Fast Evaluation of Far-Field Signals for Time-Domain Wave Propagation

Abstract

Time-domain simulation of wave phenomena on a finite computational domain often requires a fictitious outer boundary. An important practical issue is the specification of appropriate boundary conditions on this boundary, often conditions of complete transparency. Attention to this issue has been paid elsewhere, and here we consider a different, although related, issue: far-field signal recovery. Namely, from smooth data recorded on the outer boundary we wish to recover the far-field signal which would reach arbitrarily large distances. These signals encode information about interior scatterers and often correspond to actual measurements. This article expresses far-field signal recovery in terms of time-domain convolutions, each between a solution multipole moment recorded at the boundary and a sum-of-exponentials kernel. Each exponential corresponds to a pole term in the Laplace transform of the kernel, a finite sum of simple poles. Greengard, Hagstrom, and Jiang have derived the large-$\ell$ (spherical-harmonic index) asymptotic expansion for the pole residues, and their analysis shows that, when expressed in terms of the exact sum-of-exponentials, large-$\ell$ signal recovery is plagued by cancellation errors. Nevertheless, through an alternative integral representation of the kernel and its subsequent approximation by a smaller number of exponential terms (kernel compression), we are able to alleviate these errors and achieve accurate signal recovery. We empirically examine scaling relations between the parameters which determine a compressed kernel, and perform numerical tests of signal “teleportation” from one radial value $r_1$ to another $r_2$, including the case $r_2=\infty$. We conclude with a brief discussion on application to other hyperbolic equations posed on non-flat geometries where waves undergo backscatter.

Appendices

Appendix 1: Residues of the Frequency Domain Kernel

This first appendix derives the alternative expression (26) for the residues $a_{\ell j}(r_1,r_2)$ in (21). We start with

$$\begin{aligned} a_{\ell j}(r_1,r_2)&= \frac{W_\ell (b_{\ell j}r_2/r_1)}{r_1W_\ell '(b_{\ell j})} , \end{aligned}$$

(54)

a formula which agrees with (21). We now write $a_{\ell j}(r_1,r_2)$ in terms of standard special functions, starting with MacDonald’s function. From (10), we get

$$\begin{aligned} W_\ell (b_{\ell j} r_2/r_1)&= \sqrt{\frac{2 b_{\ell j}}{\pi }\frac{r_2}{r_1}} e^{b_{\ell j} r_2/r_1} K_{\ell +1/2}(b_{\ell j} r_2/r_1)\end{aligned}$$

(55a)

$$\begin{aligned} W_\ell '(b_{\ell j})&= \sqrt{\frac{2 b_{\ell j}}{\pi }} e^{b_{\ell j}} K_{\ell +1/2}'(b_{\ell j}). \end{aligned}$$

(55b)

Note that $K_{\ell + 1/2}(z)$ is defined only on the slit plane (due to the branch associated with the square root factor). Therefore, for odd $\ell$ the purely real root $b_{\ell ,1+(\ell -1)/2}$ is not in the domain of analyticity. Nevertheless, with appropriate cancellation of square-root factors, the following expression is valid even for this root

$$\begin{aligned} a_{\ell j}(r_1,r_2) = \frac{1}{r_1}\sqrt{\frac{r_2}{r_1}}e^{(r_2/r_1 - 1)b_{\ell j}} \frac{K_{\ell +1/2}(b_{\ell j} r_2/r_1)}{K_{\ell +1/2}'(b_{\ell j})}. \end{aligned}$$

(56)

To use Olver’s uniform asymptotic formulas for Bessel functions of large order and argument (see in particular AS 9.3.37 and 9.3.45), we need to express (56) in terms of the first Hankel function. To this end, we start with the first equation in AS 10.2.15 (which involves the first spherical Hankel function defined in AS 10.1.16):

$$\begin{aligned} \sqrt{\frac{\pi }{2z}} K_{\ell + 1/2}(z) = \frac{\mathrm {i}\pi }{2} e^{\mathrm {i}(\ell + 1)\pi /2} h_{\ell }^{(1)}(z e^{\mathrm {i}\pi /2}), \qquad -\pi < \arg z \le \frac{1}{2}\pi . \end{aligned}$$

(57)

Next, using the relationship between spherical and cylindrical Hankel functions given in AS 10.1.1, we get

$$\begin{aligned} \sqrt{\frac{\pi }{2z}} K_{\ell + 1/2}(z) = \frac{\mathrm {i}\pi }{2} e^{\mathrm {i}(2\ell + 1)\pi /4} \sqrt{\frac{\pi }{2z}} H_{\ell +1/2}^{(1)}(z e^{\mathrm {i}\pi /2}), \qquad -\pi < \arg z \le \frac{1}{2}\pi . \end{aligned}$$

(58)

There are two branch cuts associated with the right-hand expression. The first is the usual cut along the negative real axis (in the $z$-plane) associated with the square root. The second results from the branch associated with $H_{\ell +1/2}^{(1)}(\bullet )$; due to the rotated argument this cut is the positive imaginary axis (in the $z$-plane). Across both cuts the right-hand expression in (58) jumps by a sign, whereas, due to (57) and the domain of analyticity for $h_{\ell }^{(1)}(\bullet )$, the left-hand expression in (58) is analytic on the origin-punctured $z$-plane. Therefore, we work with the expression

$$\begin{aligned} K_{\ell + 1/2}(z) = \epsilon \frac{\mathrm {i}\pi }{2} e^{\mathrm {i}(2\ell + 1)\pi /4} H_{\ell +1/2}^{(1)}(\mathrm {i}z), \end{aligned}$$

(59)

where $\epsilon = -1$ in the second quadrant and $\epsilon = 1$ otherwise. Using this result, we cast (56) into the form

$$\begin{aligned} a_{\ell j}(r_1,r_2) = -\frac{\mathrm {i}}{r_1}\sqrt{\frac{r_2}{r_1}} e^{(r_2/r_1 - 1)b_{\ell j}} \frac{ H_{\ell +1/2}^{(1)}(\mathrm {i}b_{\ell j} r_2/r_1) }{ H_{\ell +1/2}^{(1)\prime }(\mathrm {i}b_{\ell j}) }. \end{aligned}$$

(60)

The last expression leads directly to (26) upon introduction of the scaled zeros. In both the last expression and (26) the derivative in the denominator can be eliminated with the identity $2H_{\nu }^{(1)\prime }(z) = H_{\nu -1}^{(1)}(z)-H_{\nu +1}^{(1)}(z)$.

Appendix 2: Conditioning and Error Bounds of Teleportation with Respect to Data Perturbation

We now consider conditioning of teleportation with respect to data perturbation and related error bounds. In this appendix $(\ell ,m)$ indices and, often, radial arguments $r_1,r_2$ on $\varPsi$ and $\varPhi$ are suppressed. Let

$$\begin{aligned} \varPhi * \delta \varPsi (t) = \int _0^t \varPhi (t-t') \delta \varPsi (t') dt', \qquad t \in [0,T]. \end{aligned}$$

(61)

Here $\delta \varPsi (t',r_1) = \varPsi (t',r_1) - \varPsi _h(t',r_1)$ is the time-series error due to numerical discretization, and $T$ is the final time. One version of Young’s convolution inequality yields

$$\begin{aligned} \Vert \varPhi * \delta \varPsi \Vert _{L_\infty (0,T) } \le \Vert \varPhi \Vert _{L_\infty (0,T) } \cdot \Vert \delta \varPsi \Vert _{L_1(0,T)}, \end{aligned}$$

(62)

which with (16) immediately gives (upon replacing the radial arguments)

$$\begin{aligned}&\big \Vert \varPsi (\cdot +(r_2-r_1),r_2) - \varPsi _h(\cdot +(r_2-r_1),r_2)\big \Vert _{L_\infty (0,T)}\nonumber \\&\quad \le \Vert \varPhi (\cdot ,r_1,r_2)\Vert _{L_\infty (0,T) } \cdot \Vert \delta \varPsi (\cdot ,r_1)\Vert _{L_1(0,T)} + \Vert \delta \varPsi (\cdot ,r_1)\Vert _{L_{\infty }(0,T)}. \end{aligned}$$

(63)

Numerical evidence suggests [4] that $\left| \varPhi _\ell (t,r_1,r_2)\right|$ is maximal at $t=0$. Furthermore,

$$\begin{aligned} \varPhi _\ell (0,r_1,r_2) = \frac{1}{r_1} \left[ \frac{\ell \left( \ell +1 \right) }{2} \left( \frac{r_1}{r_2} -1 \right) \right] , \end{aligned}$$

(64)

which follows from Lemma 1 of Ref. [4] and the scaling relation (22).