Gwynne A. Evans

A comparison of some methods for the evaluation of highly oscillatory integrals

Abstract In Part I the extended Clenshaw–Curtis method for finite Fourier integrals is discussed, and a number of timed comparisons are made between the various implementations which appear in the literature. Part II deals with irregular oscillatory integrals and outlines the various methods which have been proposed for their evaluation. Their effectiveness is discussed in the light of a set of test examples.

A high order, progressive method for the evaluation of irregular oscillatory integrals

Abstract A method is presented for the evaluation of rapidly oscillatory integrals. The method is a variation of Levins method, and involves forming a quadrature rule which is exact for a certain set of functions. It is shown that the choice of exact functions and, more importantly, the integration abscissae are crucial to the convergence and numerical stability of the method. The computation of the integration weights is also discussed. Comparisons are made with alternative methods, in particular with Levins original implementation.

A method to generate generalized quadrature rule for oscillatory integrals

Abstract In a recent paper Evans and Webster produced a quadrature rule for oscillatory integrals with a trigonometric kernel. Their method is a modification of an earlier method due to Levin, and involves making a quadrature rule exact for a set of functions for which the modified moments have a simple closed form. In this paper we extend their method to deal with any oscillatory kernel, provided only that it satisfies a (known) linear ordinary differential equation. Some specific examples involving Bessel functions and Bessel–Trig combinations are given to illustrate the practical use of the method.

An expansion method for irregular oscillatory integrals

Many current problems in applied mathematics require the numerical integration of irregular oscillatory integrals. Few methods have been specifically found for these problems. An alternative method based on expansion is presented here and this method is compared with other approaches. Tests are carried out on a representative set of examples, and the algorithms are applied to problems with large oscillatory factors.

Some new thoughts on Gauss-Laguerre quadrature

Gauss–Laguerre quadrature is notoriously poor for integrating non-exponentially decreasing infinite-range integrals. A method is proposed based on the non-standard basis set 1/x i and this is implemented for the weight functions 1.0, log x and sin x. A series of tests demonstrate the effectiveness of the approach, and an algorithm is presented to achieve convergence to a specified accuracy.

Laplace transform inversions using optimal contours in the complex plane

A numerical method for computing inverse Laplace transforms is proposed. In this method, the complex contour integral defining the inverse transform is computed over an equivalent contour as proposed by Talbot and Evans. Special contours, called optimal contours, are constructed so that the transformed real integrand decreases exponentially to zero as z runs along such a contour to infinity. The efficient Clenshaw-Curtis quadrature is employed for the final evaluation. The presented method is competitive and compares favourably with those of Talbot and Evans.

Planar gunn diode characterisation and resonator elements to realise oscillator circuits

The paper describes the planar Gunn diode, which is a device well suited to provide milli-metric and tera-hertz sources using microwave monolithic integrated circuit technologies. Different planar Gunn electrode geometries are described along with DC, RF and thermal characterisation. To realize the planar high frequency sources there is requirement for high frequency planar resonators, the paper will describe both the radial and new diamond shaped geometries.

Improved infrared (IR) microscope measurements for the micro-electronics industry

Infrared (IR) measurements of the surface temperature of electronic devices have improved over the last decade. However, to obtain more accurate surface temperatures the devices are often coated with a high emissivity coating leading to temperature averaging across the device surface and damage to the device. This paper will look at the problems of making accurate surface temperature measurements particularly on areas of semiconductor and will address the surface emissivity correction problem using novel measurement approaches.

Numerical Simulation of Oldroyd-B Fluid in a Contraction Channel

A spectral element method coupled with the elastic viscous split stress technique for computing viscoelastic flows is presented. The rate of deformation tensor is introduced as an additional variable in the momentum equation, but not in the constitutive equation. The nonlinear rheological model, Oldroyd-B, is chosen to simulate the flow of a viscoelastic fluid. Numerical solutions are investigated based on a planar 4:1 abrupt contraction channel flow benchmark problem with different Weissenberg numbers. The results show a good agreement with other numerical predictions.

The evaluation of infinite range oscillatory integrals using optimal contours in the complex plane

A method is proposed for the evaluation of infinite range irregular oscillatory integrals of the form which is based on finding a contour in complex space which removes the oscillatory part. The resulting integral can then be evaluated conventionally. Conditions on f are found for this process to be valid, and comparisons made with alternative methods. A highly competive algorithm is generated.