RBC and AWE Problem Descriptions

Radiation boundary conditions. Wave simulation on finite computational domains requires artificial boundaries at which one must specify boundary conditions [1]. Ideally, these boundary conditions stem from exact reduction of an infinite domain, allowing for radiation flux off the finite computational domain. For the ordinary wave equation such radiation boundary conditions (RBC) specify a transparent or nonreflecting boundary. Specification and numerical implementation of RBC is a fundamental problem in computational mathematics with broad application in the sciences. RBC for engineering applications typically involve domains which are slender or otherwise complicated by edges and corners. See [2] for recent breakthroughs on such formulations. For the Einstein equations (or linearizations thereof), the artificial outer boundary of the computational domain is often a sphere, a rather simple boundary. Nevertheless, wave propagation in the presence of spacetime curvature entails added complications, such as backscatter. RBC for gravitational waves should capture such effects.

For the wave and Maxwell equations, successful RBC exist for planar, spherical, and cylindrical boundaries [3,4,5]. For example, exploiting a spherical boundary and the rotational invariance of the wave equation, Alpert, Greengard, and Hagstrom (AGH) [3] work with spherical harmonic multipoles \(\Psi_{\ell m}(t,r)\) of the wave field \(r^{-1}\psi(t,x,y,z)\). The exact RBC are then expressed \(\ell\)-by-\(\ell\), with \(\ell\) the standard "orbital" index, and feature an integral convolution \((\Psi_{\ell m} * \Omega_\ell)(t)\) in time between each multipole \(\Psi_{\ell m}(t,r)\) and a time-domain nonreflecting kernel \(\Omega_\ell(t,r)\). The kernel is a sum of \(\ell\) exponential functions, whence the cost of evaluating the convolution grows with \(\ell\). The situation is worse for the analogous kernels appropriate for 2+1 wave propagation and a circular boundary, since then even the low wave-number kernels are expensive to evaluate. To circumvent the high cost of implementing the exact RBC, AGH have introduced the technique of kernel compression which produces an approximate nonreflecting kernel \(\Xi_\ell(t,r)\), and in turn an approximate convolution \((\Psi_{\ell m} *\Xi_\ell)(t)\). The kernel \(\Xi_\ell(t,r)\) is a finite sum of fewer exponential functions. With \(\varepsilon\) a long-time bound on the associated relative convolution error, here chosen as a fixed tolerance, AGH have shown that the number \(d\) of exponential functions in \(\Xi_\ell(t,r)\) scales like \(d = O\big(\log\nu\log(1/\varepsilon)+\log^2\nu+\nu^{-1}\log^2(1/\varepsilon)\big)\), where \(\nu = \ell+1/2\). A more complicated scaling holds for 2+1 waves. The AGH approach has been generalized to both the Schr\"{o}dinger equation by Jiang [6] and to the Regge-Wheeler and Zerilli equations in second reference on the main page of this website (and works cited therein).

Teleportation and Asymptotic Waveform Evaluation. Let the outer computational boundary be the sphere of radius \(r = r_b\), in which case the RBC expressions considered in the previous paragraph should be expressed in terms of \(r_b\), for example, \(\Omega_{\ell m}(t,r) \rightarrow \Omega_{\ell m}(t,r_b)\). Suppose there is a "detector" at \(r = r_1 \leq r_b\) which records the multipole solution \(\Psi_{\ell m}(t,r_1)\) as a time series. Most often in simulations \(r_1 = r_b\), i.e. the detector is placed where the RBC are enforced, but we need not make this assumption. From the recorded time series \(\Psi_{\ell m}(t,r_1)\) we want to recover the signal \(\Psi_{\ell m}(t+x_2-x_1,r_2)\), where \( r_1 < r_2 \leq \infty \) the time shift \(x_2-x_1\) is \(r_2-r_1\) for flatspace multipoles, but the corresponding difference in tortoise coordinate for gravitational multipoles. We achieve this \(r_1 \rightarrow r_2\) conversion of the signal through teleportation based on integral convolution with a sum-of-exponentials kernel \(\Xi^E_\ell(t,r_1,r_2)\). Here \(E\) stands for evaluation, and this kernel (specified by a numerical table) approximates the true analytic kernel \(\Omega^E_\ell(t,r_1,r_2)\). For large \(r_2\) the signal \(\Psi_{\ell m}(t+x_2-x_1,r_2)\) is far-field or asymptotic. Therefore, for large \(r_2\) we view teleportation as a method for asymptotic waveform evaluation (AWE).

For the wave equation Greengard, Hagstrom, and Jiang [7] have shown that teleportation is exponentially ill-conditioned in the following sense. The Laplace transform \(\widehat{\Omega}{}^E_\ell(s,r_1,r_2)\) is a sum of simple poles, where the largest residue depends exponentially on \(\ell\). Nevertheless, product formulas [7] or quadrature expressions (see the Benedict-Field-Lau reference on the main page) yield well-conditioned evaluations of \(\widehat{\Omega}{}^E_\ell(s,r_1,r_2)\) for \(s\in \mathrm{i}\mathbb{R}\). Moreover, for the gravitational case we have demonstrated that accurate results are obtained for \(\ell \leq 64\) and double precision/sum-of-poles representations of the approximate kernel \(\widehat{\Xi}{}^E_\ell(s,r_1,r_2)\). Subsequent investigations indicate that single precision tolerences \(\varepsilon \simeq 10^{-7}\) are possible through \(\ell = 256\). Likely, \(\ell \leq 64\) is sufficient for GR applications.

Examples. Consider the following generic 1+1 wave equation:

\(\partial_t^2 \Psi - \partial_x^2 \Psi + V(r) \Psi = 0\).
Such a wave equation arises upon (possibly vector or tensor) spherical harmonic decomposition of the ordinary 3+1 wave, Maxwell, and linearized Einstein equations. Here we consider the following cases.
  1. The ordinary wave equation
    \(\partial_t^2 \psi - \partial_x^2 \psi - \partial_y^2 \psi - \partial_z^2 \psi = 0\).
    Here \(x = r\), \(r^{-1}\psi(t,x,y,z)\) is decomposed in terms of scalar spherical harmonics with \(\Psi_{\ell m}(t,r)\) coefficients, and the effective potential is
    \(V_{\ell}(r) = r^{-2}\ell(\ell+1).\)
  2. The Regge-Wheeler equation which arises as a linearization of the Einstein equation about a non-spinning black hole of mass M. Here the multipole solutions \(\Psi_{\ell m}\) describe odd parity metric perturbations, and the potential is
    \( V_\ell(r) = r^{-2}f(r)\left[\ell(\ell+1) - 6M/r\right] \),
    where \(f(r) = 1 - 2M/r\).
  3. The Zerilli equation also arises as a linearization of the Einstein equation about a non-spinning black hole. Here the multipole solutions \(\Psi_{\ell m}\) describe even parity metric perturbations and
    \( V_{\ell}(r) = 2(n_\ell r + 3 M)^{-2}f(r) \left[n_\ell^2 \left(1 + n_\ell + 3M/r\right) + 9M^2 r^{-2} \left(n_\ell + M/r\right)\right] \),
    where \(f(r) = 1 - 2M/r\) and \( n_\ell = (\ell+2)(\ell-1)/2 \).

[1] T. Hagstrom, Radiation boundary conditions for the simulation of waves, Acta Numerica 8, pp. 47-106 (1999).
[2] T. Hagstrom, T. Warburton, and D. Givoli, Radiation boundary conditions for time-dependent waves based on complete plane wave expansions, J. Comput. Appl. Math. 234, Issue 6, pp. 1988-1995 (2010).
[3] B. Alpert, L. Greengard, and T. Hagstrom, Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation, SIAM J. Numer. Anal. 37, pp. 1138-1164 (2000).
[4] B. Alpert, L. Greengard, and T. Hagstrom, Nonreflecting Boundary Conditions for the Time-Dependent Wave Equation, J. Comput. Phys. 180, pp. 270-296 (2002).
[5] T. Hagstrom and S. Lau, Radiation Boundary Conditions for Maxwell's Equations: A Review of Accurate Time-Domain Formulations, J. Comput. Math. 25, pp. 305-336 (2007).
[6] S. Jiang, Fast Evaluation of Nonreflecting Boundary Conditions for the Schroedinger Equation, Ph.D. dissertation, New York University, New York, 2001.
[7] L. Greengard, T. Hagstrom, S. Jiang, The solution of the scalar wave equation in the exterior of a sphere, 28 page preprint, August 2013, arXiv:1308.0643 [math.NA].