Abstract
In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.
Highlights
Oscillatory integrals frequently arise in acoustic scattering [1], computational physical optics [2], computational electromagnetics [3], and related fields
Let us consider the computation of oscillatory integral without nearly singular integrands, that is, Î [ F, G1, G2, ω ] =
2DSC-Levin does a little better than 2D-Cheb–Levin quadrature when CPU time is considered. Since both approaches consist of approximations to a series of one-dimensional integrals, we show the consuming time of both approaches for computing the final highly oscillatory integrals in Table 5, where it can be seen that 2DSC-Levin is slightly faster than
Summary
Oscillatory integrals frequently arise in acoustic scattering [1], computational physical optics [2], computational electromagnetics [3], and related fields. By transforming the oscillatory integration problem into a special ordinary differential equation, one could get an efficient approximation to the generalized. In [10], Li et al proposed a stable and high-order Levin quadrature rule by employing the spectral Chebyshev collocation method and truncated singular value decomposition technique. By solving the transformed equations numerically, they obtained an efficient Levin quadrature rule for weakly singular integrals with highly oscillatory Fourier kernels. By introducing a multivariate ordinary differential equation, Levin found a non-oscillatory approximation to the integrand in [7], which led to an efficient algorithm for computing oscillatory integrals over rectangular regions. With the help of delaminating quadrature, the spectral Levin-type method for calculation of highly oscillatory integrals over rectangular regions was constructed. We present an innovative composite Levin quadrature rule for solving nearly singular and highly oscillatory problems.
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