The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

Levin methods for highly oscillatory integrals with singularities

In this paper,new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities.To avoid singularities,the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs.The solutions of one can be obtained explicitly,while the other kind of ODEs can be solved efficiently by collocation methods.The proposed methods can attain arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points.Several numerical experiments are presented to validate the efficiency of the proposed methods.     

Quadrature rules using first derivatives for oscillatory integrands

We consider the integral of a function and its approximation by a quadrature rule of the form

i.e., by a rule which uses the values of both y and its derivative at nodes of the quadrature rule. We examine the cases when the integrand is either a smooth function or an ω dependent function of the form y(x)=f₁(x) sin(ωx)+f₂(x) cos(ωx) with smoothly varying f₁ and f₂. In the latter case, the weights w_k and α_k are ω dependent. We establish some general properties of the weights and present some numerical illustrations.     

一些高振荡积分、高振荡积分方程的高性能计算

电磁、声波散射问题的研究涉及一类数学物理问题, 此类问题具有深刻的理论价值和重要的应用背景, 亟待解决. 高振荡微分、积分方程是刻画这些问题的重要的数学模型, 其数值计算存在许多挑战性研究课题. 本文从积分方程解法角度出发, 综述了求解这类高振荡问题的一些最新进展, 特别是针对广义Fourier 变换、Bessel 变换的高效算法、高振荡核Volterra 积分方程的数值解法作了详细介绍. 这些数值方法共有特点是振荡频率越高算法精度愈高, 且可望为电磁计算的研究提供一些新的高效算法.     

On the computation of highly oscillatory multivariate integrals with stationary points

We consider two types of highly oscillatory bivariate integrals with a nondegenerate stationary point. In each case we produce an asymptotic expansion and two kinds of quadrature algorithms: an asymptotic method and a Filon-type method. Our results emphasize the crucial role played by the behaviour at the stationary point and by the geometry of the boundary of the underlying domain. In memory of Germund Dahlquist (1925–2005). AMS subject classification (2000) Primary 65D32     

Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands

Numerical Algorithms - This paper focus on the numerical evaluation of the Cauchy principal value integrals with oscillatory integrands where α, β > ??1,??1...     

Numerical approximation of vector-valued highly oscillatory integrals

We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based on the asymptotic expansion results in an approximation with significantly higher asymptotic order. AMS subject classification (2000)  65D30     

Efficient quadrature for highly oscillatory integrals involving critical points

This paper based on the Levin collocation method and Levin-type method together with composite two-point Gauss–Legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points. The effectiveness and accuracy of the quadrature are tested.     

Efficient integration for a class of highly oscillatory integrals

This paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polynomial and can be evaluated efficiently in O(N log N) operations, where N + 1 is the number of Clenshaw-Curtis points in the interval of integration. In addition, we derive the corresponding error bound in inverse powers of the frequency ω for the Clenshaw-Curtis-Filon-type method for the class of highly oscillatory integrals. The efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results.     

Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: part I

This work presents methods of efficient numerical approximation for linear and nonlinear systems of highly oscillatory ordinary differential equations. We show how an appropriate choice of quadrature rule improves the accuracy of approximation as the frequency of oscillation grows. We present asymptotic and Filon-type methods to solve highly oscillatory linear systems of ODEs, and WRF method, representing a special combination of Filon-type methods and waveform relaxation methods, for nonlinear systems. Numerical examples support this paper. Dedicated to the memory of Rudolf Khanamiryan. AMS subject classification (2000)  65L05, 34E05, 34C15     

Quadrature methods for highly oscillatory linear and non-linear systems of ordinary differential equations: part II

In this paper we present efficient numerical approximation for systems of highly oscillatory ordinary differential equations with matrices of variable coefficients. We assume that the spectrum of the matrix is purely imaginary and the frequency of oscillation grows large. We develop the asymptotic and the Filon-type methods for linear systems with time dependent matrices and we integrate oscillatory quadrature rules with waveform relaxation methods employing the WRF method for non-linear systems. We solve matrix exponential in Lie groups employing Magnus expansion. The methods are illustrated in several numerical examples of interest.     

Note on the homotopy perturbation method for multivariate vector-value oscillatory integrals

This paper considers a homotopy perturbation method for approximating multivariate vector-value highly oscillatory integrals. The asymptotic formulae of the integrals and the asymptotic order of the asymptotic method are presented. Numerical examples show the efficiency of the approximation method.     

Asymptotic expansions for oscillatory integrals using inverse functions

We treat finite oscillatory integrals of the form ∫ _a ^b F(x)e ^ikG(x) dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient ∫ _a ^b φ(t)e ^ikt dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when , −1<σ ₁<σ ₂<⋅⋅⋅. The authors were supported by the Office of Advanced Scientific Computing Research, Office of Science, US Department of Energy, under Contract DE-AC02-06CH11357.     

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.     

Multi-revolution composition methods for highly oscillatory differential equations

We introduce a new class of multi-revolution composition methods for the approximation of the \(N\) th-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods.     

Quadrature formulas for oscillatory integral transforms

Summary Quadrature formulas are obtained for the Fourier and Bessel transforms which correspond to the well-known Gauss-Laguerre formula for the Laplace transform. These formulas provide effective asymptotic approximations, complete with error bounds. Comparison is also made between the quadrature formulas and the asymptotic expansions of these transforms.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Contract A7359