 
Summary: RAPID EVALUATION OF NONREFLECTING BOUNDARY
KERNELS FOR TIMEDOMAIN WAVE PROPAGATION
BRADLEY ALPERT, LESLIE GREENGARD, AND THOMAS HAGSTROM§
SIAM J. NUMER. ANAL. c 2000 Society for Industrial and Applied Mathematics
Vol. 37, No. 4, pp. 11381164
Abstract. We present a systematic approach to the computation of exact nonreflecting bound
ary 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 Han
kel function H
(1)
(z) of real order can be approximated in the upper half zplane with relative error
by a rational function of degree d O(log  log 1
+log2
+1 log2 1
) as  , 0, with
slightly more complicated bounds for = 0. If N is the number of points used in the discretization of
a cylindrical (circular) boundary in two dimensions, then, assuming that < 1/N, O(N log N log 1
