Surface scattering in three dimensions: an accelerated high–order solver

We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three–dimensional space. This algorithm evaluates scattered fields through fast, high–order, accurate solution of the corresponding boundary integral equation. The high–order accuracy of our solver is achieved through use of partitions of unityI together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of high–order equivalent source approximations, which allow for fast evaluation of non–adjacent interactions by means of the three–dimensional fast Fourier transform (FFT). Our acceleration scheme has dramatically lower memory requirements and yields much higher accuracy than existing FFT–accelerated techniques. The present algorithm computes one matrix–vector multiply in O(N6/5logN) to O(N4/3logN) operations (depending on the geometric characteristics of the scattering surface), it exhibits super–algebraic convergence, and it does not suffer from accuracy breakdowns of any kind. We demonstrate the efficiency of our method through a variety of examples. In particular, we show that the present algorithm can evaluate accurately, on a personal computer, scattering from bodies of acoustical sizes (ka) of several hundreds.