. Boundary conditions, which give rise to a well-posed IBVP and achieve this goal are called absorbing, non-reflecting or radiation boundary conditions in the literature, and there has been a substantial amount of work on the construction of such conditions for wave problems in acoustics, electromagnetism, meteorology, and solid geophysics (see [206] for a review). Some recent applications to general relativity are mentioned in Sections 6 and 10.3.1.

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 .

#### 5.3.1 The one-dimensional wave equation

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 has as its kernel the right-moving solutions ; hence, the boundary condition at is transparent to these solutions. On the other hand, , which implies that at , the boundary condition requires that is constant for advanced time . A similar argument shows that the left boundary condition implies that is constant for retarded time . Together with initial conditions for and its time derivative at satisfying the compatibility conditions, Eqs. (5.135, 5.137) give rise to a well-posed IBVP. In particular, the solution is identically zero after one crossing time for initial data, which are compactly supported inside the interval .

#### 5.3.2 The three-dimensional wave equation

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,

The general solution can be decomposed into spherical harmonics according to
which yields the family of reduced equations
For this equation reduces to the one-dimensional wave equation, for which the general solution is with and two arbitrary functions. Therefore, the boundary condition
is perfectly absorbing for spherical waves. For , exact solutions can be generated from the solutions for by applying suitable differential operators to . For this, we define the operators [92]
which satisfy the operator identities
As a consequence, for each , we have
Therefore, we have the explicit in- and outgoing solutions
where and are arbitrary smooth functions with ’th derivatives and , respectively. In order to construct boundary conditions, which are perfectly absorbing for , one first notices the following identity:
for all and all sufficiently smooth functions . This identity follows easily from Eq. (5.144) and the fact that if . Therefore, given , the boundary condition
leaves the outgoing solutions with unaltered. Notice that this condition is local in the sense that its formulation does not require the decomposition of into spherical harmonics. Based on the Laplace method, it was proven in [46] (see also [369]) that each boundary condition yields a well-posed IBVP. By uniqueness this implies that initial data corresponding to a purely outgoing solution with yields a purely outgoing solution (without reflections). In this sense, the condition is perfectly absorbing for waves with . For waves with , one obtains spurious reflections; however, for monochromatic radiation with wave number , the corresponding amplitude reflection coefficients can be calculated to decay as in the wave zone  [88]. Furthermore, in most scenarios with smooth solutions, the amplitudes corresponding to the lower few ’s will dominate over the ones with high so that reflections from high ’s are unimportant. For a numerical implementation of the boundary condition via spectral methods and a possible application to general relativity see [314].

#### 5.3.3 The wave equation on a curved background

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]. 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.