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Register a new userOn the numerical quadrature of weakly singular oscillatory integral and its fast implementation
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Zhenhua Xu
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In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(Nlog N)$ operations. The method requires the accurate computation of modified moments. We first give a method for the derivation of the recurrence relation for the modified moments, which can be applied to the derivation of the recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.
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Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the optimal approximation problem, the orders of the Babenko and Kolmogorov (n-)widths of compact sets from some classes of functions have been evaluated. In conclusion we adduce some numerical illustrations for 2-D Volterra equations.
In this article, we present an $O(N log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced computational complexity, we utilize the Fast Fourier Transform (FFT) on a uniform grid of size $N$ for approximating the convolution. To facilitate this and maintain the accuracy, we primarily rely on a periodic Fourier extension of the density with a suitably large period depending on the support of the density. The rate of convergence of the method increases with increasing smoothness of the periodic extension and, in fact, approximations exhibit super-algebraic convergence when the extension is infinitely differentiable. Furthermore, when the density has jump discontinuities, we utilize a certain Fourier smoothing technique to accelerate the convergence to achieve the quadratic rate in the overall approximation. Finally, we apply the integration scheme for numerical solution of certain partial differential equations. Moreover, we apply the quadrature to obtain a fast and high-order Nystom solver for the solution of the Lippmann-Schwinger integral equation. We validate the performance of the proposed scheme in terms of accuracy as well as computational efficiency through a variety of numerical experiments.
Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(nlog n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous and, therefore, their approximations by truncated Fourier series suffer from Gibbs oscillations. In this paper, we present and analyze an $O(nlog n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with high-order but is also relatively simple to implement. We include a theoretical error analysis as well as a wide variety of numerical experiments to demonstrate its efficacy.
In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-mu},0<mu<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $L^{infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $xrightarrow x^{1/lambda}$ for a suitable real number $lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method.
A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the fast algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (1999).
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