A time-frequency method for acoustic scattering with trapping

Abstract

A Fourier transform method is introduced for a class of hybrid time-frequency methods that solve the acoustic scattering problem in regimes where the solution exhibits both highly oscillatory behavior and slow decay in time. This extends the applicability of hybrid time-frequency schemes to domains with trapping regions. A fast sinc transform technique for managing highly oscillatory behavior and long time horizons is combined with a contour integration scheme that improves smoothness properties in the integrand.

Figures (10)

• Figure 1: Plots of $|u_{tot}(x,t)|$ at times $t = 17.541$ (left), $t = 31.311$ (center), and $t = 130.00$ (right) on a "magnetron" domain.
• Figure 1: Top row: Using the domain on the right, the mean-squared error $\|u_{res}(x,t) -u_m(x,t)\|_2$ over a small set of spatial points $\{x_j = e^{2 \pi i (j-1)/20}\}_{j = 1}^{20}$ (blue dots on right) and time points $\{t_j = 90j\}_{j=0}^{9}$ is plotted on a logarithmic scale against $m$, the number of quadrature points used in computing the inverse Fourier transform, either via Gauss-Legendre quadrature (blue), or via the truncated sinc expansion (red). Here, $u_{res}$ is a highly resolved approximation to the true solution. The convergence of the damping+correction method (black) described in \ref{['sec:DC']} is shown for reference. Bottom row: The experiment repeated but with the displayed domain, which induces stronger trapping.
• Figure 1: Left: Plots of $\hat{U}(\tilde{x}, \omega + \delta i)$ for $\delta = 0$ (blue), $\delta \approx .005$ (red), $\delta \approx .015$ (yellow), and $\delta \approx .02$ (purple) over $\omega \in [ .7, 16]$, with $\tilde{x}$ selected as the center of the $C$-shaped curve in \ref{['fig:poleplot']}. The plots are stacked vertically for illustrative purposes. Right: The magnitudes of the normalized Fourier coefficients for $\hat{U}(\tilde{x}, \omega + \delta_j i)$ for each $\delta_j$. Faster decay implies that fewer terms are needed to compute the sum in \ref{['eq:complexsincsum']} and thereby evaluate $I_{\delta}$.
• Figure 1: The $2$-norm error of solutions (relative to a highly resolved solution) computed over three points in time and the 20 spatial points shown on the right (black dots) is plotted on a logarithmic scale against the parameter $\delta$. Each line shows the error behavior for a fixed choice of $m$ as $\delta$ grows.
• Figure 2: Left: A plot of the magnitude of a meromorphic extension of $\hat{U}(\tilde{x}, \omega)$, associated with the domain on the right, is shown in the complex $\omega$-plane around the frequency interval $[3, 14]$ for a single point $\tilde{x} = [0,0]^T$. The colors are displayed using a logarithmic scale (e.g., from $10^{-11}$ (deep blue) to $10^{10}$ (dark green)). The point $\tilde{x} = [0, 0]^T$ is center of the keyhole cavity depicted on the right. Poles of the extended function, which correspond to singularities of $\hat{U}(\tilde{x}, \omega)$, are depicted as orange dots. The imaginary part of the pole closest to the real line is $\approx 1 \times 10^{-5}$ in magnitude. Right: The domain where $\hat{U}$ was computed is shown for reference, along with the real part of $u_{tot}(x,t)$ at $t = 68.121$. Details on the experiment can be found in \ref{['sec:num']}.

Theorems & Definitions (2)

• Lemma 3.1
• Remark 3.2