Estimates of the error in Gauss–Legendre quadrature for double integrals

Error estimates are a very important aspect of numerical integration. It is desirable to know what level of truncation error might be expected for a given number of integration points. Here, we determine estimates for the truncation error when Gauss–Legendre quadrature is applied to the numerical evaluation of two dimensional integrals which arise in the boundary element method. Two examples are considered; one where the integrand contains poles, when its definition is extended into the complex plane, and another which contains branch points. In both cases we obtain error estimates which agree with the actual error to at least one significant digit.     

A new family of Runge–Kutta type methods for the numerical integration of perturbed oscillators

Summary. Our task in this paper is to present a new family of methods of the Runge–Kutta type for the numerical integration of perturbed oscillators. The key property is that those algorithms are able to integrate exactly, without truncation error, harmonic oscillators, and that, for perturbed problems the local error contains the perturbation parameter as a factor. Some numerical examples show the excellent behaviour when they compete with Runge–Kutta–Nystr?m type methods. Received June 12, 1997 / Revised version received July 9, 1998     

An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

In this paper, we present two families of second-order and third-order explicit methods for numerical integration of initial-value problems of ordinary differential equations. Firstly, a family of second-order methods with two free parameters is derived by considering a suitable rational approximation to the theoretical solution of the problem at some grid points. Imposing that the principal term of the local truncation error of this family vanishes, we obtain an expression for one of the parameters in terms of the other. With this approach, a new one-parameter family of third-order methods is obtained. By selecting any 3(2) pair of second and third order methods, they can be implemented as an embedded type method, thus leading to a variable step-size formulation. We have considered one 3(2) pair of second and third order methods and made a comparison of numerical results with several ode solvers which are currently used in practice. The comparison of numerical results shows that the embedded 3(2) pair outperforms the methods considered for comparison.     

Alternating group explicit method for the diffusion equation

A new alternating group explicit method is presented for the finite difference solution of the diffusion equation. The new method uses stable asymmetric approximations to the partial differential equation which, when coupled in groups of two adjacent points on the grid, result in implicit equations which can be easily converted to explicit form and which offer many advantages. By judicious alternation of this strategy on the grid points of the domain an algorithm which possesses unconditional stability is obtained. This approach also results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method are briefly discussed and the results of numerical experiments presented.     

Theoretical Analysis of the Reproducing Kernel Gradient Smoothing Integration Technique in Galerkin Meshless Methods

Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions. The reproducing kernel gradient smoothing integration (RKGSI) is one of the optimal numerical integration techniques in Galerkin meshless methods with minimum integration points. In this paper, properties, quadrature rules and the effect of the RKGSI on meshless methods are analyzed. The existence, uniqueness and error estimates of the solution of Galerkin meshless methods under numerical integration with the RKGSI are established. A procedure on how to choose quadrature rules to recover the optimal convergence rate is presented.     

Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

A family of quadrature rules for integration over tetrahedral volumes is developed. The underlying structure of the rules is based on the cubic close-packed (CCP) lattice arrangement using 1, 4, 10, 20, 35, and 56 quadrature points. The rules are characterized by rapid convergence, positive weights, and symmetry. Each rule is an optimal approximation in the sense that lower-order terms have zero contribution to the truncation error and the leading-order error term is minimized. Quadrature formulas up to order 9 are presented with relevant numerical examples.     

A fourth order method for y″ = ƒ(x,y, y′)

A fourth order numerical method for solving the general second order differential equation $\text{y″ = ?(x, y, y′)}$ is suggested. This method is self starting, P-stable (when applied to $\text{y″ = ?(x, y)}$), and it involves only two function evaluations per step. Easily calculable and simple truncation error estimates are given, which can be used for automatic error control in computations. The method is illustrated on numerical examples.     

-curved finite elements and applications to plate and shell problems

Many thin-plate and thin-shell problems are set on plane reference domains with a curved boundary. Their approximation by conforming finite-elements methods requires ¹-curved finite elements entirely compatible with the associated ¹-rectilinear finite elements. In this contribution we introduce a ¹-curved finite element compatible with the P₅-Argyris element, we study its approximation properties, and then, we use such an element to approximate the solution of thin-plate or thin-shell problems set on a plane-curved boundary domain. We prove the convergence and we get a priori asymptotic error estimates which show the very high degree of accuracy of the method. Moreover we obtain criteria to observe when choosing the numerical integration schemes in order to preserve the order of the error estimates obtained for exact integration.     

Truncation and roundoff errors in three-point approximations of first and second derivatives

We use linear combinations of Taylor expansions to develop three-point finite difference expressions for the first and second derivative of a function at a given node. We derive analytical expressions for the truncation and roundoff errors associated with these finite difference formulae. Using these error expressions, we find optimal values for the stepsize and the distribution of the three points, relative to the given node. The latter are obtained assuming that the three points are equispaced. For the first derivative approximation, the distribution of the points relative to the given node is not symmetrical, while it is so for the second derivative approximation. We illustrate these results with a numerical example in which we compute upper bounds on the roundoff error.     

The Two-dimensional Electromagnetic Scattering from Periodic Chiral Structures and Its Finite Element Approximation

In this paper, we consider the electromagnetic scattering from periodic chiral structures. The structure is periodic in one direction and invariant in another direction. The electromagnetic fields in the chiral medium are governed by the Maxwell equations together with the Drude-Born-Fedorov equations. We simplify the problem to a two-dimensional scattering problem and we show that for all but possibly a discrete set of wave numbers, there is a unique quasi-periodic weak solution to the diffraction problem. The diffraction problem can be solved by finite element method. We also establish uniform error estimates for the finite element method and the error estimates when the truncation of the nonlocal transparent boundary operators takes place.     

Influence of Gauss and Gauss‐Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain

In this article, we examine the influence of numerical integration on finite element methods using quadrilateral or hexahedral meshes in the time domain. We pay special attention to the use of Gauss‐Lobatto points to perform mass lumping for any element order. We provide some theoretical results through several error estimates that are completed by various numerical experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Some numerical quadrature schemes of a non-conforming quadrilateral finite element

Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional (2-D) and three dimensional (3-D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.     

The parameter uniform numerical method for singularly perturbed parabolic reaction–diffusion problems on equidistributed grids

This article presents the study of singularly perturbed parabolic reaction–diffusion problems with boundary layers. To solve these problems, we use a modified backward Euler finite difference scheme on layer adapted nonuniform meshes at each time level. The nonuniform meshes are obtained by equidistribution of a positive monitor function, which involves the second-order spatial derivative of the singular component of the solution. The equidistributing monitor function at each time level allows us to use this technique to non-linear parabolic problems. The truncation error and the stability analysis are obtained. Parameter–uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.     

A stochastic approach to global error estimation in ODE multistep numerical integration

In order to assess the quality of approximate solutions obtained in the numerical integration of ordinary differential equations related to initial-value problems, there are available procedures which lead to deterministic estimates of global errors. The aim of this paper is to propose a stochastic approach to estimate the global errors, especially in the situations of integration which are often met in flight mechanics and control problems. Treating the global errors in terms of their orders of magnitude, the proposed procedure models the errors through the distribution of zero-mean random variables belonging to stochastic sequences, which take into account the influence of both local truncation and round-off errors. The dispersions of these random variables, in terms of their variances, are assumed to give an estimation of the errors. The error estimation procedure is developed for Adams-Bashforth-Moulton type of multistep methods. The computational effort in integrating the variational equations to propagate the error covariance matrix associated with error magnitudes and correlations is minimized by employing a low-order (first or second) Euler method. The diagonal variances of the covariance matrix, derived using the stochastic approach developed in this paper, are found to furnish reasonably precise measures of the orders of magnitude of accumulated global errors in short-term as well as long-term orbit propagations.     

Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions

We present a priori and a posteriori estimates for the error between the Galerkin and a discretized Galerkin method for the boundary integral equation for the single layer potential on the square plate. Using piecewise constant finite elements on a rectangular mesh we study the error coming from numerical integration. The crucial point of our analysis is the estimation of some error constants, and we demonstrate that this is necessary if our methods are to be used. After the determination of these constants we are in the position to prove invertibility and quasioptimal convergence results for our numerical scheme, if the chosen numerical integration formulas are sufficiently precise. © 1992 John Wiley & Sons, Inc.     

Spectral analysis of a high-order Hermitian algorithm for structural dynamics

The spectral analysis of an efficient step-by-step direct integration algorithm for the structural dynamic equation is presented. The proposed algorithm is formulated in terms of two Hermitian finite difference operators of fifth-order local truncation error and it is unconditionally stable with no numerical damping presenting a fourth-order truncation error for period dispersion (global error). In addition, although it is in competition with higher-order algorithms presented in the literature, the computational effort is similar to that of the classical second-order Newmark’s method. The numerical application for nonlinear structural dynamic problems is also considered.     

The Fourier-series method for inverting transforms of probability distributions

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.     

An efficient and reliable quadrature algorithm for integration with respect to binomial measures

In this work numerical methods for integration with respect to binomial measures are considered. Binomial measures are examples of fractal measures and arise when multifractal properties are investigated. Interpolatory quadrature rules are considered. An automatic integrator with local quadrature rules that generalize the five points Newton Cotes formula and error estimates based on null rules is then described. Numerical tests are performed to verify the efficiency and accuracy of the method. These tests confirm that the automatic integrator turns out to be as good as one of the best known quadrature algorithms with respect to the Lebesgue measure. AMS subject classification (2000)  28A25, 60G18, 65D30, 65D32, 68M15     

An efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations

In this paper, we propose an efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations. Firstly, we carry out a truncation from a three‐dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one‐dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results.     

Nordsieck representation of DIMSIMs

A new representation for diagonally implicit multistage integration methods (DIMSIMs) is derived in which the vector of external stages directly approximates the Nordsieck vector. The methods in this formulation are zero-stable for any choice of variable mesh. They are also easy to implement since changing step-size corresponds to a simple rescaling of the vector of external approximations. The paper contains an analysis of local truncation error and of error accumulation in a variable step-size situation. This revised version was published online in June 2006 with corrections to the Cover Date.