Fast, numerically stable computation of oscillatory integrals with stationary points

We present a numerically stable way to compute oscillatory integrals of the form $\int{-1}^{1} f(x)e^{i\omega g(x)}dx$. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $\omega$ are derived for each fixed $x\geq 0$, which clarify the large $\omega$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$, we classify our discussion into three regimes, namely, $x=\mathcal{O}(1)$ or $x\gg1$, $0<x\ll 1$ and $x=0$. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $\omega$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.Comment: 32 pages, 6 figures, 4 table

An extended Filon--Clenshaw--Curtis method for high-frequency wave scattering problems in two dimensions

We study the efficient approximation of integrals involving Hankel functions of the first kind which arise in wave scattering problems on straight or convex polygonal boundaries. Filon methods have proved to be an effective way to approximate many types of highly oscillatory integrals, however finding such methods for integrals that involve non-linear oscillators and frequency-dependent singularities is subject to a significant amount of ongoing research. In this work, we demonstrate how Filon methods can be constructed for a class of integrals involving a Hankel function of the first kind. These methods allow the numerical approximation of the integral at uniform cost even when the frequency $\omega$ is large. In constructing these Filon methods we also provide a stable algorithm for computing the Chebyshev moments of the integral based on duality to spectral methods applied to a version of Bessel's equation. Our design for this algorithm has significant potential for further generalisations that would allow Filon methods to be constructed for a wide range of integrals involving special functions. These new extended Filon methods combine many favourable properties, including robustness in regard to the regularity of the integrand and fast approximation for large frequencies. As a consequence, they are of specific relevance to applications in wave scattering, and we show how they may be used in practice to assemble collocation matrices for wavelet-based collocation methods and for hybrid oscillatory approximation spaces in high-frequency wave scattering problems on convex polygonal shapes

Numerical aspects of enriched and high-order boundary element basis functions for Helmholtz problems.

In this thesis several aspects of the Partition of Unity Boundary Element Method (PUBEM) are investigated, with novel results in three main areas: 1. Enriched modelling of wave scattering from polygonal obstacles. The plane waves are augmented by a set of enrichment functions formed from fractional order Bessel functions, as informed by classical asymptotic solutions for wave fields in the vicinity of sharp corners. It is shown that the solution accuracy can be improved markedly by the addition of a very small number of these enrichment functions, with very little effect on the run time. 2. High-order formulations. Plane waves are not the only effective means of introducing oscillatory approximation spaces. High-Order Lagrange polynomials and high-order Non-Uniform Rational B-Splines (NURBS) also exhibit oscillation and these are tested and compared against PUBEM. It is found that these high-order functions significantly outperform the corresponding low-order (typically quadratic) polynomials and NURBS that are commonly used, and that for large problems the highest order tested (11th) has potential to be competitive with PUBEM without the associated ill-conditioning. 3. Integration. The accuracy of PUBEM traditionally comes at the cost of the requirement to evaluate many highly-oscillatory integrals. Several candidate integration strategies are investigated with the aim of find- ing a robust, accurate and efficient approach. Schemes tested include the Filon and asymptotic methods, as well as the Method of Stationary Phase (MSP). Although these schemes are found to be spectacularly successful for many cases, they fail for a sufficient number of situations to cause a complete PUBEM analysis based on these methods to lack robustness. Conclusions are drawn about the effective use of more traditional quadrature for robust implementations

Analytical and numerical techniques for wave scattering

In this thesis, we study the mathematical solution of wave scattering problems which describe the behaviour of waves incident on obstacles and are highly relevant to a raft of applications in the aerospace industry. The techniques considered in the present work can be broadly classed into two categories: analytically based methods which use special transforms and functions to provide a near-complete mathematical description of the scattering process, and numerical techniques which select an approximate solution from a general finite-dimensional space of possible candidates. The first part of this thesis addresses an analytical approach to the scattering of acoustic and vortical waves on an infinite periodic arrangement of finite-length flat blades in parallel mean flow. This geometry serves as an unwrapped model of the fan components in turbo-machinery. Our contributions include a novel semi-analytical solution based on the Wiener–Hopf technique that extends previous work by lifting the restriction that adjacent blades overlap, and a comprehensive study of the composition of the outgoing energy flux for acoustic wave scattering on this array of blades. These results provide an insight into the importance of energy conversion between the unsteady vorticity shed from the trailing edges of the cascade blades and the acoustic field. Furthermore, we show that the balance of incoming and outgoing energy fluxes of the unsteady field provides a convenient tool for understanding several interesting scattering symmetries on this geometry. In the second part of the thesis, we focus on numerical techniques based on the boundary integral method which allows us to write the governing equations for zero mean flow in the form of Fredholm integral equations. We study the solution of these integral equations using collocation methods for two-dimensional scatterers with smooth and Lipschitz boundaries. Our contributions are as follows: Firstly, we explore the extent to which least-squares oversampling can improve collocation. We provide rigorous analysis that proves guaranteed convergence for small amounts of oversampling and shows that superlinear oversampling can ensure faster asymptotic convergence rates of the method. Secondly, we examine the computation of the entries in the discrete linear system representing the continuous integral equation in collocation methods for hybrid numerical-asymptotic basis spaces on simple geometric shapes in the context of high-frequency wave scattering. This requires the computation of singular highly-oscillatory integrals and we develop efficient numerical methods that can compute these integrals at frequency-independent cost. Finally, we provide a general result that allows the construction of recurrences for the efficient computation of quadrature moments in a broad class of Filon quadrature methods, and we show how this framework can also be used to accelerate certain Levin quadrature methods.Supported by EPSRC grant EP/L016516/

Studies in numerical quadrature

Various types of quadrature formulae for oscillatory integrals are studied with a view to improving the accuracy of existing techniques. Concentration is directed towards the production of practical algorithms which facilitate the efficient evaluation of integrals of this type arising in applications. [Continues.

Boundary integral methods in high frequency scattering

In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources