The Computation of Conical Diffraction Coefficients in High-Frequency Acoustic Wave Scattering

Bonner, Bradley David; Graham, Ivan G.; Smyshlyaev, V. P.

doi:10.1137/040603358

Cited by 21 publications

(13 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Their approach is to extract the far-field using properties of the Laplace-Beltrami operator on the unit sphere and the far-field is represented as a contour integral involving the spherical Green's function of the Laplace-Beltrami operator. Numerical evaluation of the far-field is not easy [10,11], and can be time-consuming. As noted by one of us [12], for a flat quarter plane computations are optimised using embedding formulae.…”

Section: Introductionmentioning

confidence: 99%

Embedding formulae for scattering by three-dimensional structures

2010

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

Embedding formulae for scattering by three-dimensional structures

2010

View full text Add to dashboard Cite

“…Extensions to more general scattering geometries may have to take account of more complicated asymptotics, such as diffraction from edges or corner points (see [6,8,9] and the references therein) or multiple scattering [11,12,20]. The idea of this paper is to show that k-robust numerical methods with a complete error analysis are possible, and to prepare the ground for further developments.…”

Section: Introductionmentioning

confidence: 99%

A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering

2007

Self Cite

View full text Add to dashboard Cite

We propose a new robust method for the computation of scattering of highfrequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L 2 , with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L 2 -coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k 1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k → ∞, for a fixed number of degrees

show abstract

“…Moreover the underlying asymptotic theory is much more challenging (e.g. [10] gives a numerical approach to a 3D "canonical problem" and contains extensive references to the asymptotic theory).…”

Section: Hybrid Approximation Spacesmentioning

confidence: 99%