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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Greengard, Leslie Alpert, Bradley Hagstrom, Thomas
Copyright Year	2000
Abstract	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.
Starting Page	1138
Ending Page	1164
Page Count	27
File Format	PDF
ISSN	00361429
DOI	10.1137/S0036142998336916
e-ISSN	10957170
Issue Number	4
Volume Number	37
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	Bessel function approximation wave equation high-order convergence Maxwell's equations absorbing boundary condition Computation of special functions, construction of tables Approximation by rational functions Convolution nonreflecting boundary condition Laplace transform radiation boundary condition Bessel and Airy functions, cylinder functions, ${}_0F_1$
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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