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

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 by Bruno and Yin [Math. Comp. 93 (2024), pp. 551–587], 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.