Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation

We present a systematic approach to the computation of exact nonreflecting boundary conditions for the wave equation. In both two and three dimensions, the critical step in our analysis involves convolution with the inverse Laplace transform of the logarithmic derivative of a Hankel function. The main technical result in this paper is that the logarithmic derivative of the Hankel function $H_\nu^{(1)}(z)$ of real order $\nu$ can be approximated in the upper half $z$-plane with relative error $\varepsilon$ by a rational function of degree $d \sim O (\log|\nu|\log\frac{1}{\varepsilon}+ \log^2 |\nu| + | \nu |^{-1} \log^2\frac{1}{\varepsilon} )$ as $|\nu|\rightarrow\infty$, $\varepsilon\rightarrow 0$, with slightly more complicated bounds for $\nu=0$. If N is the number of points used in the discretization of a cylindrical (circular) boundary in two dimensions, then, assuming that $\varepsilon < 1/N$, $O(N \log N\log\frac{1}{\varepsilon})$ work is required at each time step. This is comparable to the work required for the Fourier transform on the boundary. In three dimensions, the cost is proportional to $N^2 \log^2 N + N^2 \log N\log\frac{1}{\varepsilon}$ for a spherical boundary with N2 points, the first term coming from the calculation of a spherical harmonic transform at each time step. In short, nonreflecting boundary conditions can be imposed to any desired accuracy, at a cost dominated by the interior grid work, which scales like N3 in two dimensions and N2 in three dimensions.