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

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form ∫ 1 −1 f (s) exp(iks) ds. We use a Filon approach based on interpolating f at the classical Clenshaw-Curtis points cos(jπ/N), j = 0,. .. , N. The rule may be implemented in O(N log N) operations. We prove error estimates which show explicitly how the error depends both on the parameters k and N and on the Sobolev regularity of f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k → ∞. We also describe a method for implementating the method and prove its stability both when N ≤ k and N > k. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.

In this paper we propose and analyze composite Filon-Clenshaw-Curtis quadrature rules for integrals of the form Iwhere k ≥ 0, f may have integrable singularities, and g may have stationary points. Our composite rule is defined on a mesh with M subintervals and requires MN + 1 evaluations of f . It satisfies an error estimate of the formwhere r is determined by the strength of any singularity in f and the order of any stationary points in g and C N is a constant which is independent of k and M but depends on N .The regularity requirements on f and g are explicit in the error estimates. For fixed k, the rate of convergence of the rule as M → ∞ is the same as would be obtained if f was smooth. Moreover, the quadrature error decays at least as fast as k → ∞ as does the original integral I [a,b] k (f, g). For the case of nonlinear oscillators g, the algorithm requires the evaluation of g −1 at nonstationary points. Numerical results demonstrate the sharpness of the theory. An application to the implementation of boundary integral methods for the high-frequency Helmholtz equation is given.

We present a comparison between the performance of solvers based on Nyström discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems (1) the classical first kind integral equations for transmission problems [13], (2) the classical second kind integral equations for transmission problems [25], (3) the single integral equation formulations [21], and (4) certain direct counterparts of recently introduced Generalized Combined Source Integral Equations [4,5]. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established [13,36]. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nyström solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.

In this work we establish some new estimates for layer potentials of the acoustic wave equation in the time domain, and for their associated retarded integral operators. These estimates are proven using time-domain estimates based on theory of evolution equations and improve known estimates that use the Laplace transform.