On a novel numerical quadrature based on cycle index of symmetric group for the Hadamard finite-part integrals

To evaluate the Hadamard finite-part integrals accurately, a novel interpolatory-type quadrature is proposed in this article. In our approach, numerical divided difference is utilized to represent the high order derivatives of the integrated function, which make it possible to reduced the numerical quadrature into a concise formula based on the cycle index for symmetric group. In addition, convergence analysis is presented and the error estimation is given. Numerical results are presented on cases with different weight functions, which substantiate the performance of the proposed method

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel cot((x-s)/2) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the 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/

An adaptive space-time boundary element method for impulsive wave propagation in elastodynamics

Wave propagation in natural or man-made bodies is an important problem in civil engineering, electronic engineering and ocean engineering etc. Common examples of wave problems include earthquake wave modeling, ocean wave modeling, soil- structure interaction, geological prospecting, and acoustic or radio wave diffraction. The Boundary Element Method (BEM) is a widely-used numerical method to solve such problems in both science and engineering fields. However, conventional BEM modeling of wave problems encounters many difficulties. Firstly, the method is expensive since influence matrices are computed at each time step and BEM solutions at every former time step have to be stored. Secondly, if large time steps are used, inaccuracies arise in BEM solutions; but if small time steps are used, computational costs become impractical. Thirdly, the dimensionless space-time ratio must be limited to a narrow range to produce a stable solution. In this thesis, we attack these problems by introducing adaptive schemes and mesh refinement. Instead of using uniform meshes and uniform time steps, error indicators are employed to locate high-gradient areas; then mesh refinement in space-time is used to improve the resolution in those areas only. Another strategy is to introduce the space-time concept to track moving wave fronts. In wave problems, wave fronts move in space-time, and high gradients arise both in space and in time. It is thus inadequate to refine the mesh in space only because there are high gradients in time as well. Hence, besides a locally mesh refinement scheme employed in space, local time stepping is also used to improve the accuracy and efficiency of the algorithm. This adaptive scheme is implemented in the C language and used to solve scalar and electrodynamic 2D and 3D wave propagation problems in a open and closed field. Gradient-based and resolution-based error indicators are employed to locate these moving high-gradient areas. A space mesh refinement scheme and the local time stepping is used to refine the area to achieve higher accuracy. The adaptive BEM solver is 1.4 ~ 1.8 times faster than the conventional BEM solver. It is also more stable than the conventional BEM. We also parallelize the BEM solver to further improve its efficiency. Compared with the non-parallel code, using a 8-processor Linux cluster, a speed-up factor of four is achieved. This suggests that substantial further gains can be obtained if a larger parallel computer is available. July 11. 2007

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described