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Multi-patch/multiple-scattering frequency-time hybrid solver for interior and exterior wave equation problems

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves, under a conjectured Huygens condition, that the wave trapped in a closed cavity equals a sum of multiple-scattering solutions off overlapping open arcs, for arbitrarily long times.

desk verdict Genuine N-patch generalization of the FTH solver with a clean induction proof, but the central identity rests on an unproved Huygens condition that the authors should either prove or clearly label as an assumption. read the letter →

arxiv 2507.05725 v1 pith:OUJ2KKOZ submitted 2025-07-08 math.NA cs.NA

Shuai Pan , Gang Bao , Tao Yin , Oscar P. Bruno This is my paper · ORCID
classification math.NAcs.NA MSC 65M3865R2035L05
keywords waveequationmultiplescatteringfrequency-timehybridmethodboundaryintegralopenarcsHelmholtzHuygensprincipletime-windowing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish one identity: the solution of the wave equation inside a closed two-dimensional cavity equals the multiple-scattering sum $u_M = \sum_{m=1}^M \sum_{j=1}^N v_{j,m}$ of solutions of scattering problems on $N$ overlapping open arcs that cover the boundary, valid up to any prescribed time provided enough scattering rounds are used (and the conjectured Huygens propagation condition holds). The payoff is that a trapped interior problem, whose frequency-domain Helmholtz version is not uniquely solvable at resonant frequencies, is decomposed into open-arc subproblems that are uniquely solvable at every real frequency. The same machinery also handles exterior scattering by groups of closed curves and open arcs, including open cavities that trap waves, and it removes the two-patch limitation of the earlier ping-pong method. If the identity is right, long-time trapped-wave simulations reduce to small, parallel boundary-integral solves with spectral time accuracy and no accumulating dispersion.

What carries the argument

The machinery is the recursive multiple-scattering sum together with the Huygens condition that makes it agree with the true field. The boundary $\Gamma$ is covered by $N$ overlapping open arcs $\Gamma_j$ carrying a smooth partition of unity $\chi_j$; the recursion starts with $g_{j,1} = -\chi_j u^i$ and forms each later datum $g_{j,m+1} = -\chi_j \sum_{k\neq j} \tilde{v}_{k,j,m}$ from the fields arriving from the other patches, with the overlap parts set to zero so the data stay continuous and vanish at the truncated-arc endpoints. Condition 2.3, the restricted Huygens domain-of-influence condition, proven only for a flat arc and numerically demonstrated for curved arcs, asserts that data on a sub-arc away from the endpoints produce no field on the rest of the curve before time $c^{-1}\,\mathrm{dist}(C_{\text{inc}}, \text{endpoints})$; Lemma 3.1 converts this into the statement that a field from one patch cannot reach another patch before $\delta_{\min}/c$, so each scattering round extends the exactness window by exactly $\delta_{\min}/c$, and Theorem 3.5 assembles the induction (Lemma 3.4) into the identity $u = u_M$. Numerically, each open-arc round is solved by the FTH machinery of windowing-and-recentering plus high-frequency Fourier transforms, and a cosine-type change of variables (4.8) turns the endpoint and overlap-induced density singularities into smooth unknowns.

What would settle it

Place a probe on a curved open arc and illuminate only a sub-arc strictly away from both endpoints; if the scattered field at the probe turns nonzero before the elapsed time reaches $c^{-1}$ times the distance from the probe to the illuminated sub-arc, Condition 2.3 is violated and the induction in Theorem 3.5 loses its footing. A ready-made check in the paper's own test suite: in a unit disc with plane-wave Dirichlet data, where the exact solution is known, compare $u_M$ pointwise with the exact field at times slightly below $T(M) = M\delta_{\min}/c$; with fine discretization, any departure above round-off before that horizon would refute the identity, while agreement there and divergence after would confirm the time-stepping mechanism.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 3.5, is a conditional identity: for any positive integer $M$, the multiple-scattering sum $u_M(x,t) = \sum_{m=1}^M \sum_{j=1}^N v_{j,m}(x,t)$ equals the unique solution $u$ of the interior Dirichlet wave problem on $D \times (-\infty, T(M)]$ with $T(M) = M\delta_{\min}/c$, provided the boundary patches satisfy the conjectured restricted Huygens condition (Condition 2.3) and the data have the regularity of Theorem 2.4. Each $v_{j,m}$ is itself a solution of the wave equation outside the single open arc $\Gamma_j$, with boundary data built recursively from the fields arriving from the other patches through a partition of unity; because each subproblem is reduced by Fourier transform to an open-arc Helmholtz problem and the open-arc single-layer integral equation is uniquely solvable at all real frequencies, the non-uniqueness that plagues interior Helmholtz problems at eigenvalues never enters. The paper states plainly that Condition 2.3 is assumed valid throughout, and it extends the same decomposition to exterior problems for clusters of closed curves and open arcs.

Load-bearing premise

The method rests on the conjectured Huygens condition that boundary data on one part of an arc cannot produce a field on the rest of the arc, or on a neighbouring arc, before the time it takes sound to cross the separating distance, and the paper assumes this holds for general curved arcs even though it is proven only for a straight arc.

Editorial extensions

If this is right

  • Interior cavities, however trapping, can be simulated by solving only open-arc Helmholtz equations, which are uniquely solvable at every real frequency; the interior resonance frequencies of the cavity never appear in the computation.
  • For any target final time $T$, choosing $M$ with $M\delta_{\min}/c \geq T$ makes the multiple-scattering sum equal to the exact solution on the whole simulation interval, and the numerical tests show the error decreasing rapidly as $M$ grows.
  • The number of patches $N$ is a free parameter: the paper demonstrates $N = 3$ and $N = 6$ for a disk, $N = 4$ for an H-shaped cavity, and $N = 6$ and $N = 10$ for open-arc cavities, and shows that decomposing a trapping circular cavity into six patches reduces GMRES iteration counts.
  • Because every patch solve within a scattering round is independent, the method parallelizes embarrassingly across patches and time windows; for non-trapping patches, Remark 3.7 argues that only finitely many multiple-scattering terms remain significant as $t$ grows, so the per-time cost stays bounded.
  • The exterior version handles collections of closed curves and open arcs alike, including a highly trapping open circular cavity and a rocket-like open cavity, a capability the previous two-patch method lacked because its endpoint-singularity handling excluded open-arc scatterers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not draw the trade-off consequence of its own formula: since the exactness horizon per round is $\delta_{\min}/c$, the number of scattering rounds needed for a fixed final time is $cT/\delta_{\min}$, so choosing patch overlaps tunes the balance between the number of frequency-domain solves and the difficulty of each solve.
  • The foundation is narrow in a specific way the authors acknowledge but do not quantify: Condition 2.3 is proven only for a flat arc, so the entire multi-patch identity currently rests on that single calculation plus curved-arc numerics; a counterexample search on a strongly concave or non-smooth arc would directly test the method's scope.
  • The reduction to open-arc Helmholtz problems is not acoustic-specific: any linear wave phenomenon with finite propagation speed, a Fourier-reducible frequency problem, and uniquely solvable open-boundary integral equations fits the same template, which is the natural reading of the authors' stated plan to extend to elasticity, electromagnetics, and layered media.
  • A separation the numerical section cannot itself establish is that the identity $u_M = u$ and the conjectured singularity form (4.7) are logically independent, so the observed accuracy of the solver is joint evidence for both; if (4.7) failed for some geometry, the scheme's high-order accuracy would break while the theorem could still hold.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper presents a multi-patch, multiple-scattering frequency-time hybrid (FTH-MS) integral equation method for the two-dimensional wave equation in interior and exterior domains. The boundary is partitioned into overlapping open arcs Γj with a smooth partition of unity; the Dirichlet problem is decomposed into a sequence of exterior open-arc wave problems, each solved in the frequency domain by uniquely solvable single-layer (or combined-field) integral equations. The main correctness result, Theorem 3.5, asserts that the M-th multiple-scattering sum u_M equals the exact solution u on D×(-∞,T(M)] with T(M)=M δ_min/c, provided the patches satisfy the conjectured restricted Huygens condition (Condition 2.3). The paper also describes a smoothing change of variables for endpoint and induced density singularities, discusses parallelization, and reports numerical experiments for discs, H-shaped cavities, multiple closed obstacles, open arcs, and highly trapping cavities.

Significance. If Condition 2.3 is granted, the paper provides a transparent reduction of trapped interior wave problems to a sequence of uniquely solvable open-arc Helmholtz problems. The induction proof of Lemma 3.4 and the boundary verification in Theorem 3.5 are clean and do not involve fitted parameters or circular normalizations; the method generalizes the earlier two-patch solver [22] to arbitrary patch numbers, handles open-arc scatterers, and is naturally parallel. The numerical examples are broad and support the claimed accuracy and long-time stability. The main limitation is that the central theoretical identity is conditional on an unproved conjecture (Condition 2.3), and the high-order numerical treatment relies on a conjectural singularity structure (4.7). These two points are the principal barriers to accepting the paper in its present form.

major comments (3)
  1. [Section 2.3, Lemma 3.1, Lemma 3.4, Theorem 3.5] Condition 2.3 is explicitly conjectural ('The proof is left for future work') and is assumed throughout the paper, yet it is load-bearing: Lemma 3.1 uses it to obtain (3.7), the inductive proof of Lemma 3.4 uses Lemma 3.1 to establish (3.18), and Theorem 3.5 uses those relations to verify the boundary condition (3.23). If Condition 2.3 fails for a curved or closed arc, the identity u=u_M on D×(-∞,T(M)] is not established. I therefore request that the manuscript either (a) prove Condition 2.3 for the classes of arcs used in Section 5, or (b) state Theorem 3.5 with Condition 2.3 as an explicit hypothesis and revise the concluding claim in Section 6 that the identity has been 'established rigorously' so that the conditional character is transparent. The induction argument itself appears sound conditional on this hypothesis.
  2. [Section 4.3] The change-of-variables (4.8) and the resulting claim of high-order accuracy for open-arc FTH-MS problems rest on the assertion (4.7) that every endpoint and induced singularity of the densities Ψj,m,q has the form f(d_p^{1/2})/d_p^{1/2} with f smooth. The paper states that this is preliminary analysis, that a proof is an open question, and that the only evidence is numerical smoothness of eΨj,m,q for one configuration. Since the CoV is used in all numerical examples of Section 5, the numerical accuracy claims are conditional on this unproved singularity structure. Please either prove (4.7) for the relevant settings or add resolution/convergence studies that directly confirm the predicted order for the actual FTH-MS densities, and state the assumption explicitly wherever the CoV is used.
  3. [Section 3.1 / Lemma 3.1] Lemma 3.1 is stated without proof, but it is the bridge between Condition 2.3 and the vanishing statements used in Lemma 3.4. Because Condition 2.3 is formulated with a time restriction t ≤ c^{-1} dist(Cinc,{e1,e2}), a time-shift argument is needed to obtain (3.7) for general T0. Please include a concise proof of Lemma 3.1 (or state it as following from the unrestricted form of Condition 2.3, if that is intended).
minor comments (6)
  1. [Abstract] Please fix 'arbitrary of number' in the abstract; it should read 'arbitrary number'.
  2. [Lemma 3.4] In the proof around (3.20), the notation v_{k,j,L} is used where the definition (3.12) gives ev_{k,j,L}; align the notation.
  3. [Figure 11] The caption refers to incident field u_i^1 but the text in Section 5 (Geometry 3) describes results for u_i^2; please correct.
  4. [Section 2.2] In the paragraph following (2.12), 'the time-domain solution U(x,t)' should be 'u(x,t)'.
  5. [Equation (4.7)] The display of (4.7) is difficult to read in the present formatting; please typeset the fraction properly and state explicitly that d_p(x) is the Euclidean distance from x to the endpoint p.
  6. [Section 3.3] The exterior extension is described as following with 'minimal modifications', but no formal analogue of Theorem 3.5 is stated for E=De; please state the exact multiple-scattering identity and its hypotheses for the exterior case, or explicitly mark it as a conjecture.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the FTH-MS sum identity follows from the recursive definitions and the explicit Huygens condition; the main caveat is the unproved, self-authored Condition 2.3, which is an explicit assumption rather than a fitted or self-referential input.

full rationale

The derivation chain is self-contained in the sense relevant to circularity. Definition 3.2 recursively builds the boundary data g_{j,m+1} from the previous open-arc iterates v_{k,m}; Lemma 3.4 proves the relations (3.16)-(3.18) by induction using only the partition-of-unity property (3.1)-(3.2), the boundary conditions (3.9), and the truncation rule (3.12). Theorem 3.5 combines these algebraic identities with Lemma 3.1, which is a geometric specialization of Condition 2.3, to verify the boundary equation (3.23); uniqueness of the interior wave-equation problem then gives (3.22). No parameter is fitted to the target solution, and the numerical accuracy claims are checked against exact solutions u=-u_i (Geometries 1 and 3) or independent fine-discretization references, so the predictions are not self-confirming. The only caveat touching the theorem's logical status is Condition 2.3 itself: Section 2.3 states that it is conjectural, was proved only for a flat arc, was numerically demonstrated for curved arcs in [22], and that 'throughout this paper Condition 2.3 is assumed to be valid.' This hypothesis is genuinely load-bearing, since Lemma 3.1, the inductive step (3.18), and the boundary verification (3.23) all use it. However, this is a correctness/rigor gap rather than circularity: the paper does not define the target solution in terms of the condition, does not fit u_M to u, and does not cite [22] as a proof of Condition 2.3—it explicitly labels the condition as an assumption. The central multiple-scattering algebra remains independent of any fitted or constructed input. Score 1 reflects the self-authored, conjectural nature of the unproved condition while the core derivation is not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants enter the proof of Theorem 3.5; user-chosen values such as c0=1/3, c1=2/3, H=10 and frequency grids affect accuracy and runtime but are not fitted to reproduce the reference solution. The central derivation depends on the conjectural Huygens Condition 2.3, the conjectural singularity form (4.7), and a decay assertion without proof.

assumptions (4)
  • domain assumption Condition 2.3 (restricted Huygens domain-of-influence): for open/closed Lipschitz curves, solutions propagate along the boundary at sound speed, so data supported on Cinc produce zero field before t <= c^-1 dist(Cinc, endpoints).
    Stated as a conjecture in Section 2.3; the paper says 'throughout this paper Condition 2.3 is assumed to be valid'. Proven only for flat arcs, demonstrated numerically for curved arcs in [22]. Lemma 3.1 and Theorem 3.5 rely on it directly.
  • ad hoc to paper Singularity structure (4.7): all endpoint and induced endpoint singularities of open-arc densities take the form f(d_p^{1/2}) / d_p^{1/2} with f smooth.
    Section 4.3 states a rigorous proof 'remains an open question' and the paper relies on this form to design the cosine change-of-variables used in all numerical examples; if false, the discretization loses accuracy.
  • domain assumption Rapid decay of multiple-scattering iterates vj,m on non-trapping patches, keeping the number of active terms bounded as time grows.
    Remark 3.7 asserts this based on 'integration by parts arguments similar to [5]' and an 'in preparation' reference [7]; no proof is given. It justifies the complexity claim that cost per unit time is uniformly bounded.
  • standard math Standard time-domain well-posedness and regularity of wave problems on closed curves and open arcs (Theorem 2.4).
    Taken from [23,51]; standard in time-domain boundary integral equation literature; used to set function spaces for boundary data and solutions in Section 2.4.

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Pith. "Pith review of Multi-patch/multiple-scattering frequency-time hybrid solver for interior and exterior wave equation problems." pith.science (2026). https://pith.science/paper/OUJ2KKOZ

@misc{pith2026250705725,
  author       = {Pith},
  title        = {Pith review of: Multi-patch/multiple-scattering frequency-time hybrid solver for interior and exterior wave equation problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUJ2KKOZ}},
  note         = {Machine review of arXiv:2507.05725}
}
read the original abstract

This paper proposes a new multiple-scattering frequency-time hybrid (FTH-MS) integral equation solver for problems of wave scattering by obstacles in two dimensional space, including interior problems in closed cavities and problems exterior to a set of disconnected open or closed scattering obstacles. The multiple-scattering FTH-MS method is based on a partition of the domain boundary into a user-prescribed set of overlapping open arcs, along with a corresponding sequence of multiple-scattering problems that effectively decompose the interior problem into a series of open-arc wave equation subproblems. The new strategy provides a significant extension of the original FTH-MS algorithm originally presented in [22], in that (1) By allowing for use of an arbitrary of number of component arcs, and not just two as in the previous contribution, the new approach affords (1a) A significantly increased geometric flexibility, as well as, (1b) The use of partitions for which each open arc leads to small numbers of iterations if iterative linear-algebra solvers are employed; and, (2) It facilitates parallelization -- as the subproblem solutions that are needed at each multiple scattering step can be evaluated in an embarrassingly parallel fashion. Utilizing a suitably-implemented Fourier transformation, each sub-problem is reduced to a Helmholtz frequency-domain problem that is tackled via a uniquely-solvable boundary integral equation. Similar FTH-MS methods are also presented for problems exterior to a number of bounded obstacles. All of the algorithms considered incorporate the previously introduced ``time-windowing and recentering'' methodology (that enables both treatment of incident signals of long duration and long time simulation), as well as a high-frequency Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for arbitrarily long times.

Figures

Figures reproduced from arXiv: 2507.05725 by the authors.

Figure 1
Figure 1. Illustration of the overlapping-patch decomposition of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Notations used for the boundary partitioning (a) and windowing functions (b). [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Singularities of the functions Gj,m,q and Ψj,m,q on the arc Γj = Γov j−1,j ∪ Γ tov j ∪ Γ ov j,j+1. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Comparison of the solutions Ψj,m,q and Ψej,m,q (normalized values). the proposed numerical implementation, and, in particular, are used in all of the numerical examples presented in Section 5. Depending on the location of the singularities of the function Ψj,m,q within…
Figure 5
Figure 5. Figure 5: Scattered field and errors obtained for the problem considered in Geometry 1 with [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Scattered field and errors obtained for the problem considered in Example 1 with [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Long-time propagation illustration. Real parts of the numerical and exact solutions at [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Real part of the total field u tot for the problems of scattering associated with the Geometry 2, with point source u i 1 located at z0 = (0, 0)⊤. Upper row:w0 = 15, Lower row: w0 = 50. Fields at time t = 4, 6, 8 and 10 are displayed from left to right in each row. Geo…
Figure 9
Figure 9. Figure 9: Real part of the scattered field (b) at x = (−1, 1)⊤ resulting from the incident field u i 1 with z0 = (0, 2)⊤ and numerical errors (c) as functions of time t for various values of M obtained for the exterior wave equation problem with 3 bounded obstacles (a) considere…
Figure 10
Figure 10. Figure 10: Real part of the scattered field (b) at x = (2, 2)⊤ resulting from the incident field u i 1 with z0 = (0, 0)⊤ and numerical errors (c) as functions of time t for various values of M obtained for the exterior wave equation problem with 9 bounded obstacles (a) considere…
Figure 11
Figure 11. Figure 11: Real part of the total fields for the scattering of plane incidence [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Real part of the scattered field (b) at x = (−1, 0)⊤ resulting from the incident field u i 2 and numerical errors (c) as functions of time t for various values of M obtained for the exterior wave equation problem with 3 open-arcs (a) considered in Geometry 4 [PITH_FU…
Figure 13
Figure 13. Figure 13: Real part of the total fields for the scattering of plane wave [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Real part of the scattered field (a) at x = (0.5, 0)⊤ resulting from the incident field u i 2 with θ inc = −π. and numerical errors (b) as functions of time t for various values of M. 6 Concluding remarks In significant extension of a previous two-patch interior multi…
Figure 15
Figure 15. Figure 15: Real part of the total fields for the scattering of plane wave [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: GMRES iteration numbers for different frequencies while solving the integral equation (3.31) [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: Real part of the total fields for the scattering of plane wave [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]

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Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [22]

    O.P.Bruno, T.Yin, Multiple-scatteringfrequency-timehybridsolverforthewaveequationininterior domains, Math. Comput. 93(346) (2024) 551-587

    work page 2024

  2. [1]

    A. Aimi, M. Diligenti, C. Guardasoni, On the energetic Galerkin boundary element method applied to interior wave propagation problems, J. Comput. Appl. Math. 235(7) (2011) 1746-1754

    work page 2011

  3. [2]

    Akhmetgaliyev and O

    E. Akhmetgaliyev and O. Bruno, Regularized integral formulation of mixed Dirichlet-Neumann prob- lems, Journal of Integral Equations and applications 29 (2017) 493-529

    work page 2017

  4. [3]

    Alpert, L

    B. Alpert, L. Greengard, T. Hagstrom, Nonreflecting boundary conditions for the time-dependent wave equation, J. Comput. Phys. 180 (2002) 270-296

    work page 2002

  5. [4]

    Amlani, O

    F. Amlani, O. P. Bruno, An FC-based spectral solver for elastodynamic problems in general three- dimensional domains, J. Comput. Phys. 307 (2016) 333-354

    work page 2016

  6. [5]

    "Bootstrap Domain of Dependence": Bounds and Time Decay of Solutions of the Wave Equation

    T.G. Anderson and O. P. Bruno, “Bootstrap Domain of Dependence”: Bounds and Time Decay of Solutions of the Wave Equation, arXiv preprint arXiv:2010.09002 (2022)

    work page Pith review arXiv 2022

  7. [6]

    fast-hybrid

    T.G. Anderson, O. P. Bruno, M. Lyon, High-order, dispersionless “fast-hybrid” wave equation solver. Part I: O(1) sampling cost via incident-field windowing and recentering, SIAM J. Sci. Comput. 422 (2020) A1348-A1379

    work page 2020

  8. [7]

    T.G.Anderson, O.P.Bruno, M.Lyon, FastHybridwaveequationsolverPartII:3Dwindowtracking, 2D asymptotics tracking, general incident fields and remote field evaluation, In preparation

    work page

Show all 51 references
  1. [8]

    Babuška, S.A

    I.M. Babuška, S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34 (1997) 2392-2423. 25

    1997

  2. [9]

    Bamberger, T

    A. Bamberger, T. Ha-Duong, Formulation variationelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique, Math. Methods Appl. Sci. 8 (1986) 405-435

    1986

  3. [10]

    Banjai, M

    L. Banjai, M. Kachanovska, Fast convolution quadrature for the wave equation in three dimensions, J. Comput. Phy. 279 (2014) 103-126

    2014

  4. [11]

    Bansal, A

    P. Bansal, A. Moiola, I. Perugia, C. Schwab, Space-time discontinuous Galerkin approximation of acoustic waves with point singularities, IMA J. Numer. Anal. 41 (2021) 2056-2109

    2021

  5. [12]

    Barnett, L

    A.H. Barnett, L. Greengard, T. Hagstrom, High-order discretization of a stable time-domain integral equation for 3D acoustic scattering, J. Comput. Phy. 402 (2020) 109047

    2020

  6. [13]

    Interpolated Factored Green Function

    C. Bauinger, O.P. Bruno, “Interpolated Factored Green Function” method for accelerated solution of scattering problems, J. Comput. Phy. 430 (2021) 110095

    2021

  7. [14]

    Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J

    J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phy. 114 (1994) 185-200

    1994

  8. [15]

    Bruno, M

    O.P. Bruno, M. Lyon, C. Pérez-Arancibia, C. Turc, Windowed Green function method for layered- media scattering, SIAM J. Appl. Math. 76(5) (2016) 1871-1898

    2016

  9. [16]

    Bruno, E

    O.P. Bruno, E. Garza, A Chebyshev-based rectangular-polar integral solver for scattering by general geometries described by non-overlapping patches, J. Comput. Phys. 421 (2020) 109740

    2020

  10. [17]

    Bruno, E

    O.P. Bruno, E. Garza, C. Pérez-Arancibia, Windowed Green function method for nonuniform open- waveguide problems, IEEE Trans. Antenn. Propag. 65 (2017) 4684-4692

    2017

  11. [18]

    Bruno, S

    O.P. Bruno, S. Lintner, Second-kind integral solvers for TE and TM problems of diffraction by open arcs, Radio Sci. 47 (6) (2012)

    2012

  12. [19]

    Bruno, M

    O.P. Bruno, M. Lyon, High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements, J. Comput. Phys. 229 (6) (2010) 2009-2033

    2010

  13. [20]

    Bruno, T

    O.P. Bruno, T. Yin, Regularized integral equation methods for the elastic scattering problems in three dimensions, J. Comput. Phys. 410 (2020) 109350

    2020

  14. [21]

    Bruno, T

    O.P. Bruno, T. Yin, A windowed Green function method for elastic scattering problems on a half- space, Comput. Methods Appl. Mech. Engrg. 376 (2021) 113651

    2021

  15. [23]

    Q. Chen, H. Haddar, A. Lechleiter, P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems 8 (2010) 85001-85017

    2010

  16. [24]

    Colton and R

    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Berlin, Springer, 1998

    1998

  17. [25]

    Costabel, M

    M. Costabel, M. Dauge, R. Duduchava, Asymptotics without logarithmic terms for crack problems, Commun. Partial Differ. Equ. 28 (2003) 869-926

    2003

  18. [26]

    Douglas, J.E

    J. Douglas, J.E. Santos, D. Sheen, L.S. Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci. 3 (1993) 171-194

    1993

  19. [27]

    Ainsworth et al., eds., Springer Berlin Heidelberg, Berlin, Heidelberg, 2003, pp

    T.Ha-Duong, Onretardedpotentialboundaryintegralequationsandtheirdiscretisation, inTopicsin Computational Wave Propagation: Direct and Inverse Problems, M. Ainsworth et al., eds., Springer Berlin Heidelberg, Berlin, Heidelberg, 2003, pp. 301–336. 26

    2003

  20. [28]

    Engquist, A

    B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977) 629–629

    1977

  21. [29]

    Evans, Partial Differential Equations (2nd edition), Graduate Studies in Mathematics, Vol

    L.C. Evans, Partial Differential Equations (2nd edition), Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 2010

    2010

  22. [30]

    French, T.E

    D.A. French, T.E. Peterson, A continuous space-time finite element method for the wave equation, Math. Comp. 65 (1996) 491-506

    1996

  23. [31]

    Grote, A

    M.J. Grote, A. Schneebeli, D. Schötzau, Dicontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal. 44 (2006) 2408-2431

    2006

  24. [32]

    Hassell, T

    M. Hassell, T. Qiu, T. Sánchez-Vizuet, F.-J. Sayas, A new and improved analysis of the time domain boundary integral operators for acoustics, J. Integral Equations Applications 29(1) (2017) 107-136

    2017

  25. [33]

    Labarca, R

    I. Labarca, R. Hiptmair, Acoustic scattering problems with convolution quadrature and the method of fundamental solutions, Commun. Comput. Phys. 30 (2021) 985-1008

    2021

  26. [34]

    Laliena, F.-J

    A.R. Laliena, F.-J. Sayas, Theoretical aspects of the application of convolution quadrature to scat- tering of acoustic waves, Numer. Math. 112 (2009) 637-678

    2009

  27. [35]

    Liu, Fast Multipole Boundary Element Method, Cambridge University Press, New York, 2009

    Y. Liu, Fast Multipole Boundary Element Method, Cambridge University Press, New York, 2009

    2009

  28. [36]

    Löscher, O

    R. Löscher, O. Steinbach, M. Zank, Numerical results for an unconditionally stable space-time finite element method for the wave equation, Domain Decomposition Methods in Science and Engineering XXVI, 625-632

  29. [37]

    Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer

    C. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math. 67 (1994) 365-389

    1994

  30. [38]

    R. C. MacCamy, Low frequency acoustic oscillations, Q. Appl. Math. 23 (1965) 247-255

    1965

  31. [39]

    E.Mecocci, L.Misici, M.C.Recchioni, F.Zirilli, Anewformalismfortime-dependentwavescattering from a bounded obstacle, J. Acoust. Soc. Am. 107 (2000) 1825-1840

    2000

  32. [40]

    J. M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation, SIAM J. Numer. Anal. 49 (2011) 1210–1243

    2011

  33. [41]

    Sayas, Retarded Potentials andc Time Domain Boundary Integral Equations, Springer Inter- national Publishing, 2016

    F.-J. Sayas, Retarded Potentials andc Time Domain Boundary Integral Equations, Springer Inter- national Publishing, 2016

    2016

  34. [42]

    Shen and T

    J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Beijing Science Press, 2006

    2006

  35. [43]

    Stephan, W.L

    E.P. Stephan, W.L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Anal. 3 (1984) 183-219

    1984

  36. [44]

    Steinbach, C

    O. Steinbach, C. Urzúa-Torres, A new approach to space-time boundary integral equations for the wave equation, SIAM J. Math. Anal. 54(2) (2022) 1370-1392

    2022

  37. [45]

    Steinbach, C

    O. Steinbach, C. Urzúa-Torres, M. Zank, Towards coercive boundary element methods for the wave equation, arXiv:2106.01646

    arXiv

  38. [46]

    Stratton, Electromagnetic theory, McGraw-Hill, New York, 1941

    J.A. Stratton, Electromagnetic theory, McGraw-Hill, New York, 1941

    1941

  39. [47]

    Z. Sun, Y. Xing, Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes, Math. Comput. 90 (2021) 1741-1772. 27

    2021

  40. [48]

    Taflove, Computational electrodynamics: the finite-difference time-domain method, Artech House, Boston, 2000

    A. Taflove, Computational electrodynamics: the finite-difference time-domain method, Artech House, Boston, 2000

    2000

  41. [49]

    Werner, Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces, Math

    P. Werner, Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces, Math. Meth. Appl. Sci. 8 (1986): 134-156

    1986

  42. [50]

    Xing, C.-S

    Y. Xing, C.-S. Chou, C.-W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Probl. Imag. 7 (2013) 967-986

    2013

  43. [51]

    Y. Yue, F. Ma, B. Chen, Time domain linear sampling method for inverse scattering problems with cracks, E. Asian J. Appl. Math. 12(1) (2022) 96-110. 28

    2022

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