A high-order accurate, collocated boundary element method for wave propagation in layered media

The ultimate goal of this research is to construct a hybrid model for sound propagation in layered underwater environments with curved boundaries by employing a di↵erential formulation for inhomogeneous layers and a boundary integral formulation for homogeneous layers. The discretization of the new hybrid model is a combination of a finite di↵erence method for the Helmholtz equation for inhomogeneous media and a collocated boundary element method (BEM) for the integral equation for homogeneous media, while taking special care of the open boundaries and the common interface. Our focus is on sound wave propagation in layered piecewise homogeneous fluid media. A boundary integral formulation of the Helmholtz equation governing the acoustic pressure is employed. A fourth-order accurate, collocated BEM is constructed for this equation in a systematic way, facilitating its generalization to even higher orders of accuracy. A novel approach (for boundary element techniques) is proposed for modelling the open vertical boundaries. We introduce artificial nearand far-field boundaries and apply nonlocal radiation boundary conditions at these. The collocated BEM is implemented in Matlab and the numerical experiments show the expected convergence rate. A strong benefit of the collocated BEM is that only the boundary is discretized, thus, reducing the number of dimensions by one. By a comparison with a fourth-order finite di↵erence method (FD) it is illustrated that both methods have memory requirements of the same order, however, the number of unknowns in the collocated BEM is an order of magnitude less than in FD and the ratio grows with the problem size.