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We develop two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points. The first class of the quadrature rules has a polynomial order of convergence and the second class has an exponential order of convergence. We first modify the moment-free Filon-type method for the oscillatory integrals without a singularity or a stationary point to accelerate their convergence. We then consider the oscillatory integrals without a singularity or a stationary point and then those with singularities and stationary points. The composite quadrature rules are developed based on partitioning the integration domain according to the wave number and the singularity of the integrand. The integral defined on a subinterval has either a weak singularity without rapid oscillation or oscillation without a singularity. The classical quadrature rules for weakly singular integrals using graded points are employed for the singular integral without rapid oscillation and the modified moment-free Filon-type method is used for the oscillatory integrals without a singularity. Unlike the existing methods, the proposed methods do not have to compute the inverse of the oscillator. Numerical experiments are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods outperform methods published most recently.
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We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solution, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.
In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel ({em J. Math. Phys.} {bf 34} (1956), 298--307) and Hunter and Regan ({em Math. Comp.} {bf 26} (1972), 539--541). We construct approximations which we prove are exponentially convergent as a function of $N$, the number of quadrature points, obtaining explicit error bounds which show that accuracies of $10^{-15}$ uniformly on the real line are achieved with N=12, this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.
In this paper, we consider the Clenshaw-Curtis-Filon method for the highly oscillatory Bessel transform $int_0^1x^alpha (1-x)^beta f(x) J_{ u}(omega x)dx$, where $f$ is a smooth function on $[0, 1]$, and $ ugeq0.$ The method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments. We give a recurrence relation for the modified moments and present an efficient method for the evaluation of modified moments by using recurrence relation. Moreover, the corresponding error bound in inverse powers of $N$ for this method for the integral is presented. Numerical examples are provided to support our analysis and show the efficiency and accuracy of the method.
We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to the integral normally lead to poor accuracy due to the rapid change in high order derivatives of its integrand when the standard deviation is small. The proposed quadrature schemes are based on graded meshes designed according to the standard deviation so that the quadrature errors on the resulting subintervals are approximately equal. The integral in each subinterval is then computed by considering the Gaussian function as a weight function and interpolating the smooth factor of the integrand at the Chebyshev points of the first kind. For a finite order differentiable factor, we design a quadrature scheme having accuracy of a polynomial order and for an infinitely differentiable factor of the integrand, we design a quadrature scheme having accuracy of an exponential order. Numerical results are presented to confirm the accuracy of these proposed quadrature schemes.
We introduce a new method for evaluating the oscillatory integrals which describe natural interference patterns. As an illustrative example of contemporary interest, we consider astrophysical plasma lensing of coherent sources like pulsars and fast radio bursts in radioastronomy. Plasma lenses are known to occur near the source, in the interstellar medium, as well as in the solar wind and the earths ionosphere. Such lensing is strongest at long wavelengths hence it is generally important to go beyond geometric optics and into the full wave optics regime. Our computational method is a spinoff of new techniques two of us, and our collaborators, have developed for defining and performing Lorentzian path integrals. Cauchys theorem allows one to transform a computationally fragile and expensive, highly oscillatory integral into an exactly equivalent sum of absolutely and rapidly convergent integrals which can be evaluated in polynomial time. We require only that it is possible to analytically continue the lensing phase, expressed in the integrated coordinates, into the complex domain. We give a first-principles derivation of the Fresnel-Kirchhoff integral, starting from Feynmans path integral for a massless particle in a refractive medium. We then demonstrate the effectiveness of our method by computing the interference patterns of Thoms caustic catastrophes, both in their normal forms and within a variety of more realistic, local lens models, over all wavelengths. Our numerical method, implemented in a freely downloadable code, provides a fast, accurate tool for modeling interference patterns in radioastronomy and other fields of physics.
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