A high-order algorithm for time-domain scattering in three dimensions

Abstract

The ubiquitous scattering of time-dependent waves in free-space exterior to bounded configurations is fundamental for numerous applications. Simulation of time-domain scattering in the unbounded exterior region, without artificial domain truncation, facilitates understanding of the wave propagation process in the entire exterior region. The space-time hyperbolic partial differential equation (PDE) for the unknown scalar scattered field in the free-space can be reformulated as a retarded surface integral equation (SIE) on the boundary of the configuration, using a retarded potential ansatz for the field. The unknown surface density in the ansatz satisfies the SIE, and hence the exterior scattering problem reduces to the SIE model. The weakly- or hyper-singular complexity of the SIE depends on the (Dirichlet/Neumann/Robin) condition on the boundary of the configuration in the PDE model. In this work, we develop a fully discrete high-order algorithm for efficient and stable simulation of the time-domain scattering weakly- and hyper-singular SIE models. Our algorithm is a hybrid of high-order convolution quadrature (CQ) discretization in time and spectrally accurate approximation in space. We demonstrate computational efficiency of the algorithm using a gallery of configurations with Dirichlet/Neumann/Robin boundary conditions, and compare with CQ-based benchmarks and recent results in the literature.

Appendices

Appendix A: Analytical solutions to TD-SIEs on the sphere

In this appendix, we give the analytical expressions of the density solution to the time-domain surface integral equations modeling the wave equation outside a spherical domain with either a Dirichlet, a Neumann or a Robin separable boundary condition defined using a general time-variable function and a spherical harmonic. The results are derived for general radius $R>0$ and sound speed $c>0$ following details of results in [39, Chapter 2] and [20]. Since spherical harmonics form a basis for $L^2(\mathbb {S}^2)$, the results hold for any square-integrable separable boundary condition on [0, T] $\times \mathbb {S}^2$.

Proposition A.1

The spherical harmonics $Y_{pq}$ [7, Eq. (2.37)] are eigenfunctions of the time-domain surface integral operators S, D, $D'$ and N. For the Laplace-transformed operators, with s being the Laplace-domain variable, the following results hold:

$$\begin{aligned}{} & {} \mathcal {L}(S)Y_{pq}(s)=\lambda _{p}^{(\hat{S})}(s)Y_{pq}\quad \textrm{ with}\,\, \lambda _{p}^{(\hat{S})}(s)=-R^2\tfrac{s}{c}j_{p}(iR\tfrac{s}{c})h_{p}^{(1)}(iR\tfrac{s}{c}),\\{} & {} \mathcal {L}(D)Y_{pq}(s)=\lambda _{p}^{(\hat{D})}(s)Y_{pq}\quad \textrm{ with}\,\, \lambda _{p}^{(\hat{D})}(s)=-\frac{1}{2}-iR^2(\tfrac{s}{c})^2j'_{p}(iR\tfrac{s}{c})h_{p}^{(1)}(iR\tfrac{s}{c}),\\{} & {} \mathcal {L}(D')Y_{pq}(s)\!=\!\lambda _{p}^{(\hat{D'})}(s)Y_{pq}\!\!\quad \textrm{ with}\,\, \lambda _{p}^{(\hat{D'})}(s)=\lambda _{p}^{(\hat{D})}(s)=\frac{1}{2}-iR^2(\tfrac{s}{c})^2j_{p}(iR\tfrac{s}{c})h_{p}^{(1)'}(iR\tfrac{s}{c}),\\{} & {} \mathcal {L}(N)Y_{pq}(s)=\lambda _{p}^{(\hat{N})}(s)Y_{pq}\quad \textrm{ with}\,\, \lambda _{p}^{(\hat{S})}(s)=R^2s^3j'_{p}(iR\tfrac{s}{c})h_{p}^{(1)'}(iR\tfrac{s}{c}), \end{aligned}$$

where

$$\begin{aligned} j_{p}(is)\!= & {} \!(-1)^{p}i^{p}\frac{e^s}{2s^{p+1}}\left( \chi _{p}(\!-s)\!-\!e^{-2s}\chi _{p}(s)\right) ,\,\,\! j_{p}'(is)\!=-\!j_{p+1}(is)\!+\frac{p}{is}j_{p}(is),\\ h_{p}^{(1)}(is)= & {} -i^{-n}\frac{e^{-s}}{s^{p+1}}\chi _{p}(s),\qquad \qquad \qquad \qquad h^{(1)'}_{p}(is)=-h^{(1)}_{p+1}(is)+\frac{p}{is}h^{(1)}_{p}(is),\\ \chi _{p}(s)= & {} \sum _{q=0}^{p}(p,q)s^{p-q},\qquad \qquad \qquad \quad \qquad (p,q)=\frac{(p+q)!}{q!2^q(p-q)!}. \end{aligned}$$

1.1 A.1 Exact solution to the Dirichlet problem (first-kind TD-SIE on ${\displaystyle \Gamma } = \mathbb {S}^2$): $S\varphi =-\tilde{f}(t)Y_{pq}$

The solution of Eq. (5) is given by

$$\begin{aligned} \varphi _{pq}^{(\hat{S})}(t,\textbf{x})=Y_{pq}(\hat{x})\int _{0}^t\tilde{f}(t-\tau )\mathscr {L}^{-1}\left( \frac{1}{\lambda ^{(\hat{S})}_{p}(s)}\right) (\tau )d\tau , \end{aligned}$$

where

$$\begin{aligned} \frac{1}{\lambda ^{(\hat{S})}_{p}(s)}=\frac{(-1)^{p}}{R}\frac{2(\frac{sR}{c})^{2p+1}}{\chi _{p}(\tfrac{sR}{c})\chi _{p}(-\tfrac{sR}{c})-\chi ^2_{p}(\tfrac{sR}{c})e^{-2\tfrac{sR}{c}}}. \end{aligned}$$

For $p=0,1$, we obtain

$$\begin{aligned} \varphi ^{(\hat{S})}_{00}(t,\textbf{x})= & {} Y_{00}(\hat{x})\left( \frac{2}{c}\sum _{k=0}^{\lfloor {\tfrac{ct}{2R}}\rfloor }\tilde{f}'(t-\tfrac{2R}{c}k)\right) ,\\ \varphi ^{(\hat{S})}_{1q}(t,\textbf{x})= & {} Y_{1q}(\widehat{\textbf{x}}) \Bigg ( \frac{2}{c}\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }(-1)^{\ell }{\tilde{f}}'(t-\tfrac{2\ell R}{c})+\frac{2}{R}\int _{0}^t{\text {sinh}}(\tfrac{c}{R}\tau ){\tilde{f}}'(t-\tau )\,d\tau \\{} & {} -\frac{2}{R}\sum _{\ell =1}^{\lfloor \frac{ct}{2R}\rfloor }\sum _{j=1}^{\ell } (-1)^{\ell +1}\int _{\frac{2\ell R}{c}}^{t}\big ({\omega ^{(2)}_{\ell j}}+{\omega _{\ell j}^{(3)}}(\tfrac{c\tau }{R}-2\ell )\big )\\{} & {} \times (\tfrac{c\tau }{R}-2\ell )^{j-1}e^{\tfrac{c\tau }{R}-2\ell }{\tilde{f}}'(t-\tau )\,d\tau \Bigg ), \end{aligned}$$

with

$$\begin{aligned} {\omega ^{(3)}_{\ell ,j}}=\begin{pmatrix}\ell -1\\ j-1\end{pmatrix}\frac{2^{j-1}}{j!}\quad \textrm{and}\quad {\omega ^{(2)}_{\ell ,j}}=\begin{pmatrix}\ell \\ j\end{pmatrix}\frac{2^j}{(j-1)!}. \end{aligned}$$

1.2 A.2 Exact solution to the Dirichlet problem (second-kind TD-SIE on $\Gamma = \mathbb {S}^2$): $\frac{1}{2}\varphi +D\varphi =\tilde{f}(t)Y_{pq}$

The solution of Eq. (6) is given by

$$\begin{aligned} \psi _{pq}^{(\hat{D})}(t,\textbf{x})=-Y_{pq}(\widehat{\textbf{x}})\int _{0}^t\tilde{f}(t-\tau )\mathscr {L}^{-1}\left( \frac{1}{\tfrac{1}{2}+\lambda ^{(\hat{D})}_{p}(s)}\right) (\tau )d\tau , \end{aligned}$$

where

$$\begin{aligned}{} & {} \frac{1}{\tfrac{1}{2}+\lambda ^{(\hat{D})}_{p}(s)}\\= & {} \frac{(-1)^{p+1}2(\tfrac{sR}{c})^{2p+1}}{\chi _{p}(\tfrac{sR}{c})\big (\chi _{p+1}(-\tfrac{sR}{c})-p\chi _{p}(-\tfrac{sR}{c})\big )-\chi _{p}(\tfrac{sR}{c})\big (\chi _{p+1}(\tfrac{sR}{c})-p\chi _{p}(\tfrac{sR}{c})\big )e^{-2\tfrac{sR}{c}}}. \end{aligned}$$

For $p=0,1$, we obtain

$$\begin{aligned} \psi ^{(\hat{D})}_{00}(t,\textbf{x}\!= & {} \!Y_{00}(\widehat{\textbf{x}}) \left( \! -2\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }\sum _{j=0}^{\ell } (\!-\!1)^{\ell }\!\int _{\frac{2\ell R}{c}}^{t}{\omega _{\ell , j}}(\tfrac{c\tau }{R}-2\ell )^{j}e^{\tfrac{c\tau }{R}\!-2\ell }{\tilde{f}}'(t-\tau )\,d\tau \right) \!\textrm{with}\\ {\omega _{\ell ,j}}= & {} \begin{pmatrix}\ell \\ j\end{pmatrix}\frac{2^j}{j!}. \end{aligned}$$

Let set $F_{\ell }=\mathscr {L}^{-1}\left( \frac{\left( \tfrac{R}{c}s\right) ^3}{\tfrac{R}{c}s+1}\tfrac{\left( \left( \tfrac{R}{c}s\right) ^2+2\tfrac{R}{c}s+2\right) ^{\ell }}{\left( \left( \tfrac{R}{c}s\right) ^2-2\tfrac{R}{c}s+2\right) ^{\ell +1}}-1\right)$, then

$$\begin{aligned} \psi ^{(\hat{D})}_{1q}(t,\textbf{x})\!=\!Y_{1q}(\widehat{\textbf{x}}) \left( -2\sum _{\ell =0}^{\lfloor {\tfrac{ct}{2R}}\rfloor }\tilde{f}(t\!-\tfrac{2R}{c}\ell )\! -2\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }\int _{\frac{2\ell R}{c}}^{t}F_{\ell }(\tau \!-\tfrac{2R}{c}\ell ){\tilde{f}}(t-\tau )\,d\tau \right) . \end{aligned}$$

The functions $F_{\ell }$ involving the inverse Laplace transform (used above and below) are evaluated with established packages, such as MATLAB routines ilaplace and matlabFunction. The Dirac term in the integrals needs to be evaluated separately.

1.3 A.3 Exact solution to the impedance problem (second-kind TD-SIE on $\Gamma = \mathbb {S}^2$): $W\varphi := -\frac{1}{2}\varphi +D'\varphi -\frac{Z}{c}S(\partial _{t}\varphi )=-\tilde{g}(t)Y_{pq}$

The solution of Eq. (7) is given by

$$\begin{aligned} \varphi _{pq}^{(\hat{W})}(t,\textbf{x})=Y_{pq}(\widehat{\textbf{x}})\int _{0}^t\tilde{g}(t-\tau )\mathscr {L}^{-1}\left( \frac{1}{-\tfrac{1}{2}+\lambda ^{(\hat{D'})}_{p}(s)-\frac{ Z }{c}s\lambda ^{(\hat{S})}_{p}(s)}\right) (\tau )d\tau , \end{aligned}$$

where

$$\begin{aligned}{} & {} \frac{1}{\lambda ^{(\hat{W})}_{p}(s)}\\= & {} \frac{(-1)^{p+1}2(\tfrac{sR}{c})^{2p+1}}{\chi _{p}(-\tfrac{sR}{c})\big (\chi _{p+1}(\tfrac{sR}{c})-(p-\tfrac{ Z Rs}{c})\chi _{p}(\tfrac{sR}{c})\big )-\chi _{p}(\tfrac{sR}{c})\big (\chi _{p+1}(\tfrac{sR}{c})-(p-\tfrac{ Z Rs}{c})\chi _{p}(\tfrac{sR}{c})\big )e^{-2\tfrac{sR}{c}}}. \end{aligned}$$

For $p=0,1$, we obtain

$$\begin{aligned} \varphi ^{(\hat{W})}_{00}(t,\textbf{x})=Y_{00}(\widehat{\textbf{x}}) \Bigg ( -\frac{2}{1+ Z }\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }\tilde{g}(t-\tfrac{2\ell R}{c}) +\frac{2c}{(1+ Z )^2R}\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }\int _{\frac{2\ell R}{c}}^{t}e^{-\frac{(\tfrac{c\tau }{R}-2\ell )}{(1+ Z )}}\tilde{g}(t-\tau )\,d\tau \Bigg ). \end{aligned}$$

Let set $F_{\ell }=\mathscr {L}^{-1}\left( \frac{(\tfrac{R}{c}s)^3}{((1+ Z )(\tfrac{R}{c}s)^2+(2+ Z )\tfrac{R}{c}s+2)}\tfrac{\left( 1+\tfrac{R}{c}s\right) ^{\ell }}{\left( 1-\tfrac{R}{c}s\right) {\ell +1}}+\frac{(-1)^{\ell }}{1+ Z }\right)$, then

$$\begin{aligned} \begin{aligned} \varphi ^{(\hat{W})}_{1q}(t,\textbf{x})\!=\!Y_{1q}(\widehat{\textbf{x}})\times&\left( -\frac{2}{1+ Z }\sum _{\ell =0}^{\lfloor {\tfrac{ct}{2R}}\rfloor }(-1)^{\ell }\tilde{g}(t-\tfrac{2R}{c}\ell ) \!+2\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }\int _{\frac{2\ell R}{c}}^{t}F_{\ell }(\tau -\tfrac{2R}{c}\ell ){\tilde{g}}(t-\tau )\,d\tau \right) . \end{aligned} \end{aligned}$$

1.4 A.4 Exact solution to the Neumann problem (first-kind TD-SIE on $\Gamma = \mathbb {S}^2$): $N\varphi =\tilde{g}(t)Y_{pq}$

The solution of Eq. (8) is given by

$$\begin{aligned} \psi _{pq}^{(\hat{N})}(t,\textbf{x}=-Y_{pq}(\widehat{\textbf{x}})\int _{0}^t\tilde{g}(t-\tau )\mathscr {L}^{-1}\left( \frac{1}{\lambda ^{(\hat{N})}_{p}(s)}\right) (\tau )d\tau , \end{aligned}$$

where

$$\begin{aligned}{} & {} \frac{1}{\lambda ^{(\hat{N})}_{p}(s)}\\= & {} \frac{(-1)^{p}2R(\tfrac{R}{c}s)^{2p+1}}{\big (\chi _{p+1}(\tfrac{sR}{c})-p\chi _{p}(\tfrac{sR}{c})\big )\big (\chi _{p+1}(-\tfrac{sR}{c})-p\chi _{p}(-\tfrac{sR}{c})\big )-\big (\chi _{p+1}(\tfrac{sR}{c})-p\chi _{p}(\tfrac{sR}{c})\big )^2e^{-2\tfrac{sR}{c}}}. \end{aligned}$$

For $p=0,1$, we obtain

$$\begin{aligned} \psi ^{(\hat{N})}_{00}(t,\textbf{x})= & {} Y_{00}(\widehat{\textbf{x}}) \Bigg (2c\int _{0}^t{\text {cosh}}(\tfrac{c}{R}\tau ){\tilde{g}}(t-\tau )\,d\tau \\{} & {} -2R\sum _{\ell =1}^{\lfloor \frac{ct}{2R}\rfloor }\sum _{j=1}^{\ell } (-1)^{\ell +1}\int _{\frac{2\ell R}{c}}^{t}{\omega _{\ell j}^{(3)}}(\tfrac{c\tau }{R}-2\ell )^{j}e^{\tfrac{c\tau }{R}-2\ell }{\tilde{g}}'(t-\tau )\,d\tau \Bigg ). \end{aligned}$$

Let set $F_{\ell }=\mathscr {L}^{-1}\left( \frac{2(\tfrac{R}{c}s)^3}{\left( (\tfrac{R}{c}s)^2+2\tfrac{R}{c}s+2\right) }\frac{\left( (\tfrac{R}{c}s)^2+2\tfrac{R}{c}s+2\right) ^{\ell }}{\left( (\tfrac{R}{c}s)^2-2\tfrac{R}{c}s+2\right) ^{\ell +1}}\right)$, then

$$\begin{aligned} \begin{aligned} \psi ^{(\hat{N})}_{1q}(t,\textbf{x})=Y_{1q}(\widehat{\textbf{x}})&\left( 2R\sum _{\ell =0}^{\lfloor \frac{ct}{2R}\rfloor }\int _{\frac{2\ell R}{c}}^{t}F_{\ell }(\tau -\tfrac{2R}{c}\ell ){\tilde{g}}(t-\tau )\,d\tau \right) . \end{aligned} \end{aligned}$$

Appendix B: Multiple scatterer configurations: numerical experiments

1.1 B.1 Multiple scattering: Dirichlet TD wave propagation model

In addition to the five single obstacle TD scattering experiments in Section 4.2 comparing in Table 10 with established Dirichlet TD scattering benchmark results [15], we simulated counterparts of these experiments with five distinct multiple scatterers configurations each comprising two identical scatterers located at the positions ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,0,0)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,-3,0)$ for the ball and bean shapes, ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,1,0)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,-1,0)$ for the ellipsoid and ogive shapes and ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,0.5,0)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,-1.5,0)$ for the NASA almond shape. These Dirichlet problem multiple scattering configurations are shown in Fig. 14. As in Table 10, we report the corresponding multiple scattering simulation errors in Table 11, and we observe similar accuracies as with the single obstacle case. The near scattered field solutions for the multiple scattering Dirichlet models are visualized in Fig. 15.

Fig. 14

Geometrical configurations for the exterior Dirichlet problem. Top row (left to right): 2 $\times$ ball(1) & 2 $\times$ bean(1). Bottom row (left to right): 2 $\times$ ellipsoid(1,0.5,0.5), 2 $\times$ ogive(2.5), & 2 $\times$ Nasa_alm(2.484)

Table 11 Multiple TD scattering counterparts of Table 10 experiments on time interval [0,12.5]

Fig. 15

The near scattered field solution $u(t,\textbf{x}_{*})$, evaluated at $\textbf{x}_{*}=(2.5,0,0)$, to the wave equation with a Dirichlet boundary condition on the couples of spheres (top left), beans (top right), ellipsoids (bottom left), ogives (bottom middle) and the NASA almonds (bottom right) with $\sigma =0.1$ and incident field (32)

We are not aware of any multiple obstacle exterior TD scattering experiments in the literature with configurations comprising smooth or non-smooth convex/non-convex obstacles. Accordingly, results in this section (for Dirichlet, Neumann, and impedance TD scattering models) are expected to serve as a new class of benchmarks for such simulations.

In addition to several tabulated results, in this section, we also provide a gallery of visualizations of TD scattering obtained using a large number of spatial points exterior to ten distinct multiple scattering configurations (five for the Dirichlet, and five distinct ones for the Neumann and impedance boundary conditions). Snapshots at chosen discrete time steps can be seen by pausing the visualization movies to observe interesting scattering patterns that depend on the shape and conditions on the boundary of the configurations.

The following (mp4 format) movies (in the five hyperlinks) display the Dirichlet multiple scattering TD model exterior total field $u+u^{inc}_{\textsc {b}}$ evaluated in a rectangular box B for the case $\sigma =0.1$. The hyperlink names for the multiple scattering movies in this section are of the form $\mathrm {TotWaves\_xxx\_2yyy}$ with $\textrm{xxx}$ denoting the Dirichlet ($\textrm{Dir}$), Neumann ($\textrm{Neu}$), and Robin ($\textrm{Imp}$) boundary conditions, respectively, and $\textrm{yyy}$ denoting one of the five chosen shapes ($\textrm{sphere}$, $\textrm{ellipsoid}$, $\textrm{bean}$, $\textrm{ogive}$, $\mathrm {NASA~ almond}$).

	TotWaves_Dir_2spheres.mp4
	TotWaves_Dir_2ellipsoids.mp4
	TotWaves_Dir_2beans.mp4
	TotWaves_Dir_2ogives.mp4
	TotWaves_Dir_2NASAalmonds.mp4

The box B used for the movies in this section include the field visualization planes: Recall that the multiple scatterer configuration $\Omega$ is the union of two identical objects with the boundary $\Gamma =\Gamma _{\!1}\cup \Gamma _{2}$, where

$$\begin{aligned} \Gamma _{\!\ell }={\scriptstyle \varvec{\mathcal {O}}}_{_{\ell }}+\varvec{q}_{_{\ell }}(\mathbb {S}^2), \textrm{ for }\ell =1,2, \end{aligned}$$

and ${{{\scriptstyle \varvec{\mathcal {O}}}_{\ell }}}$ is the location of the $\ell$-th scatterer and the parametrizations $\varvec{q}_{_{\ell }}$ are $\mathscr {C}^1$-diffeomorphisms as described in Section 3.2.1. The total field is evaluated at points of the form $\textbf{x} = (x_1,x_2, x_3)$ on three different planes with either $x_1$ or $x_2$ fixed, and other two components varying in the (YZ or XZ) plane located inside a rectangular box $B=[a_1,a_2]\times [b_1,b_2]\times [d_1,d_2]$ and outside the volume $\Omega _{\varepsilon }$ delimited by the closed surfaces

$$\begin{aligned} \Gamma ^{\varepsilon }_{_{\!\ell }}={{\scriptstyle \varvec{\mathcal {O}}}_{\ell }}+(1+\varepsilon )\varvec{q}_{_{\ell }}(\mathbb {S}^2), \textrm{ for }\ell =1,2, \end{aligned}$$

for a given thickness $\varepsilon =5\%$. For the couple of ball and bean shaped objects located at the positions ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,0,0)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,-3,0)$, the associated box B is chosen as $[-2,2]\times [-4.5,1.5]\times [-2,2]$. For the couple of ellipsoid and ogive shaped objects located at the positions ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,-1,0)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,1,0)$ and for the couple of NASAalmond shaped objects located at the postions ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,-1.5,0)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,0.5,0)$, the associated box B is chosen as $[-2,2]^3$. In all cases, the three visualization planes are defined by the implicit equations $x_2={\scriptstyle {\mathcal {O}}}^2_{\ell =1}$, $x_2={\scriptstyle {\mathcal {O}}}^2_{\ell =2}$ and $x_1=0$.

1.2 B.2 Multiple scattering: Neumann and impedance TD wave propagation models

In this subsection we implement our algorithm for simulating multiple TD scattering wave propagation with Neumann and impedance boundary conditions by solving the TD-SIE Eq. (7) with $Z=0$ and $Z=1$, respectively, and using five distinct configurations shown in Fig. 16.

For these non-Dirichlet models, the incident plane wave is as defined in (33) with its incident direction induced by ${\varvec{d}}=(-0.2,0.1,-1)$ and with the values $\sigma =0.15$ and 0.07. The latter case near field solutions are shown in Figs. 17 and 18.

The associated ($\textrm{mp4}$ format) movies of our simulations, that demonstrate propagation of the total field induced by the incident Gaussian pulse (33) impinging on the 5 impenetrable configurations, for the case $\sigma =0.07$, are in the following 10 hyperlinks.

Fig. 16

Geometrical configurations for the exterior Neumann and impedance problems. From left side to right side, and top to bottom, visualization of $2\times$ball(1), $2\times$bean(1), $2\times$ellipsoid(1,0.5,0.5), $2\times$ogive(2.5), $2\times$Nasa_alm(2.484)

Fig. 17

The near scattered field solution $u(t,\textbf{x}_{*})$, evaluated at $\textbf{x}_{*}=(0,0,1)$, to the wave equation with a Neumann boundary condition on the couples of spheres (top left), beans (top right), ellipsoids (bottom left), ogives (bottom middle) and the NASA almonds (bottom right) with $\sigma =0.07$ and incident field (33) with ${\varvec{d}}=(-0.2,0.1,-1)$

Fig. 18

The near scattered field solution $u(t,\textbf{x}_{*})$ to the wave equation with an impedance boundary condition on the couples of spheres (top left) evaluated at $\textbf{x}_{*}=(-1.5,-2,1)$, beans (top right) evaluated at $\textbf{x}_{*}=(-1,-2,1)$, ellipsoids (bottom left), ogives (bottom middle) and the NASA almonds (bottom right) evaluated at $\textbf{x}_{*}=(0,1,0)$ with $\sigma =0.07$ and incident field (33) with ${\varvec{d}}=(-0.2,0.1,-1)$

TotWaves_Neu_2spheres.mp4	TotWaves_Imp_2spheres.mp4
TotWaves_Neu_2ellipsoids.mp4	TotWaves_Imp_2ellipsoids.mp4
TotWaves_Neu_2beans.mp4	TotWaves_Imp_2beans.mp4
TotWaves_Neu_2ogives.mp4	TotWaves_Imp_2ogives.mp4
TotWaves_Neu_2NASAalmonds.mp4	TotWaves_Imp_2NASAalmonds.mp4

For the couple of ball and bean shaped objects located at the positions ${\scriptstyle \varvec{\mathcal {O}}}_{1}=(0,-2,1)$ and ${\scriptstyle \varvec{\mathcal {O}}}_{2}=(0,1,-1)$, the three visualization planes are defined by the implicit equations $x_2={\scriptstyle {\mathcal {O}}}^2_{\ell =1}$, $x_2={\scriptstyle {\mathcal {O}}}^2_{\ell =2}$ and $x_1=0$. The associated box B is chosen as $[-2,2]\times [-4.5,1.5]\times [-2.5,2.5]$. For the three others couples of objects that are now aligned along the $x_1$-axis located at ${\scriptstyle \varvec{\mathcal {O}}}_1={}^T{(0,0,0)}$ and ${\scriptstyle \varvec{\mathcal {O}}}_2={}^T{(-3,0,0)}$, the three planes are defined by the implicit equations $x_1={\scriptstyle {\mathcal {O}}}^1_{\ell =1}$, $x_1={\scriptstyle {\mathcal {O}}}^1_{\ell =2}$ and $x_2=0$. The box B chosen for these cases is $[-4.5,1.5]\times [-2,2]^2$.

For all the above displayed movies we use the results obtained for the fixed time step $h=0.0625$ and spherical harmonics order $n_{_{\textsc {sh}}}=40$ meaning that the ratio is again $(hc)n_{_{\textsc {sh}}}=2.5$.

Table 12 Relative Euclidean errors (B.1) for multiple TD scattering with $t \in [0,5]$, and three boundary conditions

1.3 B.3 Multiple scattering space-time errors: Dirichlet, Neumann and impedance TD models

Next we consider space-time relative Euclidean norm errors, with $t\in [0, 5]$, for the numerical total field solutions for the exterior multiple TD scattering problems described in this Appendix. For the error computations, a total number of $N_{slice}$ ($>5000$) spatial sampling points $\textbf{x}_i, i = 1, \hdots , N_{slice}$ was chosen over the visualized three planes outside $\Omega _{\varepsilon }$. We then evaluate the relative Euclidean norm errors:

$$\begin{aligned} \textbf{Rerr}_{2}=\frac{\left( \sum \limits _{n=0}^{\lfloor \frac{5}{h} \rfloor }\sum \limits _{i=1}^{N_{slice}}|u^h_\textrm{num}(nh,\textbf{x}_i)-u^{h/\sqrt{2}}_\textrm{num}(nh,\textbf{x}_i)|^2\right) ^{\frac{1}{2}}}{\left( \sum \limits _{n=0}^{\lfloor \frac{5}{h}\rfloor }\sum \limits _{i=1}^{N_{slice}}|u^{h/\sqrt{2}}_\textrm{num}(nh,\textbf{x}_i)|^2\right) ^{\frac{1}{2}}}. \end{aligned}$$

(B.1)

Results in Table 12 demonstrate relative error accuracies of the simulated scattered field solutions of the TD multiple scattering models from multiple scatterer configurations (comprising smooth and non-smooth convex/non-convex obstacles with curved boundaries), and wide-band initial states.