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

Efficient computation of highly oscillatory integrals by using QTT tensor approximation

We propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter $\omega \geq 0$, typically varying in a large interval. Our approach is based, for fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain $\omega$-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals and are itself oscillatory which makes them both difficult to evaluate and to approximate. Here we use the quantized-tensor train (QTT) approximation method for functional $m$-vectors of logarithmic complexity in $m$ in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.Comment: 20 page

Asymptotic and numerical methods for high-frequency scattering problems

This thesis is concerned with the development, analysis and implementation of efficient and accurate numerical methods for solving high-frequency acoustic scattering problems. Classical boundary or finite element methods that are based on approximating the solution by polynomials can be effective for small and moderate frequencies. However, as the frequency increases, the solution to the scattering problem becomes more oscillatory and classical numerical methods cope very badly with high oscillation. For example, for two-dimensional scattering problems, classical numerical methods require their number of degrees of freedom to grow at least linearly with frequency to capture the oscillatory behaviour of the solution accurately. Therefore, at large frequencies, classical numerical methods become essentially numerically intractable. In order to overcome the limitations of classical methods, one can seek to incorporate the known asymptotic behaviour of the solution in the numerical method. This involves using asymptotic theory to determine the oscillatory part of the solution and then using classical numerical methods to approximate the slowly varying remainder. Such methods are often referred to as hybrid numerical-asymptotic methods. Determining the high frequency asymptotics of acoustic scattering problems is a classic problem in applied mathematics, with methods such as geometrical optics or the geometrical theory of diffraction providing asymptotic expansions of the solutions. Considerable amount of research has been directed towards both constructing these asymptotic expansions and proving error bounds for truncated asymptotic series of the solution, notably by Buslaev [23], Morawetz and Ludwig [78], and Melrose and Taylor [75], among others. Often, the oscillatory component of the solution can be determined explicitly from these asymptotic expansions. This can then be used in designing ecient hybrid methods. Furthermore, from the asymptotic expansions, frequency-dependent bounds on the slowly-varying remainder and its derivatives can be obtained (in some cases these follow directly from classical results, in other cases some additional work is required). The frequency-dependent bounds are the key results used in the frequency-explicit numerical error analysis of the approximation of the slowly-varying remainder. This thesis presents a rigorous justification of one of the key result using only elementary techniques. Hybrid numerical-asymptotic methods have been shown in theory to be substantially more efficient than classical numerical methods alone. For example, [40] presented a hybrid numerical-asymptotic method in the context of boundary integral equations (BIEs) for solving the problem of high-frequency scattering by smooth, convex obstacles in two dimensions. It was proved in [40] that in order to maintain the accuracy as the frequency increases, the hybrid BIE method requires the number of degrees of freedom to grow slightly faster than k1=9, where k is a parameter proportional to the frequency. This is a substantial improvement from the classical boundary integral methods that require O(k) number of degrees of freedom to achieve the same accuracy for this problem. Despite this slow growth in the number of degrees of freedom, hybrid numerical-asymptotic methods lead to stiffness matrices with entries that are highly-oscillatory singular integrals that can not be computed exactly. Thus, without efficient and accurate numerical treatment of these integrals, the hybrid numerical-asymptotic methods, regardless of their attractive theoretical accuracy, can not be efficiently implemented in practice. In order to resolve this difficulty, this thesis develops a methodology for approximating the integrals arising from hybrid methods in the context of BIEs. The integrals are transformed under a change of variables into integrals amenable to Filon-type quadratures. Filon-type quadratures are designed to cope well with high oscillations in the integrands. Then, graded meshes are used to capture the singularities accurately. Along with k-explicit error bounds for the integration methods, this thesis derives k-explicit error bounds for the hybrid BIE methods that incorporate the error of the inexact approximation of the entries of the stiffness matrix. The error bounds suggest that, with an appropriate choice of parameters of Filon quadrature and mesh grading, the overall error of the hybrid method does not deteriorate due to inexact approximation of the stiffness matrix, therefore preserving its attractive theoretical convergence properties.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Error analysis of the extended Filon-type method for highly oscillatory integrals

Abstract We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay between the two varies for different frequencies. We are particularly concerned with two strategies for the choice of inner nodes: Clenshaw–Curtis points and zeros of an appropriate Jacobi polynomial. Once the frequency $$\omega$$ ω is large, the asymptotic error dominates, but the situation is altogether different when $$\omega \ge 0$$ ω ≥ 0 is small. In the first regime, our optimal error bounds indicate that Clenshaw–Curtis points are always marginally better, but this is reversed for small $$\omega$$ ω ; then, Jacobi points enjoy an advantage. The main tool in our analysis is the Peano Kernel theorem. While the main part of the paper addresses integrals without stationary points, we indicate how to extend this work to the case when stationary points are present. Numerical experiments are provided to illustrate theoretical analysis

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/

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: O(1)\mathcal{O}(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now including direct comparisons to existing CQ and TDIE solver implementations) (Part I of II

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.