Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Poetry on Pre-Raphaelite principles: science, nature and knowledge in William Michael Rossetti’s 'Fancies at Leisure' and 'Mrs. Holmes Grey'

This article demonstrates how early Pre-Raphaelite poetry worked according to the principle that art should be modelled on science theorised by the Pre-Raphaelites in their early essays. As the main theorists (rather than practitioners) of Pre-Raphaelite art, F. G. Stephens and William Michael Rossetti defined the Pre-Raphaelite project in terms of observation, investigation, experiment, the “adherence to fact” and the “search after truth”. In the hands of the early Pre-Raphaelite poets, and particularly Rossetti himself, poetry too becomes a mode of scientific enquiry into the natural world, the nature of observation, human psychology and medical practice.