A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications

Oscar P. Bruno, Leonid A. Kunyansky

Research output: Contribution to journalArticlepeer-review

190 Scopus citations


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 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 a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds.

Original languageEnglish (US)
Pages (from-to)80-110
Number of pages31
JournalJournal of Computational Physics
Issue number1
StatePublished - May 1 2001

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications'. Together they form a unique fingerprint.

Cite this