Analogue of Newton–Cotes formulas for numerical integration of functions with a boundary-layer component

The numerical integration of functions with a boundary-layer component whose derivatives are not uniformly bounded is investigated. The Newton–Cotes formulas as applied to such functions can lead to significant errors. An analogue of Newton–Cotes formulas that is exact for the boundary-layer component is constructed. For the resulting formula, an error estimate that is uniform with respect to the boundary-layer component and its derivatives is obtained. Numerical results that agree with the error estimates are presented.     

An analogue of the four-point Newton-Cotes formula for a function with a boundary-layer component

The construction of the Newton-Cotes formulas is based on approximating the integrand by a Lagrange polynomial. The error of such quadrature formulas can be great for a function with a boundary-layer component. In this paper, an analog of the four-point Newton-Cotes rule is constructed. The construction is based on using a nonpolynomial interpolation that is exact for the boundary layer component. Error estimates of the quadrature rule independent of the boundary layer component gradients are obtained. Numerical experiments are performed.     

Quadrature formulas for functions with a boundary-layer component

Quadrature formulas for one-variable functions with a boundary-layer component are constructed and studied. It is assumed that the integrand can be represented as the sum of a regular and a boundary-layer component, the latter having high gradients that reduce the accuracy of classical quadrature formulas, such as the trapezoidal and Simpson rules. The formulas are modified so that their error is independent of the gradients of the boundary-layer component. Results of numerical experiments are presented.     

Spline interpolation on a uniform grid for functions with a boundary-layer component

Spline interpolation of functions of one variable with a boundary-layer component is examined. Functions of this type can arise in the solution of a singularly perturbed boundary value problem on an interval. Spline interpolation formulas that are exact for the boundary-layer component are constructed, and their errors are estimated. Formulas for calculating the derivative based on the constructed interpolants are obtained. Numerical results are presented.     

Cubature formulas for a two-variable function with boundary-layer components

Cubature formulas for evaluating the double integral of a two-variable function with boundary-layer components are constructed and studied. Because of the boundary-layer components, the cubature formulas based on Newton-Cotes formulas become considerably less accurate. Analogues of the trapezoidal and Simpson rules that are exact for the boundary-layer components are constructed. Error estimates for the constructed formulas are derived that are uniform in the gradients of the integrand in the boundary layers.     

一类求积公式的构造方法

本文利用 Euler-Maclaurin求和公式构造了一类求积公式 ,称为修正复合梯形公式 .它和复合梯形公式的求积节点及计算量是一样的 ,但收敛阶有很大的提高 ,特别适合于计算带有各种类型小波的数值积分 .     

An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

In this paper, the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by means of an analytic approach. Unlike perturbation techniques, this approach gives accurate analytic approximations uniformly valid for all dimensionless time. Besides, a simple but accurate analytic formula for the local skin friction is given, which agrees well with numerical results and thus is useful in the related industries. To the best of our knowledge, this type of analytic solutions has been never reported. Furthermore, the proposed analytic approach has general meaning and therefore may be applied in the similar way to other unsteady boundary-layer flows to get accurate analytic solutions valid for all time.     

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m ² nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m ? 1.     

Higher-order Newton–Cotes rules with end corrections

We present higher-order quadrature rules with end corrections for general Newton–Cotes quadrature rules. The construction is based on the Euler–Maclaurin formula for the trapezoidal rule. We present examples with 6 well-known Newton–Cotes quadrature rules. We analyze modified end corrected quadrature rules, which consist on a simple modification of the Newton–Cotes quadratures with end corrections. Numerical tests and stability estimates show the superiority of the corrected rules based on the trapezoidal and the midpoint rules.     

An analog to Gaussian quadrature implemented on a specific trigonometric basis

A new quadrature formula implemented on a nonstandard basis of trigonometric functions is constructed. The quadrature is comparable in accuracy to a Gaussian quadrature formula and is used with the same class of functions. However, this quadrature differs greatly from that for periodic functions, which is also based on trigonometric functions.     

Maximization of the lift/drag ratio of airfoils with a turbulent boundary layer: Sharp estimates,approximation, and numerical solutions

The lift/drag ratio of an airfoil placed in an incompressible attached flow is maximized taking into account the viscosity in the boundary-layer approximation. An exact solution is constructed. The situation when the resulting solutions are not in the admissible class of univalent flows is discussed. A procedure is proposed for determining physically feasible airfoils (with a univalent flow region) with a high lift/drag ratio. For this purpose, a class of airfoils is constructed that are determined by a twoparameter function approximating the found exact solution to the variational problem. For this class, the ranges of free parameters leading to physically feasible flows are found. The results are verified by computing a turbulent boundary layer using Eppler’s method, and airfoils with a high lift/drag ratio in an attached flow are detected.     

A natural interpolation formula for the numerical solution of singular integral equations with hilbert kernel

The direct quadrature method for the numerical solution of singular integral equations with Hilbert kernel is investigated and a very accurate natural interpolation formula for the approximation of the unknown function is proposed. It is further proved that this formula coincides with Nyström's natural interpolation formula for the Fredholm integral equation of the second kind equivalent to the original integral equation if the same quadrature rule is used in both cases.     

The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Summary. The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains are not subdomains of the original domain and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence for approximate eigenfunctions in the -norm is proved while in the -norm an almost optimum rate of convergence (i.e. near to is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter . Received September 18, 1993 / Revised version received September 26, 1994     

具有Markov调制的随机种群系统半驯服Euler法的指数稳定性

根据半驯服Euler法讨论了具有Markov调制的随机年龄结构种群系统的数值解. 在非局部Lipschitz条件下, 利用~Burkholder-Davis-Gundy~不等式、It\^{o} 公式和~Gronwall~引理, 证明了半驯服Euler数值解不仅强收敛阶数为~0.5, 而且这种方法在时间步长一定的条件下有很好的均方指数稳定性. 最后通过数值例子对所给的结论进行了验证.     

General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.     

On an optimal quadrature formula in the sense of Sard

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K ₂(P ₂). Using S.L. Sobolev’s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L₂⁽²⁾(0,1)L_2^{(2)}(0,1). The obtained optimal quadrature formula is exact for the trigonometric functions sinx and cosx. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.     

A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.

     

Chebyshev series method for computing weighted quadrature formulas

In this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions. The main goal is to evaluate accurately a definite integral, whose mass is highly concentrated near some points. The numerical implementation of this approach is based on the calculation of Chebyshev series and some integration formulas which are exact for polynomials. In terms of accuracy, the proposed method can be compared with rational Gauss quadrature formula.     

Second-order boundary-layer solutions for flow past a blunted wedge

The second-order effects of the longitudinal curvature and displacement for flow past a blunted wedge have been studied employing the Görtler power series in streamwise coordinate, σ. The first five terms for each of the effects have been computed. The results, in general, have a restricted region of validity in the downstream direction due to the limited radius of convergence. The results are improved by using Euler transformation technique and it is found that the results are good even as σ→∞. For a parabola, the present results are compared with the exact numerical solution of the Navier-Stokes equations and it is shown that the second-order boundary-layer equations can be usefully employed down to Reynolds number 10³.     

中心受集中载荷的固定夹支边圆板和圆底扁球壳的卡门方程的精确解

由于卡门方程的非线性性和耦合性,使得寻求精确解的困难很大。迄今为止,除了少数未从数学上严格证明其收敛性的精确解外,大多数均采用近似方法求解。本文将卡门方程化为非线性奇异耦合的积分方程组,运用迭代法求得了连续函数序列。通过证明其一致收敛性,得到了中心受集中载荷作用的固定夹紧边界的圆板和圆底扁球壳的卡门方程的精确解的解析式及其收敛性证明。