## Abstract

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 unity 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(N^{6/5} log N) to O(N^{4/3} log N) 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.

Original language | English (US) |
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Pages (from-to) | 2921-2934 |

Number of pages | 14 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 457 |

Issue number | 2016 |

DOIs | |

State | Published - Dec 8 2001 |

## Keywords

- Equivalent sources
- Fast Fourier transform
- Fast algorithm
- Integral equation
- Spherical wave expansion
- Wave scattering

## ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)