4,588 research outputs found

    A Short Tale of Long Tail Integration

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    Integration of the form ∫a∞f(x)w(x)dx\int_a^\infty {f(x)w(x)dx} , where w(x)w(x) is either sin⁑(Ο‰x)\sin (\omega {\kern 1pt} x) or cos⁑(Ο‰x)\cos (\omega {\kern 1pt} x), is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration over a finite domain (a, b)(a,\,b), leaving a truncation error equal to the tail integration ∫b∞f(x)w(x)dx\int_b^\infty {f(x)w(x)dx} in addition to the discretization error. This paper describes a very simple, perhaps the simplest, end-point correction to approximate the tail integration, which significantly reduces the truncation error and thus increases the overall accuracy of the numerical integration, with virtually no extra computational effort. Higher order correction terms and error estimates for the end-point correction formula are also derived. The effectiveness of this one-point correction formula is demonstrated through several examples

    High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

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    Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure

    Fast, numerically stable computation of oscillatory integrals with stationary points

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    We present a numerically stable way to compute oscillatory integrals of the form βˆ«βˆ’11f(x)eiΟ‰g(x)dx\int{-1}^{1} f(x)e^{i\omega g(x)}dx. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane

    Asymptotic expansions and fast computation of oscillatory Hilbert transforms

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    In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form H+(f(t)eiΟ‰t)(x)=βˆ’int0∞eiΟ‰tf(t)tβˆ’xdt,Ο‰>0,xβ‰₯0,H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0, where the bar indicates the Cauchy principal value and ff is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When x=0x=0, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of Ο‰\omega are derived for each fixed xβ‰₯0x\geq 0, which clarify the large Ο‰\omega behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of xx, we classify our discussion into three regimes, namely, x=O(1)x=\mathcal{O}(1) or x≫1x\gg1, 0<xβ‰ͺ10<x\ll 1 and x=0x=0. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency Ο‰\omega increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.Comment: 32 pages, 6 figures, 4 table

    Modified Filon-Clenshaw-Curtis rules for oscillatory integrals with a nonlinear oscillator

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    Filon-Clenshaw-Curtis rules are among rapid and accurate quadrature rules for computing highly oscillatory integrals. In the implementation of the Filon-Clenshaw-Curtis rules in the case when the oscillator function is not linear, its inverse should be evaluated at some points. In this paper, we solve this problem by introducing an approach based on the interpolation, which leads to a class of modifications of the original Filon-Clenshaw-Curtis rules. In the absence of stationary points, two kinds of modified Filon-Clenshaw-Curtis rules are introduced. For each kind, an error estimate is given theoretically, and then illustrated by some numerical experiments. Also, some numerical experiments are carried out for a comparison of the accuracy and the efficiency of the two rules. In the presence of stationary points, the idea is applied to the composite Filon-Clenshaw-Curtis rules on graded meshes. An error estimate is given theoretically, and then illustrated by some numerical experiments

    Fast computation of effective diffusivities using a semi-analytical solution of the homogenization boundary value problem for block locally-isotropic heterogeneous media

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    Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a grid of blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media.Comment: 29 pages, 4 figures, 5 table

    Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity

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    Real and imaginary part of the limit 2N->infinity of the integral int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over the first halve wave with series acceleration techniques for the alternating series co-phased to each of its points. The integrand is of the logarithmic kind; its branch cut limits the performance of integration techniques that rely on smooth higher order derivatives.Comment: Section 2.5 (approach of modified integration path in the complex plane) adde
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