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We present a frequency/time hybrid integral-equation method for the time-dependent wave equation in two and three-dimensional spatial domains, possibly including closed cavities. 

We present a frequency/time hybrid integral-equation method for the time-dependent wave equation in two and three-dimensional spatial domains, possibly including closed cavities. 

We present a family of numerical methods for the solution of Maxwell’s equations, with application to simulation, optimization, and design. In particular, a novel rectangular-polar integral equation solver is mentioned which can produce solutions to the time harmonic Maxwell’s equations, with high order accuracy, for general 2D and 3D structures, with an extension to time domain problems on the basis of a time re-centering synthesis technique. An effective integral equation acceleration method, the IFGF method (Interpolated Factored Green Function), is used, which evaluates the action of Green function-based integral operators for an 𝑁𝑁-point surface discretization at a computational cost of 𝑂(𝑁log𝑁) operations without recourse to the FFT—thus, lending itself to effective distributed memory parallelization. Computational illustrations include applications to photonic optimization and design. 

This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a hybrid direct/iterative combination of 1)~A differential volumetric formulation (which is based on the use of appropriate Chebyshev differentiation matrices enacting the Laplace operator) and, 2)~A second-kind boundary integral formulation (which, once again, utilizes Chebyshev discretization, but, in this case, in the boundary-integral context). The approach enjoys low dispersion and high-order accuracy for smooth refractivities, as well as second-order accuracy (while maintaining low dispersion) in the discontinuous refractivity case. The solution approach proceeds by application of Impedance-to-Impedance (ItI) maps to couple the volumetric and boundary discretizations. The volumetric linear algebra solutions are obtained by means of a multifrontal solver, and the coupling with the boundary integral formulation is achieved via an application of the iterative linear-algebra solver GMRES. In particular, the existence and uniqueness theory presented in the present paper provides an affirmative answer to an open question concerning the existence of a uniquely solvable second-kind ItI-based formulation for the overall scattering problem under consideration. Relying on a modestly-demanding scatterer-dependent precomputation stage (requiring in practice a computing cost of the order of $$O(N^{\alpha})$$ operations, with $$\alpha \approx 1.07$$, for an $$N$$-point discretization \textcolor{black}{and for the relevant Chebyshev accuracy orders $$q$$ used)}, together with fast ($O(N)$-cost) single-core runs for each incident field considered, the proposed algorithm can effectively solve scattering problems for large and complex objects possibly containing discontinuities and strong refractivity contrasts. 

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensionalinterior spatial domains. The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a giveninteriortime-domain problem into a sequence oflimited-durationtime-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence ofHelmholtz frequency-domain problems; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth“Time-windowing and recentering”methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless,spectrally-accurate time evolutionfor given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.