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 [153Jump To The Next Citation Point, 46Jump To The Next Citation Point, 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 [271Jump To The Next Citation Point, 270Jump To The Next Citation Point, 272Jump To The Next Citation Point] for the Regge–Wheeler [347Jump To The Next Citation Point] and Zerilli [453Jump To The Next Citation Point] 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 [372Jump To The Next Citation Point]. Specifically, we construct local absorbing boundary conditions for the wave equation with a spherical outer boundary at radius R > 0.

5.3.1 The one-dimensional wave equation

Consider first the one-dimensional case,

utt − uxx = 0, |x| < R, t > 0. (5.135 )
The general solution is a superposition of a left- and a right-moving solution,
u (t,x ) = f ↖(x + t) + f↗(x − t). (5.136 )
Therefore, the boundary conditions
∂ ∂ (b− u )(t,− R ) = 0, (b+u )(t,+R ) = 0, b± := ---± --, t > 0, (5.137 ) ∂t ∂x
are perfectly absorbing according to our terminology. Indeed, the operator b+ has as its kernel the right-moving solutions f↗ (x − t); hence, the boundary condition (b+u )(t,R ) = 0 at x = R is transparent to these solutions. On the other hand, ′ b+f ↖(t + x) = 2f↖ (t + x ), which implies that at x = R, the boundary condition requires that f↖ (v) = f↖ (1 ) is constant for advanced time v = t + x > R. A similar argument shows that the left boundary condition (b− u )(t,− R ) = 0 implies that f↗ (− u) = f↗ (− R ) is constant for retarded time u = t − x > R. Together with initial conditions for u and its time derivative at t = 0 satisfying the compatibility conditions, Eqs. (5.135View Equation, 5.137View Equation) give rise to a well-posed IBVP. In particular, the solution is identically zero after one crossing time t ≥ 2R for initial data, which are compactly supported inside the interval (− R, R).

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 k, which are outgoing at the boundary (k ⋅ n > 0, where n 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 [46Jump To The Next Citation Point] on the three-dimensional wave equation,

utt − Δu = 0, |x| < R, t > 0. (5.138 )
The general solution can be decomposed into spherical harmonics Y ℓm according to
1 ∑∞ ∑ ℓ ℓm u(t,r,𝜗,φ) = -- uℓm(t,r)Y (𝜗,φ ), (5.139 ) r ℓ=0 m= −ℓ
which yields the family of reduced equations
[ 2 2 ] -∂--− ∂---+ ℓ(ℓ-+-1) uℓm (t,r) = 0, 0 < r < R, t > 0. (5.140 ) ∂t2 ∂r2 r2
For ℓ = 0 this equation reduces to the one-dimensional wave equation, for which the general solution is u00(t,r) = U00↗ (r − t) + U00↖ (r + t) with U00↗ and U00↖ two arbitrary functions. Therefore, the boundary condition
( ∂ ∂ ) ℬ0 : b(ru)|r=R = 0, b := r2 --+ --- , t > 0, (5.141 ) ∂t ∂r
is perfectly absorbing for spherical waves. For ℓ ≥ 1, exact solutions can be generated from the solutions for ℓ = 0 by applying suitable differential operators to u (t,r) 00. For this, we define the operators [92]
-∂- ℓ- † -∂- ℓ- aℓ ≡ ∂r + r, a ℓ ≡ − ∂r + r, (5.142 )
which satisfy the operator identities
† † ∂2-- ℓ(ℓ +-1) aℓ+1a ℓ+1 = aℓaℓ = − ∂r2 + r2 . (5.143 )
As a consequence, for each ℓ = 1,2,3,..., we have
[ ∂2 ∂2 ℓ(ℓ + 1)] [ ∂2 ] --2 − --2-+ ----2--- a†ℓa†ℓ−1...a†1 = --2 + a†ℓaℓ a†ℓa†ℓ−1 ...a†1 ∂t ∂r r ∂t[ ] † ∂2 † † † = aℓ ∂t2-+ aℓ−1aℓ−1 aℓ− 1...a1 [ ] † † † -∂2- ∂2-- = aℓaℓ−1 ...a1 ∂t2 − ∂r2 .
Therefore, we have the explicit in- and outgoing solutions
∑ℓ (2ℓ − j)! uℓm↖ (t,r) = a†ℓa†ℓ−1...a†1Vℓm (r + t) = (− 1)j---------(2r)j−ℓVℓ(jm)(r + t), j=0 (ℓ − j)!j! ℓ † † † ∑ j-(2-ℓ −-j)! j−ℓ (j) uℓm↗ (t,r) = aℓaℓ−1...a1U ℓm (r − t) = (− 1) (ℓ − j)!j!(2r) U ℓm (r − t), (5.144 ) j=0
where V ℓm and U ℓm are arbitrary smooth functions with j’th derivatives V (j) ℓm and U (j) ℓm, respectively. In order to construct boundary conditions, which are perfectly absorbing for uℓm, one first notices the following identity:
ℓ+1 † † † b aℓaℓ−1...a 1U (r − t) = 0 (5.145 )
for all ℓ = 0,1,2,... and all sufficiently smooth functions U. This identity follows easily from Eq. (5.144View Equation) and the fact that ℓ+1 k k+ℓ+1 b (r ) = k (k + 1 ) ⋅ ⋅ ⋅ (k + ℓ)r = 0 if k ∈ {0, − 1,− 2,...,− ℓ}. Therefore, given L ∈ {1,2, 3,...}, the boundary condition
| ℬL : bL+1 (ru )|r=R = 0 (5.146 )
leaves the outgoing solutions with ℓ ≤ L unaltered. Notice that this condition is local in the sense that its formulation does not require the decomposition of u into spherical harmonics. Based on the Laplace method, it was proven in [46Jump To The Next Citation Point] (see also [369Jump To The Next Citation Point]) that each boundary condition ℬL yields a well-posed IBVP. By uniqueness this implies that initial data corresponding to a purely outgoing solution with ℓ ≤ L yields a purely outgoing solution (without reflections). In this sense, the condition ℬL is perfectly absorbing for waves with ℓ ≤ L. For waves with ℓ > L, one obtains spurious reflections; however, for monochromatic radiation with wave number k, the corresponding amplitude reflection coefficients can be calculated to decay as −2(L+1) (kR ) in the wave zone kR ≫ 1 [88Jump To The Next Citation Point]. Furthermore, in most scenarios with smooth solutions, the amplitudes corresponding to the lower few ℓ’s will dominate over the ones with high ℓ so that reflections from high ℓ’s are unimportant. For a numerical implementation of the boundary condition ℬ2 via spectral methods and a possible application to general relativity see [314Jump To The Next Citation Point].

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 M ∕R with M the ADM mass and R 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 (ℓ,m ) modes can again be computed [372Jump To The Next Citation Point].25 The M ∕R terms in the background metric induce two kind of corrections in the in- and outgoing solutions u ℓm ↖. The first is a curvature correction term, which just adds M ∕R terms to the coefficients in the sum of Eq. (5.144View Equation). This term is local and still obeys Huygens’ principle. The second term is fast decaying (it decays as R ∕rℓ+1) 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 ℬL are perfectly absorbing for outgoing waves with angular momentum number ℓ ≤ L, including their curvature corrections to first order in M ∕R. If the first-order correction terms responsible for the backscatter are taken into account, then ℬL are not perfectly absorbing anymore, but the spurious reflections arising from these correction terms have been estimated in [372Jump To The Next Citation Point] to decay at least as fast as −2 (M ∕R )(kR ) for monochromatic waves with wave number k satisfying M ≪ k− 1 ≪ R.

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 [369Jump To The Next Citation Point]. Some applications to general relativity are discussed in Sections 6 and 10.3.1.

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