Levin methods for highly oscillatory integrals with singularities

Abstract

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, 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 those of the other can be solved efficiently by collocation methods. The proposed methods can attach 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.

Oscillation-preserving Legendre-Galerkin methods for second kind integral equations with highly oscillatory kernels

“…In the last twenty years, highly oscillatory integrals were now well understood, and many highly efficiently methods have been devised so that the accuracy of the computation increases as the wave number tends to infinity. There are mainly four classes of approaches for calculating oscillatory integrals: asymptotic methods in [12,22], Filon-type methods in [8,9,12,13,32], Levin-type methods in [15,16,19,20,28] and the numerical steepest decent method in [10,11]. Recently, a new method is presented in [17,18], which combines the moment free Filon-type method with graded meshes.…”

“…Because the solutions of these equations have a clear structure, an efficient hybrid finite element method is proposed for solving these equations. Later, Xiang and Wang in [26,28] solved certain Volterra integral equations of the first kind with a highly oscillatory Bessel kernel, based on the fact that the exact solution has an explicit integral expansion. Consequently, the low-order finite element collocation method combining with a Filon-type algorithm was presented in [33,34] for solving weakly singular Volterra integral equations of second kind with a highly oscillatory Bessel kernel, based on the asymptotic analysis of their original solutions.…”

Fast linear barycentric rational interpolation for singular functions via scaled transformations

“…Example 3.4 (Highly oscillatory problem). One application considered in this example is about solving a highly oscillatory problem with a singulary kernel [38]:…”

“…x and a = 1 in (3.5), we can obtain a reference solution by calling MATLAB code quadgk equipped with setting both relative error tolerance and absolute error tolerance to 10 −14 . In [38], to avoid the singularity, the authors apply a singularity separation technique converting the singular ODE (3.6) into two kinds of non-singular ODEs. Here we will approximate the derived ODE directly with rational interpolation (1.4).…”

The quantum-mechanical Coulomb propagator in an L2 function representation

“…A survey of these methods can be found in Deaño, Huybrech and Iserles 23 , where further work on the topic is also referenced, up to and including 2017. Oscillatory integrals are the subject of ongoing research: Yang and Ma 24 , Zaman et al 25 used Levin-based approaches to calculate highly oscillatory Fourier integrals in one- and two-dimensional domains, whereas Wang und Xiang 26 applied a Levin method to singular integrands. Recent articles from Kayijuka et al 27 dealt with oscillatory Fourier integrals with singular integrands, while Zaman et al 28 used a Levin based approach to evaluate the integrals over Bessel functions.…”