Showing papers by "Leslie Greengard published in 2000"
TL;DR: The main technical result in this paper is that the logarithmic derivative of the Hankel function H_
u(1)(z) can be approximated in the upper half of the z-plane with relative error $\varepsilon$ by a rational function of degree d.
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 $H_
u^{(1)}(z)$ of real order $
u$ can be approximated in the upper half $z$-plane with relative error $\varepsilon$ by a rational function of degree $d \sim O (\log|
u|\log\frac{1}{\varepsilon}+ \log^2 |
u| + |
u |^{-1} \log^2\frac{1}{\varepsilon} )$ as $|
u|\rightarrow\infty$, $\varepsilon\rightarrow 0$, with slightly more complicated bounds for $
u=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.
216 citations
TL;DR: An algorithm which requires an amount of work of the order O(NMlog M+mlog m) and which is based on the evolution of the continuousspectrum of the solution, which generalizes an earlier technique developed by Greengard and Strain for evaluating layer potentials in bounded domains.
90 citations
TL;DR: In this article, a time-symmetric evolution formula for the scalar wave equation is presented, which is related to the classical D'Alembert or spherical means representations but applies equally well in two space dimensions.
37 citations
TL;DR: This work presents an adaptive fast multipole method for inverting the square root of the Laplacian in two dimensions and assumes only that the source distribution is discretized using an adaptive quad-tree.
Abstract: We present an adaptive fast multipole method for inverting the square root of the Laplacian in two dimensions. Solving this problem is the dominant computational cost in many applications arising in electrical engineering, geophysical fluid dynamics, and the study of thin films. It corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity. Our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad-tree. The amount of work grows linearly with the number of mesh points.
24 citations