A fourth-order Cartesian grid method for multiple acoustic scattering on closely packed obstacles

In this paper, we present a fourth-order Cartesian grid-based boundary integral method (BIM) for a multiple acoustic scattering problem on closely packed obstacles. We reformulate the exterior Helmholtz boundary value problems (BVPs) as a Fredholm boundary integral equation (BIE) of the second kind for some unknown density function. Unlike the traditional boundary integral method, a distinctive feature of our scheme is that we do not require quadratures and direct evaluations of nearly singular, singular or hyper-singular boundary integrals in the solution of BIEs. Instead, we reinterpret boundary integrals as solutions to equivalent simple interface problems in an extended rectangle domain, which can be solved efficiently by a fourth-order finite difference method coupled with numerical corrections, FFT based solution and interpolations. Extensive numerical experiments show that our method is formally high-order accurate, fast convergent and in particular insensitive to complexity of scatterers.