{"id":"c8bb0a4f-93af-4c1e-82d7-4b3ff11d676d","arxiv_id":"2507.05725","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A frequency-time hybrid wave solver is extended from two boundary patches to arbitrarily many, with a partition-of-unity multi-scattering scheme and new singularity treatment, enabling interior and exterior long-time simulations including open arcs.","lead":"A new numerical method solves two-dimensional wave scattering problems, inside cavities or around obstacles, by splitting the boundary into many overlapping pieces and summing the multiple reflections among the pieces. It promises long-time, high-accuracy simulations that run in parallel, for geometries that previously required costly fine meshes or many solver iterations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5 is conditional on the unproved Huygens Condition 2.3, which is load-bearing in Lemma 3.1, Lemma 3.4, and the boundary verification (3.23); if it fails for curved arcs, the central identity u=u_M collapses.","rationale":"The paper's main theoretical contribution is Theorem 3.5, and the proof chain from Lemma 3.1 through Lemma 3.4 to the boundary condition (3.23) invokes Condition 2.3 at every key step. The reader identified exactly this as the weakest assumption, and the manuscript itself concedes that Condition 2.3 is a conjecture for general curved arcs. I agree with that assessment. I do not recommend REJECT because the numerical experiments are extensive and consistent, and the condition is physically plausible: for an open arc, the field near an unilluminated portion should not appear before a wave from the illuminated set can travel there at speed c. Indeed, a finite-speed energy argument on the slit domain may turn the conjecture into a theorem, in which case Theorem 3.5 would be unconditional apart from the separate singularity conjecture in Section 4.3. But until such a proof is supplied, the central claim is rigorously established only conditional on Condition 2.3, so the reader's CONDITIONAL verdict is appropriate and no change is needed.","tokens_in":24965,"tokens_out":16173,"duration_ms":185995,"concrete_test":"Attempt an independent proof of Lemma 3.1 that does not invoke Condition 2.3: use standard finite-speed/domain-of-dependence energy estimates on the slit domain R^2\\Γ_j, with zero Dirichlet data on Γ_j^0 and causal data on Γ_j^tr. If the proof succeeds, Condition 2.3 is unnecessary for Theorem 3.5 and the main claim becomes unconditional, apart from the Section 4.3 singularity conjecture. If the proof reveals a counterexample near the endpoints for a circular or elliptical arc, then the time horizon T(M)=Mδmin/c must be revisited.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identity (3.22) depends on Lemma 3.1, which uses Condition 2.3 to assert that a solution of the open-arc problem (3.5) with data vanishing on Γ_j^0 stays zero on Γ\\Γ_j up to time T0+δmin/c (equation (3.7)). This lemma is then used in the inductive proof of Lemma 3.4, especially in establishing (3.18), and again in Theorem 3.5 to reduce u_M on the boundary to the algebraic relations (3.16)-(3.17). Condition 2.3 is explicitly conjectural: Section 2.3 states that it has been established only for a flat arc, demonstrated numerically for curved arcs in [22], and that 'throughout this paper Condition 2.3 is assumed to be valid.' Therefore the theorem is not a proof of correctness under the stated hypotheses; it is a reduction of correctness to an unproved physical hypothesis. The conjecture is plausible and likely provable by standard finite-speed/domain-of-dependence estimates on the slit domain, but as written the main claim is conditional. A secondary conjectural element, the singularity form (4.7) in Section 4.3, affects numerical accuracy rather than the theoretical identity (3.22).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25285,"tokens_out":14162,"duration_ms":155609,"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":[{"comment":"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.","section":"Section 2.3, Lemma 3.1, Lemma 3.4, Theorem 3.5"},{"comment":"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.","section":"Section 4.3"},{"comment":"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).","section":"Section 3.1 / Lemma 3.1"}],"minor_comments":[{"comment":"Please fix 'arbitrary of number' in the abstract; it should read 'arbitrary number'.","section":"Abstract"},{"comment":"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.","section":"Lemma 3.4"},{"comment":"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.","section":"Figure 11"},{"comment":"In the paragraph following (2.12), 'the time-domain solution U(x,t)' should be 'u(x,t)'.","section":"Section 2.2"},{"comment":"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.","section":"Equation (4.7)"},{"comment":"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.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is a solid methods contribution, and conditional on its two explicitly identified conjectures the mathematics is coherent. I recommend major revision rather than rejection because the unproved Huygens condition is disclosed rather than hidden; nevertheless it controls the central theorem and should be dealt with in the revision, either by proof for the geometric classes used in the numerics or by an unambiguous conditional framing. The singularity-structure conjecture should also be addressed, since the proposed CoV is the basis for the claimed high-order accuracy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a real extension of the two-patch FTH solver. The new pieces are the boundary POU that allows an arbitrary number of overlapping patches, the cosine change-of-variables treatment of the induced endpoint singularities, and the open-arc exterior capability. The induction proof of Theorem 3.5 is clear and, given Condition 2.3, it works: the boundary relations (3.16)-(3.17) plus finite-speed propagation give the identity u_M = u. I also see no fitted parameters or circular claims; the method builds on prior published algorithms, and the numerical results are strong, including long-time simulations and an explicit demonstration of GMRES iteration counts dropping with multi-patch decompositions.\n\nThe soft spot is exactly where the reader and stress-test put it. Condition 2.3 is conjectural, proven only for a flat arc and numerically demonstrated for curved arcs. It is load-bearing in Lemma 3.1 and again in Theorem 3.5. So the paper does not prove the correctness of the method under the stated hypotheses; it reduces correctness to a physically plausible but unproved domain-of-influence property. The authors are honest about this in Section 2.3, but the concluding remarks overstate by saying the result was established rigorously. The singularity form (4.7) is also conjectural, though it mainly affects numerical accuracy rather than the theoretical identity; the numerics suggest the CoV approach works.\n\nThese concerns are real but not disqualifying. Condition 2.3 is a natural finite-speed propagation property, likely provable for smooth curved arcs with the right domain-of-dependence estimates. The paper would be stronger if the authors proved it for a nontrivial curved case, or at minimum softened the language and stated Theorem 3.5 as conditional on an explicit conjecture. This is a fixable issue, not a fundamental flaw in the algorithm's design.\n\nThe paper deserves a serious referee. It will be useful to researchers working on frequency-time hybrid methods, boundary integral equations, and scattering in trapping geometries. I would bring it to a reading group and would probably cite it. My recommendation: send it to peer review, and ask the referee to focus on Condition 2.3 and on clarifying the theorem's conditional status.","headline":"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.","tokens_in":25745,"tokens_out":1294,"would_cite":true,"duration_ms":17002,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M38","65R20","35L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["wave equation","multiple scattering","frequency-time hybrid method","boundary integral equation","open arcs","Helmholtz equation","Huygens principle","time-windowing"],"falsifier":"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.","tokens_in":24782,"feed_emoji":"🌊","tokens_out":21022,"duration_ms":202421,"temperature":0.7,"pith_summary":"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.","feed_headline":"Solve trapped cavity waves by summing open-arc fields","feed_subtitle":"Resonant frequencies of the cavity never appear, so trapped waves stay computable for long times.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The two-patch ping-pong predecessor that this paper generalizes; it introduced the multiple-scattering strategy, the restricted Huygens Condition 2.3, and the previous endpoint-singularity technique that the new change-of-variables replaces.","marker":"[22]"},{"why":"The frequency-time hybrid (FTH) solver whose windowing-and-recentering and high-frequency Fourier transform algorithms solve every open-arc time-domain subproblem.","marker":"[6]"},{"why":"Supplies Theorem 2.1 (retarded-Green-function domain of influence, Proposition 3.6.2), the classical Huygens result that Condition 2.3 refines for propagation along the boundary.","marker":"[41]"},{"why":"Source of the spatio-temporal Sobolev spaces and the well-posedness result quoted as Theorem 2.4 for the interior problem, on which Theorem 3.5's data regularity rests.","marker":"[23]"},{"why":"Second regularity reference in Theorem 2.4, providing the corresponding well-posedness for the open-arc exterior problem used at each scattering round.","marker":"[51]"},{"why":"Establishes unique solvability of the open-arc single-layer boundary integral equation (3.31) at all real frequencies, the well-posedness foundation of every frequency-domain subproblem.","marker":"[43]"},{"why":"The time-decay result invoked in Remark 3.7 to argue that the number of significant multiple-scattering terms stays bounded as time grows, keeping per-time cost uniform.","marker":"[5]"},{"why":"Open-arc density singularity treatment ($\\psi = \\alpha/w$ with square-root endpoint factor) on which the new cosine-type change-of-variables handling builds.","marker":"[18]"},{"why":"Chebyshev-based rectangular-polar spectral discretization method into which the cosine change of variables (4.8) is inserted for the frequency-domain open-arc solves.","marker":"[21]"},{"why":"Rigorous establishment of the singularity form (4.7) at a Dirichlet-Neumann junction, the evidence cited for the conjectured induced-singularity structure on which the whole change-of-variables solve relies.","marker":"[2]"}],"fun_headline_variants":["Open arcs dodge cavity eigenvalues for long-time wave solving","Multiple-scattering solver bypasses trapped-mode failure in cavities","Wave solver splits cavity into open arcs to avoid resonant breakdown","New hybrid method keeps trapped waves computable without eigenvalues","Interior wave problems solved by open-arc decomposition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Open arcs dodge cavity eigenvalues for long-time wave solving","Multiple-scattering solver bypasses trapped-mode failure in cavities","Wave solver splits cavity into open arcs to avoid resonant breakdown","New hybrid method keeps trapped waves computable without eigenvalues","Interior wave problems solved by open-arc decomposition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000454,"raw_usage":{"total_tokens":2371,"prompt_tokens":1123,"completion_tokens":1248,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":1169}},"tokens_in":739,"tokens_out":1248,"duration_ms":9315,"temperature":1.0,"reasoning_tokens":1169,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:19:03.119661+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The two-patch ping-pong predecessor that this paper generalizes; it introduced the multiple-scattering strategy, the restricted Huygens Condition 2.3, and the previous endpoint-singularity technique that the new change-of-variables replaces."},{"cited_title":"fast-hybrid","cited_arxiv_id":null,"evidence_quote":"The frequency-time hybrid (FTH) solver whose windowing-and-recentering and high-frequency Fourier transform algorithms solve every open-arc time-domain subproblem."},{"cited_title":"Sayas, Retarded Potentials andc Time Domain Boundary Integral Equations, Springer Inter- national Publishing, 2016","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.1 (retarded-Green-function domain of influence, Proposition 3.6.2), the classical Huygens result that Condition 2.3 refines for propagation along the boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the spatio-temporal Sobolev spaces and the well-posedness result quoted as Theorem 2.4 for the interior problem, on which Theorem 3.5's data regularity rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Second regularity reference in Theorem 2.4, providing the corresponding well-posedness for the open-arc exterior problem used at each scattering round."},{"cited_title":"Stephan, W.L","cited_arxiv_id":null,"evidence_quote":"Establishes unique solvability of the open-arc single-layer boundary integral equation (3.31) at all real frequencies, the well-posedness foundation of every frequency-domain subproblem."},{"cited_title":"\"Bootstrap Domain of Dependence\": Bounds and Time Decay of Solutions of the Wave Equation","cited_arxiv_id":"2010.09002","evidence_quote":"The time-decay result invoked in Remark 3.7 to argue that the number of significant multiple-scattering terms stays bounded as time grows, keeping per-time cost uniform."},{"cited_title":"Bruno, S","cited_arxiv_id":null,"evidence_quote":"Open-arc density singularity treatment ($\\psi = \\alpha/w$ with square-root endpoint factor) on which the new cosine-type change-of-variables handling builds."},{"cited_title":"Bruno, T","cited_arxiv_id":null,"evidence_quote":"Chebyshev-based rectangular-polar spectral discretization method into which the cosine change of variables (4.8) is inserted for the frequency-domain open-arc solves."},{"cited_title":"Akhmetgaliyev and O","cited_arxiv_id":null,"evidence_quote":"Rigorous establishment of the singularity form (4.7) at a Dirichlet-Neumann junction, the evidence cited for the conjectured induced-singularity structure on which the whole change-of-variables solve relies."}],"review_version":1}