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This article considers the computation of the matrix exponential eA{{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates eA{{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes eA{{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance.

This article investigates discrete versions of the projection-type and modified projection-type methods for solving Hammerstein integral equations with a weakly singular kernel. The approximating operator utilized is either the orthogonal projection or an interpolatory projection onto a space of piecewise polynomials of degree ≤ r - 1 with respect to a graded partition of [0, 1]. The study reveals that the proposed methods achieve optimal rates of convergence, demonstrating that the numerical quadrature used to estimate the integrals maintains the same rates of convergence as the continuous methods. The theoretical findings are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and NNs act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. We derive a priori error estimates in the standard |$L^2(\varOmega)$| norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g. spatial mesh size, time step size and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation. [ABSTRACT FROM AUTHOR]

This paper provides an efficient numerical scheme for approximating the solution of the time-fractional parabolic equation on 2D curved domain. Here, the solution of the problem exhibits a weak singularity at the initial time t = 0 . The method is based on applying the L1 formula to approximate the Caputo time-fractional derivative and using the isoparametric finite element method to approximate the spatial direction. A fully discrete scheme is then constructed using numerical quadrature. Since Ω h differs from Ω , the error estimates of the boundary terms on the region between Ω h and Ω and the effect of numerical quadrature are both considered. Afterward, the stability analysis and optimal error estimates for the fully discrete scheme are proved in detail. Finally, some numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Numerical methods for the computation of the parabolic cylinder function U (a , z) for real a and complex z are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main methods can be complemented with Maclaurin series and a Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with 5 × 10 − 13 relative accuracy in double precision floating point arithmetic. [ABSTRACT FROM AUTHOR]

Balanced singular perturbation approximation (SPA) is a model order reduction method for linear time-invariant systems that guarantees asymptotic stability and for which there exists an a priori error bound. In that respect, it is similar to balanced truncation (BT). However, the reduced models obtained by SPA generally introduce better approximation in the lower frequency range and near steady-states, whereas BT is better suited for the higher frequency range. Even so, independently of the frequency range of interest, BT and its variants are more often applied in practice, since there exist more efficient algorithmic realizations thereof. In this paper, we aim at closing this practically relevant gap for SPA. We propose two novel and efficient algorithms that are adapted for different settings. First, we derive a low-rank implementation of SPA that is applicable in the large-scale setting. Second, a data-driven reinterpretation of the method is proposed that only requires input-output data and thus is realization-free. A main tool for our derivations is the reciprocal transformation, which induces a distinct view on implementing the method. While the reciprocal transformation and the characterization of SPA are not new, their significance for the practical algorithmic realization has been overlooked in the literature. Our proposed algorithms have well-established counterparts for BT and, as such, a comparable computational complexity. The numerical performance of the two novel implementations is tested for several numerical benchmarks, and comparisons to their counterparts for BT as well as to existing implementations of SPA are made. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop an efficient and accurate procedure of electromagnetic multipole decomposition by using the Lebedev and Gaussian quadrature methods to perform the numerical integration. Firstly, we briefly review the principles of multipole decomposition, highlighting two numerical projection methods including surface and volume integration. Secondly, we discuss the Lebedev and Gaussian quadrature methods, provide a detailed recipe to select the quadrature points and the corresponding weighting factor, and illustrate the integration accuracy and numerical efficiency (that is, with very few sampling points) using a unit sphere surface and regular tetrahedron. In the demonstrations of an isotropic dielectric nanosphere, a symmetric scatterer, and an anisotropic nanosphere, we perform multipole decomposition and validate our numerical projection procedure. The obtained results from our procedure are all consistent with those from Mie theory, symmetry constraints, and finite element simulations. Graphical Abstract

We consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed. [ABSTRACT FROM AUTHOR]

Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow. [ABSTRACT FROM AUTHOR]

In this paper, discrete versions of the modified projectiontype methods are studied for solving the Hammerstein integral equations with a smooth kernel. The approximating operator is either the orthogonal projection or an interpolatory projection at Gauss points onto a space of piecewise polynomials of degree = r - 1. The orders of convergence that one obtains for the proposed methods shows that the used numerical quadrature for approximating the integrals preserves the orders of convergence of continuous methods. Numerical results are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

A classical theorem due to Chebyshev, Markov and Stieltjes states that the Gauss-Legendre quadrature of a generic function f is a Riemann sum of f. In this note we prove an analogue of this theorem for Romberg quadrature. [ABSTRACT FROM AUTHOR]

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter ensures stability of the scheme, a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for the mapping parameter, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that a static choice of the map’s parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of Hadamard Finite Part (HFP) integrals K m (t ; u) = ⨎ 0 T S m (π (x − t) T) u (x) d x , 0 < t < T , m ∈ { 1 , 2 , ... } , where u (x) is T -periodic and sufficiently differentiable and S 2 r − 1 (y) = cos ⁡ y sin 2 r − 1 ⁡ y , S 2 r (y) = 1 sin 2 r ⁡ y , r = 1 , 2 , 3 , .... For each m , we regularize the HFP integral K m (t ; u) and show that K m (t ; u) = K 0 (t ; U m) ≡ ∫ 0 T (log ⁡ | sin ⁡ π (x − t) T |) U m (x) d x , U m (x) being some linear combination of the first m derivatives of u (x). We then propose to approximate K m (t ; u) by the quadrature formula Q m , n (t ; u) ≡ K m (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q m , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q m , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We illustrate the effectiveness of Q m , n (t ; u) with numerical examples for m = 0 , 1 , ... , 5. We also show that the HFP integral ⨎ 0 T f (x , t) d x of any T -periodic integrand f (x , t) that has m th order poles at x = t + k T , k = 0 , ± 1 , ± 2 , ... , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed in terms of the K s (t ; u (⋅ , t)) , where u (x , t) is a T -periodic and sufficiently differentiable function in x on R that can be computed from f (x , t). Therefore, ⨎ 0 T f (x , t) d x can be computed efficiently using our new numerical quadrature formulas Q s , n (t ; u (⋅ , t)) on the individual K s (t ; u (⋅ , t)). Again, only 2 n function evaluations, namely, u (x n , k , t) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]

Based on the Euler–Maclaurin formula, the Romberg quadrature method extrapolates trapezoidal values to improve their accuracy when computing the integral of smooth functions from equispaced samples. It has been known at least since an article of Lyness in 1971 that the Euler–Maclaurin formula may be extended to accommodate functions with jumps. In the present work, we develop an extrapolation method, based on this extended formula, for the quadrature of such discontinuous functions. We illustrate the method with numerical examples, using one as well as several sample vectors. [ABSTRACT FROM AUTHOR]

We propose an integral representation for the higher-order Fréchet derivative of analytic matrix functions f (A) which unifies known results for the first-order Fréchet derivative of general analytic matrix functions and for higher-order Fréchet derivatives of A − 1. We highlight two applications of this integral representation: On the one hand, it allows to find the exact value of the level-2 condition number (i.e., the condition number of the condition number) of f (A) for a large class of functions f when A is Hermitian. On the other hand, it also allows to use numerical quadrature methods to approximate higher-order Fréchet derivatives. We demonstrate that in certain situations—in particular when the derivative order k is moderate and the direction terms in the derivative have low-rank structure—the resulting algorithm can outperform established methods from the literature by a large margin. [ABSTRACT FROM AUTHOR]

In this work, we propose the application of the eXtended Finite Element Method (XFEM) in the context of the coupling between three-dimensional and one-dimensional elliptic problems. In particular, we consider the case in which the 3D–1D coupled problem arises from the geometrical model reduction of a fully three-dimensional problem, characterized by thin tubular inclusions embedded in a much wider domain. In the 3D–1D coupling framework, the use of non conforming meshes is widely adopted. However, since the inclusions typically behave as singular sinks or sources for the 3D problem, mesh adaptation near the embedded 1D domains may be necessary to enhance solution accuracy and recover optimal convergence rates. An alternative to mesh adaptation is represented by the XFEM, which we here propose to enhance the approximation capabilities of an optimization-based 3D–1D coupling approach. An effective quadrature strategy is devised to integrate the enrichment functions and numerical tests on single and multiple segments are proposed to demonstrate the effectiveness of the approach. • Accurate resolution of 3D–1D coupled problems on non conforming coarse meshes with the XFEM. • Use of the XFEM in presence of multiple intersecting 1D inclusions. • Definition of a novel quadrature strategy for the used enrichment functions. • Numerical examples are used to validate the method also for realistic configurations. [ABSTRACT FROM AUTHOR]

Let assuming that g ∈ C ∞ [ a , b ] such that f(x) is T-periodic, T = b - a , and f (x) ∈ C ∞ (R t) with R t = R \ { t + k T } k = - ∞ ∞ . Here stands for the Hadamard Finite Part (HFP) of the singular integral ∫ a b f (x) d x that does not exist in the regular sense. In a recent work, we unified the treatment of these HFP integrals by using a generalization of the Euler–Maclaurin expansion due to the author and developed a number of numerical quadrature formulas T ^ m , n (s) [ f ] of trapezoidal type for I[f] for all m. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3 , and these are T ^ 3 , n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , h = T n , T ^ 3 , n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , h = T n , T ^ 3 , n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4) , h = T n. For all m and s, we showed that all of the numerical quadrature formulas T ^ m , n (s) [ f ] have spectral accuracy; that is, T ^ m , n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In this work, we continue our study of convergence and extend it to functions f(x) that possess certain analyticity properties. Specifically, we assume that f(z), as a function of the complex variable z, is also analytic in the infinite strip | Im z | < σ for some σ > 0 , excluding the poles of order m at the points t + k T , k = 0 , ± 1 , ± 2 , …. For m = 1 , 2 , 3 , 4 and relevant s, we prove that T ^ m , n (s) [ f ] - I [ f ] = O (exp (- 2 π n ρ / T)) as n → ∞ ∀ ρ < σ. [ABSTRACT FROM AUTHOR]

In this paper, we design efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three‐dimensional settings. In this work, we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two‐ and three‐dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient FE assembly. [ABSTRACT FROM AUTHOR]

When solving the wave equation with finite elements, mass lumping allows for explicit time stepping, avoiding the cost of a lower-upper decomposition of the large sparse mass matrix. Mass lumping on the reference element amounts to numerical quadrature. The weights should be positive for stable time stepping and preserve numerical accuracy. The standard triangular polynomial elements, except for the linear element, do not have these properties. Accuracy can be preserved by augmenting them with higher-degree polynomials in the interior. This leaves the search for elements with positive weights, which were found up to degree 9 by various authors. The classic accuracy condition, however, is too restrictive. A sharper, less restrictive condition recently led to new mass-lumped tetrahedral elements up to degree 4. Compared to the known ones up to degree 3, they have less nodes and are computationally more efficient. The same criterion is applied here to the construction of triangular elements. For degrees 2 to 4, these turn out to be identical to the known ones. For degree 5, the number of nodes is the same as for the known element, but now there are infinitely many solutions. Some of these have a considerably larger stability limit for time stepping. For degree 6, two elements are found with less nodes than the known ones. For degree 7, one element with less nodes was found but with a negative weight, making it useless for time stepping with the wave equation. If the number of nodes is the same as for the classic element, there are now infinitely many solutions. Numerical tests for a homogeneous wave-propagation problem with a point source confirm the expected accuracy of the new elements. Some of them require less compute time than those obtained with the more restrictive accuracy criterion. [ABSTRACT FROM AUTHOR]

The equilibrium shape of a thin inextensible membrane subject to solar radiation pressure under given boundary constraints is studied. The membrane is assumed to be insusceptible to elastic deformation and to have negligible bending resistance, and its steady-state shape is therefore described by a developable surface (i.e., a surface of zero Gaussian curvature), resulting from an equilibrium between radiation pressure and membrane tension forces. A quantitative understanding of the mechanics of such membranes is essential in characterizing the dynamics of solar sail spacecraft that use sail wing tip displacement as an attitude control mode. The analysis in this paper develops a theoretical foundation for the billowed wing shape. Under reasonable simplifying assumptions, the key result is that solar radiation pressure and a given wing tip displacement yield a billowed solar sail wing with the shape of a generalized cylinder (i.e., a developable ruled surface, whose rulings are all parallel, rather than a general developable with variable ruling directions). The base curve geometry for the solar sail is also determined as the solution to a boundary value problem. The results presented herein allow the shape of the billowed membrane to be computed to any desired precision, for any given tip displacement.

The equilibrium shape of a thin inextensible membrane subject to solar radiation pressure under given boundary constraints is studied. The membrane is assumed to be insusceptible to elastic deformation and to have negligible bending resistance, and its steady-state shape is therefore described by a developable surface (i.e., a surface of zero Gaussian curvature), resulting from an equilibrium between radiation pressure and membrane tension forces. A quantitative understanding of the mechanics of such membranes is essential in characterizing the dynamics of solar sail spacecraft that use sail wing tip displacement as an attitude control mode. The analysis in this paper develops a theoretical foundation for the billowed wing shape. Under reasonable simplifying assumptions, the key result is that solar radiation pressure and a given wing tip displacement yield a billowed solar sail wing with the shape of a generalized cylinder (i.e., a developable ruled surface, whose rulings are all parallel, rather than a general developable with variable ruling directions). The base curve geometry for the solar sail is also determined as the solution to a boundary value problem. The results presented herein allow the shape of the billowed membrane to be computed to any desired precision, for any given tip displacement.

The singular integrals in the Galerkin Boundary Element Method are usually treated using singularity removing transformations. The regularized integrals are approximated by tensor product Gaussian quadrature rules. Although these integrals are smooth, the convergence rate depends strongly on the type of singularity and the aspect ratio of the triangulation. Here an adaptive quadrature scheme is presented that ensures a convergence at a user-specified rate. The effectiveness of the resulting quadrature scheme is compared with the non-adaptive approach. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of I [ f ] = ∫ = a b f (x) d x , the Hadamard finite part of the finite-range singular integral ∫ a b f (x) d x , f (x) = g (x) / (x - t) m with a < t < b and m ∈ { 1 , 2 , ... } , assuming that (i) g ∈ C ∞ (a , b) and (ii) g(x) is allowed to have arbitrary integrable singularities at the endpoints x = a and x = b . We first prove that ∫ = a b f (x) d x is invariant under any legitimate variable transformation x = ψ (ξ) , ψ : [ α , β ] → [ a , b ] , hence there holds ∫ = α β F (ξ) d ξ = ∫ = a b f (x) d x , where F (ξ) = f (ψ (ξ)) ψ ′ (ξ) . Based on this result, we next choose ψ (ξ) such that F (ξ) , the T -periodic extension of F (ξ) , T = β - α , is sufficiently smooth, and prove, with the help of some recent extension/generalization of the Euler–Maclaurin expansion, that we can apply to ∫ = α β F (ξ) d ξ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author: [A. Sidi, "Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions." Calcolo, 58, 2021. Article number 22]. We give a whole family of numerical quadrature formulas for ∫ = α β F (ξ) d ξ for each m, which we denote T ^ m , n (s) [ F ] . Letting G (ξ) = (ξ - τ) m F (ξ) , with τ ∈ (α , β) determined from t = ψ (τ) , and letting h = T / n , for m = 3 , for example, we have the three formulas T ^ 3 , n (0) [ F ] = h ∑ j = 1 n - 1 F (τ + j h) - π 2 3 G ′ (τ) h - 1 + 1 6 G ′ ′ ′ (τ) h , T ^ 3 , n (1) [ F ] = h ∑ j = 1 n F (τ + j h - h / 2) - π 2 G ′ (τ) h - 1 , T ^ 3 , n (2) [ F ] = 2 h ∑ j = 1 n F (τ + j h - h / 2) - h 2 ∑ j = 1 2 n F (τ + j h / 2 - h / 4). We show that all of the formulas T ^ m , n (s) [ F ] converge to I[f] as n → ∞ ; indeed, if ψ (ξ) is chosen such that F (i) (α) = F (i) (β) = 0 , i = 0 , 1 , ... , q - 1 , and F (q) (ξ) is absolutely integrable in every closed interval not containing ξ = τ , then T ^ m , n (s) [ F ] - I [ f ] = O (n - q) as n → ∞ , where q is a positive integer determined by the behavior of g(x) at x = a and x = b and also by ψ (ξ) . As such, q can be increased arbitrarily (even to q = ∞ , thus inducing spectral convergence) by choosing ψ (ξ) suitably. We provide several numerical examples involving nonperiodic integrands and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in the presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provides valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called quadrature by parity asymptotic expansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples. [ABSTRACT FROM AUTHOR]

Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve - and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.

Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve - and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals I [ f ] = ∫ = a b f (x) d x , where f (x) = g (x) / (x - t) 3 , assuming that g ∈ C ∞ [ a , b ] and f(x) is T-periodic, T = b - a . With h = T / n , these numerical quadrature formulas read T ^ n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , T ^ n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , T ^ n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4). We also showed that these formulas have spectral accuracy; that is, T ^ n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In the present work, we continue our study of these formulas for the special case in which f (x) = cos π (x - t) T sin 3 π (x - t) T u (x) , where u(x) is in C ∞ (R) and is T-periodic. Actually, we prove that T ^ n (s) [ f ] , s = 0 , 1 , 2 , are exact for a class of singular integrals involving T-periodic trigonometric polynomials of degree at most n - 1 ; that is, T ^ n (s) [ f ] = I [ f ] when f (x) = cos π (x - t) T sin 3 π (x - t) T ∑ m = - (n - 1) n - 1 c m exp (i 2 m π x / T). We also prove that, when u(z) is analytic in a strip | Im z | < σ of the complex z-plane, the errors in all three T ^ n (s) [ f ] are O (e - 2 n π σ / T) as n → ∞ , for all practical purposes. [ABSTRACT FROM AUTHOR]

In this work, a novel gas kinetic flux solver (GKFS) is presented for simulation of compressible and incompressible flows in the continuum and slip regimes. The finite volume method is adopted to discretize the governing differential equations, and inviscid and viscous fluxes at the cell interface are evaluated simultaneously by the local solution to the Boltzmann equation. Different from the conventional GKFS, in which the local solution to the Boltzmann equation is divided into the equilibrium part and the nonequilibrium part and the nonequilibrium distribution function is approximated by the difference of equilibrium distribution functions at the cell interface and its surrounding points, the present solver evaluates the local solution by integrating the Boltzmann equation along the characteristic line. As a result, the local solution to the Boltzmann equation consists of the equilibrium distribution function at the cell interface and the initial distribution function at the surrounding points. In the present work, the initial distribution function is given from the first‐order Chapman–Enskog expansion. Finally, the distribution function at the cell interface is only relevant to the macroscopic variables and their spatial derivatives. Accordingly, the fluxes across the cell interface can be evaluated by the moments of the distribution function at the cell interface. Test results show that the present solver can provide accurate numerical predictions for flows in both continuum and slip regimes. [ABSTRACT FROM AUTHOR]

A Ciarlet-Raviart type isoparametric mixed finite element method (MFEM) is constructed and analyzed for solving a class of fourth-order elliptic equation with Navier boundary condition defined on a curved domain in R 2 , and numerical quadrature is also considered in the scheme. The existence and uniqueness of the numerical solutions are proved under certain numerical quadrature. With the help of the special technically analysis, the optimal error estimates with H 1 norm are obtained in Ω h , which yields better accuracy than using a convex polygonal domain to approximate the curved domain. For either constant coefficients or nonconstant coefficients problem, numerical examples are listed to confirm theoretical analysis respectively. [ABSTRACT FROM AUTHOR]

The numerical integration over a planar domain that is cut by an implicitly defined boundary curve is an important problem that arises, for example, in unfitted finite element methods and in isogeometric analysis on trimmed computational domains. In this paper, we introduce a a very general version of the transport theorem for moving domains defined by implicitly defined curves and use it to establish an efficient and accurate quadrature rule for this class of domains. In numerical experiments it is shown that the method achieves high orders of convergence. Our approach is suited for high-order geometrically unfitted finite element methods as well as for high-order trimmed isogeometric analysis. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of finite-range singular integrals that are defined in the sense of Hadamard Finite Part, assuming that g ∈ C ∞ [ a , b ] and f (x) ∈ C ∞ (R t) is T-periodic with f ∈ C ∞ (R t) , R t = R \ { t + k T } k = - ∞ ∞ , T = b - a . Using a generalization of the Euler–Maclaurin expansion developed in [A. Sidi, Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp., 81:2159–2173, 2012], we unify the treatment of these integrals. For each m, we develop a number of numerical quadrature formulas T ^ m , n (s) [ f ] of trapezoidal type for I[f]. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3 , and these are T ^ 3 , n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , h = T n , T ^ 3 , n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , h = T n , T ^ 3 , n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4) , h = T n. For all m and s, we show that all of the numerical quadrature formulas T ^ m , n (s) [ f ] have spectral accuracy; that is, T ^ m , n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. We provide a numerical example involving a periodic integrand with m = 3 that confirms our convergence theory. We also show how the formulas T ^ 3 , n (s) [ f ] can be used in an efficient manner for solving supersingular integral equations whose kernels have a (x - t) - 3 singularity. A similar approach can be applied for all m. [ABSTRACT FROM AUTHOR]

An accurate method is proposed to calculate volume integrals arising from integral equation (IE) analysis using the Schaubert–Wilton–Glisson (SWG) basis function (BF). The volume integral is transformed into a summation of some of line integrals. The dimension reduction is achieved using the divergence theorem twice. Finally, each line integral is evaluated using an accuracy-controllable quadrature rule. We compare our results with those generated by Mathematica. Good agreement is found, which verifies the effectiveness and correctness of our method. [ABSTRACT FROM AUTHOR]

Considering the both effect of boundary approximation and numerical quadrature, a second‐order isoparametric element method is given to solve the homogeneous isotropic plane linear elasticity problem in domain Ω with curved boundary. By using technically analysis, the optimal error estimate with ‖u˜→−u→h‖1,Ωh=O(h2) is obtained, where the function u˜→ is an extension of the true solution u→ to Ω˜. It yields better accuracy than traditional quadratic finite element method. Finally, two numerical examples are presented, which further illustrate the analytical result and show the scheme is effective. [ABSTRACT FROM AUTHOR]

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

In map projection theory, it is usual to utilize numerical quadrature rules to estimate the overall map distortion. However, it is not known which method is the most efficient to approximate this integral. In this paper, overall map distortion is calculated analytically by a computer algebra system. Various integration methods are compared to the exact results. Some calculations are also performed on irregular spherical polygons. Considering the experiments, the author suggests utilizing the first-order Gaussian quadrature as it always gave reasonable results, although it is not the best for all cases. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of Hadamard Finite Part (HFP) integrals H σ (t ; u) = ⨎ 0 T | sin π (x − t) T | σ u (x) d x , 0 < t < T , σ < − 1 , σ ⁄ ∈ Z , u (x) being sufficiently differentiable and T -periodic on R. Thus σ = − (m + δ) , m ∈ { 1 , 2 , 3 , ... } , 0 < δ < 1. For each such σ , we regularize H σ (t ; u) , and show that H σ (t ; u) = H σ + 2 r (t ; U σ) , r = ⌊ (m + 1) / 2 ⌋ , where U σ (x) = ∑ k = 0 r a k u (2 k) (x) for some constants a k , H σ + 2 r (t ; U σ) being a regular integral. We then propose to approximate H σ (t ; u) by the quadrature formula Q σ , n (t ; u) ≡ H σ (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q σ , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q σ , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals H σ (t ; u) with σ < − 1 and σ ⁄ ∈ Z apply, with no changes, to the regular singular integrals H σ (t ; u) with σ > − 1 and σ ⁄ ∈ Z. We illustrate the effectiveness of Q σ , n (t ; u) with numerical examples both for σ < − 1 and σ > − 1. Finally, we show that the HFP or regular integral ⨎ 0 T f (x) d x of any T -periodic integrand f (x) that has algebraic singularities of the form | x − t + k T | σ , 0 < t < T , k = 0 , ± 1 , ± 2 , ... , with σ ⁄ ∈ Z , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed as H σ (t ; u) , where u (x) is a T -periodic and sufficiently differentiable function of x on R that can be computed from f (x). Therefore, ⨎ 0 T f (x) d x can be computed efficiently using our new numerical quadrature formulas Q σ , n (t ; u) on the individual H σ (t ; u). Again, only 2 n function evaluations, namely, u (x n , k) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]

This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and third- to fourth-order convergence in space. • Compute volume fractions from (complicated) hypersurfaces on Cartesian grids. • High accuracy due to local second-order approximation of hypersurface. • Third- to fourth-order convergence of global volume error with spatial resolution. • Robustness for a wide range of ratios of principal curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we consider to compute high frequency Hankel transforms numerically. An asymptotic quadrature rule is derived based on the variable upper bound integrals. The scheme is verified to be effective by testing some numerical examples. [ABSTRACT FROM AUTHOR]

Summary: With the development of the generalized/extended finite element method for fracture problems, the accurate and efficient integration of singular enrichment functions has been an open issue, especially for the 3D case. In this paper, we reveal the near singularities caused by distorted integral patch/cell shape numerically and theoretically during the implementation of generalized Duffy transformation, and the Duffy‐distance transformation is developed step by step for the 2D and 3D vertex singularities. Meanwhile, the 3D conformal preconditioning strategy is constructed to eliminate the near singularity caused by element shape distortion during the iso‐parametric transformation, which enables the Duffy‐distance transformation to be applicable for arbitrary shaped tetrahedral elements. As a result, the near singularities can be fully or partly canceled depending on the order of singularity. The implementation of the proposed scheme in existing codes is straightforward. Numerous numerical examples for arbitrary shaped triangles and tetrahedrons are presented to demonstrate its robustness and efficiency, along with comparisons to the generalized Duffy transformation. [ABSTRACT FROM AUTHOR]

Abstract The successful implementation of the Galerkin Boundary Element Method hinges on the accurate and effective quadrature of the influence coefficients. For parabolic boundary integral operators quadrature must be performed in space and time where integrals have singularities when source- and evaluation points coincide. For problems where the surface is fixed, the time integration can be performed analytically, but for moving geometries numerical quadrature in space and time must be used. For this case a set of transformations is derived that render the singular space–time integrals into smooth integrals that can be treated with standard tensor product Gauss quadrature rules. This methodology can be applied to the heat equation and to transient Stokes flow. [ABSTRACT FROM AUTHOR]