Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

We present an elegant algorithm for stably and quickly generating the weights of Fejér’s quadrature rules and of the Clenshaw–Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions. AMS subject classification (2000) 65D32, 65T20, 65Y20     

A numerical method for the integration of oscillatory functions

A new method for the calculation of the integrals $$I_1 (m) = \int\limits_a^b {f(x)\sin mxdx} andI_2 (m) = \int\limits_a^b {f(x)\cos mxdx}$$ is presented. The functionf(x) is approximated by a sum of Chebyshev polynomials. The Chebyshev coefficients are then used to calculate a Neumann series approximation forI ₁(m) andI ₂(m). The numerical examples demonstrate that this method is very accurate and efficient.     

On the effect of numerical integration in the Galerkin boundary element method

Summary. We discuss the effect of cubature errors when using the Galerkin method for approximating the solution of Fredholm integral equations in three dimensions. The accuracy of the cubature method has to be chosen such that the error resulting from this further discretization does not increase the asymptotic discretization error. We will show that the asymptotic accuracy is not influenced provided that polynomials of a certain degree are integrated exactly by the cubature method. This is done by applying the Bramble-Hilbert Lemma to the boundary element method. Received May 24, 1995     

The centroid method of numerical integration

Summary The midpoint method of integration of a function of one variable is perhaps the simplest method of numerical integration, although it is often not mentioned in textbooks. It is here generalized to any number of dimensions and the generalization is called thecentroid method. This again is a very simple method and it can be conveniently used, for example, for the integration of a function of several variables over any non-pathological region. The numerical examples include the integration of multinormal integrands.     

A multi-step method for the numerical integration of ordinary differential equations

Summary In this paper we develop a multi-step method of order nine for obtaining an approximate solution of the initial value problemy'=f(x,y),y((x₀)=y ₀. The present method makes use of the second derivatives, namely, at the grid points. A sufficient criterion for the convergence of the iteration procedure is established. Analysis of the discretization error is performed. Various numerical examples are presented to demonstrate the practical usefulness of our integration method.

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Zusammenfassung In dieser Arbeit entwickeln wir eine mehrschrittige Methode der neunten Ordnung, um eine angenäherte Lösung des Anfangswertproblemsy'=f(x, y), y(x ₀)=y ₀. zu erhalten. Diese Methode bedient sich der Ableitungen zweiter Ordnung an den Schnittpunkten, d.h. . Ein hinreichendes Kriterium für die Konvergenz des Iterationsprozesses wird aufgestellt. Eine Analyse des Diskretionsfehlers ist durchgeführt. Verschiedene numerische Beispiele sollen den praktischen Nutzen unserer Integrationsmethode beweisen.

     

On the p version of the finite element method in the presence of numerical integration

We analyze the error in the p version of the finite element method when the effect of the quadrature error is taken into account. We extend some results by Banerjee and Suri [Math. Comput. 59 , 1–20 (1992)] on the H¹-norm error to the case of the error in the L₂ norm. We investigate three sources of quadrature error that can occur: the error due to the numerical integration of the right-hand side, that due to nonconstant coefficients, and that due to the presence of mapped elements. Presented are various theoretical and computational examples regarding the sharpness of our results. In addition, we make a note on the use of numerical quadrature in conjunction with p-adaptive procedures and on the necessity of overintegration in the h version with linear elements, when the L₂ norm is of interest. © 1993 John Wiley & Sons, Inc.     

A method for numerical integration on an automatic computer

A new method for the numerical integration of a well-behaved function over a finite range of argument is described. It consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term. Illustrative examples are given, and the method is compared with the most commonly-used alternatives, namelySimpson's rule and the method ofGauss.     

A method of high-order accuracy for the numerical integration of boundary value problems

A finite-difference method for the integration of the linear two-point boundary value problem $$y'' = f(x)y + g(x), y(a) = y_a , y(b) = y_b $$ is constructed. It uses values off′,f″,g′,g″ at the grid points to obtainO(h ⁸) global error, and allows strict global error bounds, while needing only the solution of a tridiagonal system of equations. Numerical examples are presented to demonstrate the practical usefulness of our method.     

On the numerical imbedding method

Journal of Applied Mathematics and Computing - WhenF(x) = 0 is a system of nonlinear equations, we have established the parametrizied Homotopy algorithm for solvingF(x) = 0 and some theorems for...     

Regularized numerical integration of multibody dynamics with the generalized method

This paper discusses the consistent regularization property of the generalized α method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane’s equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its A-stability and same order of accuracy for positions and velocities.     

A four-step phase-fitted method for the numerical integration of second order initial-value problems

A four-step method with phase-lag of infinite order is developed for the numerical integration of second order initial-value problems. Extensive numerical testing indicates that this new method can be generally more accurate than other four-step methods.     

On numerical integration of perturbed linear oscillating systems

Zusammenfassung Es wird ein Schrittverfahren zur numerischen Integration von gestörten linearen Differentialgleichungssystemen aufgestellt. Die Grundlage dazu bildet die Definition einer Folge von ganzen Funktionen, die keine Polynome sind, aber eine einfache Potenzreihendarstellung besitzen. Im Gegensatz zu den üblichen Potenzreihenverfahren, wird nicht nach Potenzen der unabhängigen Variabeln, sondern nach diesen Funktionen entwickelt. Das Integrationsverfahren hat die Eigenschaft, dass die ungestörten Differentialgleichungen ohne Diskretisationsfehler integriert werden, und dass die Eigenwerte der Koeffizientenmatrix nicht berechnet werden müssen. Die Verbesserung gegenüber der gewöhnlichen Potenzreihenmethode wird durch asymptotische Formeln für die Residuen und durch numerische Beispiele belegt.Nach einer Einleitung über Ziel und Zweck der Arbeit wird die Methode anhand eine: gestörten linearen Differentialgleichung zweiter Ordnung illustriert. Anschliessend wird das Integrationsverfahren auf beliebige gestörte lineare Systeme verallgemeinert.     

A block method for the numerical integration of stiff systems of ordinary differential equations

A new approach to the approximate numerical integration of stiff systems of first order ordinary differential equations is developed. In this approach several different formulae are applied in a well defined cyclic order to produce highly accurate integration schemes with infinite regions of absolute stability. The efficiency of these new algorithms, compared with that of certain existing ones, is demonstrated for some particular test problems.     

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.     

A new two-step P-stable hybrid Obrechkoff method for the numerical integration of second-order IVPs

In this paper, we introduce a class of new P-stable hybrid Obrechkoff methods for the numerical solution of second-order initial value problems. We generate a two-step, symmetric, P-stable hybrid Obrechkoff method of order 8 with two off-step, symmetric, points and phase lag of order at least 2 and for special cases, this order can be increased up to 6. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability.     

On approximating the finite Hilbert transform and applications in numerical integration

In this study, approximating the finite Hilbert transform are given for absolutely continuous mappings. Then, some numerical experiments for the obtained approximation are also presented.     

On numerical integration by the shift and application to Wiener space

Since the advantages of quasi-Monte Carlo methods vanish when the dimension of the basic space increases, the question arises whether there are better methods than the classical Monte Carlo in large or infinite-dimensional basic spaces. We study here the use of the shift operator with the pointwise ergodic theorem whose implementation is particularly interesting. After recalling the theoretical results on the speed of convergence in a form useful for applications, we give sufficient criteria for the law of iterated logarithm in several cases and, in particular, in situations involving the Wiener space.