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Delays are important processes represented by continuous simulation models; however, representing queueing delays efficiently within continuous simulations merits the development of new methodology. Rothkopf and Oren introduced the concept of using a surrogate distribution, viz., the negative-binomial, as a closure approximation to the infinite set of Chapman-Kolmogorov equations representing a nonstationary M/M/s queue. The method presented in this paper uses the Polya-Eggenberger distribution as a surrogate for the true distribution of the number in the queueing system at a particular time and only requires the numerical integration of five differential equations. The paper presents numerical results comparing the Polya surrogate and Rothkopf and Oren's approximation for a number of diverse cases, and these results indicate that the Polya surrogate is, in general, more accurate, although exceptions were encountered. Moreover, queueing delays represented by a closure approximation involving a surrogate distribution, in particular, the Polya, are suitable for use within a larger continuous simulation. [ABSTRACT FROM AUTHOR]

The article presents an algorithm for the numerical integration over a finite interval automatically. The algorithm's purpose is to automatically calculate the integral over a finite interval with relative error not exceeding a specified value. It utilizes a basic integration algorithm applied under the control of algorithms which invoke adaptive or nonadaptive subdivision of the range of integration. The subdivision processes will only normally be needed on extremely difficult integrals as the basic algorithm is sufficiently powerful.

In a companion paper to this, "I Methodology and Experiences," the automatic Clenshaw-Curtis quadrature scheme was described and how each quadrature formula used in the scheme requires a cosine transformation of the integrand values was shown. The high cost of these cosine transformations has been a serious drawback in using Clenshaw-Curtis quadrature. Two other problems related to the cosine transformation have also been troublesome. First, ... conventional computation of the cosine transformation by recurrence relation is numerically unstable, particularly at the low frequencies which have the largest effect upon the integral. Second, in case the automatic scheme should require refinement of the sampling, storage is required to save the integrand values after the cosine transformation is computed. This second part of the paper shows how the cosine transformation can be computed by a modification of the fast Fourier transform and all three problems overcome. The modification is also applicable in other circumstances requiring cosine or sine transformations, such as polynomial interpolation through the Chebyshev points. [ABSTRACT FROM AUTHOR]

Clenshaw-Curtis quadrature is a particularly important automatic quadrature scheme for a variety of reasons, especially the high accuracy obtained from relatively few integrand values. However, it has received little use because it requires the computation of a cosine transformation, and the arithmetic cost of this has been prohibitive. This paper is in two parts; a companion paper, "II Computing the Cosine Transformation," shows that this objection can be overcome by computing the cosine transformation by a modification of the fast Fourier transform algorithm. This first part discusses the strategy and various error estimates, and summarizes experience with a particular implementation of the scheme. [ABSTRACT FROM AUTHOR]

Let f have n continuous derivatives on a closed interval [a, b] and let L be a linear functional. The attempt is made to approximate L (f) with L (Q) where Q is a polynomial, approximating f. Algorithms are developed for rapid computation of L (Q) for a wide class of selections of Q which includes the Lagrangian and Hermitian rules as special cases. [ABSTRACT FROM AUTHOR]

An algorithm for Filon quadrature is described. Considerable attention has been devoted to an analysis of the round-off and truncation errors. The algorithm includes an automatic error control feature. [ABSTRACT FROM AUTHOR]

In the numerical integration of an improper integral of the first kind, it is customary to truncate the integral when the change yielded by the last iteration is less than some predetermined constant. The efficiency of such integration schemes can often be improved by use of recent advances in the theory of nonlinear transformations; however, for several important integrals, e.g. integrals whose integrands are rational polynomials these transformations fail to yield much improvement. In this paper, several methods of convergence improvement are developed which greatly improve convergence of some improper integrals, including the integrals of rational polynomials. [ABSTRACT FROM AUTHOR]

It is common to need to integrate numerically functions that diverge somewhere outside the range of integration. Even if the divergence occurs quite far away, integration formulas like Simpson's, that depend on fining a polynomial, usually will be inaccurate: near a pole they will be very bad. A method is described that gives formulas that will integrate functions of this kind accurately if the orders and positions of the poles are known. Explicit formulas are given that are easy to use on an automatic computer. It is shown that they can be used for some other singularities as well as poles. If the integral converges, integration can be carried to the singularity. The accuracy of the integration with a pole of second order is discussed, and, as an example, the new formula is compared with Simpson's in the computation of ∫ 0X sec2 π ξ d ξ, 0 < X < 0.5. In this case the new formula is more accurate for X > 0.1, being 30 times as accurate as Simpson's at X = 0.3, 400 times at X = 0.4 and 104 times at X = 0.47. Thus, the new formulas are useful even far from the pole, while near the pole their advantage is overwhelming. [ABSTRACT FROM AUTHOR]

A procedure is given for continuously computing and monitoring the step size to be used by a self-starting, pth-order numerical integration method to solve an initial value problem. The procedure uses an estimate of the truncation error to calculate the step size. [ABSTRACT FROM AUTHOR]

A study has been made to determine which methods of numerical integration require the least computation time for a given amount of truncation error when applied to a particular system of ordinary differential equations where function evaluations are relatively trivial. Recent methods due to Butcher and Gear are compared with classic Runge-Kutta Kutta-Nyström and Adams methods. Some of the newer one-step methods due to Butcher are found to be slightly superior, but no one method is found to have any great advantage over the others in the application to this particular problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a new technique to get superconvergence phenomenon of spline quasi-interpolants at the knots of the partition. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. Moreover, we give an application to numerical integration. Numerical results are given to illustrate the theoretical ones. [ABSTRACT FROM AUTHOR]

The article presents an algorithm for modified Havie integration. The Havie integration method for the approximate evaluation of the definite integral is based upon the parallel generation of the Romberg table of trapezoidal values and the table of rectangular values. Like Romberg quadrature, Havie integration requires the evaluation of the rectangular values. The method, used by mathematician G. Fairweather in a modified Romberg algorithm, leads to a significant improvement in accuracy for large orders of extrapolation.

It is shown to be practical to compute Jn(x) by numerical integration of its integral representation using the trapezoidal rule. The error in this approximation was studied empirically. [ABSTRACT FROM AUTHOR]

SecDec is a program which can be used for the factorization of dimensionally regulated poles from parametric integrals, in particular multi-loop integrals, and the subsequent numerical evaluation of the finite coefficients. Here we present version 3.0 of the program, which has major improvements compared to version 2: it is faster, contains new decomposition strategies, an improved user interface and various other new features which extend the range of applicability. Program summary Program title: SecDec 3.0 Catalogue identifier: AEIR_v3_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIR_v3_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 123828 No. of bytes in distributed program, including test data, etc.: 1651026 Distribution format: tar.gz Programming language: Wolfram Mathematica, perl, Fortran/C++. Computer: From a single PC to a cluster, depending on the problem. Operating system: Unix, Linux. RAM: Depending on the complexity of the problem Classification: 4.4, 5, 11.1. Catalogue identifier of previous version: AEIR_v2_1 Journal reference of previous version: Comput. Phys. Comm. 184(2013)2552 Does the new version supersede the previous version?: Yes Nature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in gauge theories. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds). Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter, where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. Reasons for new version: • Improved user interface. • Additional new decomposition strategies. • Usage on a cluster is much improved. • Speed-up in numerical evaluation times. • Various new features (please see below). Summary of revisions: • Implementation of two new decompositions strategies based on a geometric algorithm. • Scans over large ranges of parameters are facilitated. • Linear propagators can be treated. • Propagators with negative indices are possible. • Interface to reduction programs like Reduze, Fire, LiteRed facilitated. • Option to use numerical integrator from Mathematica. • Using CQUAD for 1-dimensional integrals to improve speed of numerical evaluations. • Option to include epsilon-dependent dummy functions. Restrictions: Depending on the complexity of the problem, limited by memory and CPU time. Running time: Between a few seconds and several hours, depending on the complexity of the problem. [ABSTRACT FROM AUTHOR]

We present our numerical observations on dynamics in the system of two linearly coupled FitzHugh–Nagumo oscillators close to the destabilization of the state of rest. Under the considered parameter values the system, if integrated sufficiently accurately, converges to small-scale periodic oscillations. However, minor numerical inaccuracies, which occur already at the default precision of the standard Runge–Kutta solver, lead to a breakup of periodicity and an onset of large-scale aperiodic bursting. [ABSTRACT FROM AUTHOR]

The extended Gaussian quadrature rules are shown to be an efficient tool for numerical integration of wide class of functions with singularities of logarithmic type. The quadratures are exact for the functions pol1 n − 1 ( x ) + ln x pol2 n − 1 ( x ) , where pol1 n − 1 ( x ) and pol2 n − 1 ( x ) are two arbitrary polynomials of degree n − 1 and n is the order of the quadrature formula. We present an implementation of numerical algorithm that calculates the nodes and the weights of the quadrature formulas, provide a Fortran code for numerical integration, and test the performance of different kinds of Gaussian quadratures for functions with logarithmic singularities. Program summary Program title: GAUSEXT Catalogue identifier: AETP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AETP_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2535 No. of bytes in distributed program, including test data, etc.: 39 963 Distribution format: tar.gz Programming language: Mathematica, Fortran. Computer: PCs or higher performance computers. Operating system: Linux, Windows, MacOS. RAM: Kilobytes. Classification: 4.11. Nature of problem: Quadrature formulas for numerical integration, effective for a wide class of functions with end-point singularities of logarithmic type. Solution method: The method of solution is based on the algorithm developed in Ref. [1] with some modifications. Running time: Milliseconds to minutes. References: [1] J. Ma, V. Rokhlin, S. Wandzura, Generalized Gaussian quadrature rules for systems of arbitrary functions, Soc. Indust. Appl. Math. J. Numer. Anal. 33 (3) (1996) 971-996. [ABSTRACT FROM AUTHOR]

Abstract: A parallel implementation of the Shardlow splitting algorithm (SSA) for Dissipative Particle Dynamics (DPD) simulations is presented. The isothermal and isoenergetic SSA implementations are compared to the DPD version of the velocity-Verlet integrator in terms of numerical stability and performance. The integrator stability is assessed by monitoring temperature, pressure and total energy for both the standard and ideal DPD fluid models. The SSA requires special consideration due to its recursive nature resulting in more inter-processor communication as compared to traditional DPD integrators. Nevertheless, this work demonstrates that the SSA exhibits stability over longer time steps that justify its regular use in parallel, multi-core applications. For the computer architecture used in this study, a factor of 10–100 speedup is achieved in the overall time-to-solution for isoenergetic DPD simulations and a 15–34 speedup is achieved for the isothermal DPD simulations. [Copyright &y& Elsevier]

Abstract: This paper investigates a class of algorithms for numerical integration of a function in dimensions over a compact domain by Monte Carlo methods. We construct a histogram approximation to the function using a partition of the integration domain into a set of bins specified by some parameters. We then consider two adaptations: the first is to subtract the histogram approximation, whose integral we may easily evaluate explicitly, from the function and integrate the difference using Monte Carlo; the second is to modify the bin parameters in order to make the variance of the Monte Carlo estimate of the integral the same for all bins. This allows us to use Student’s -test as a trigger for rebinning, which we claim is more stable than the test that is commonly used for this purpose. We provide a program that we have used to study the algorithm for the case where the histogram is represented as a product of one-dimensional histograms. We discuss the assumptions and approximations made, as well as giving a pedagogical discussion of the myriad ways in which the results of any such Monte Carlo integration program can be misleading. [Copyright &y& Elsevier]

Abstract: The accurate numerical evaluation of nearly singular boundary integrals is a major concerned issue in the implementation of the boundary element method (BEM). In this paper, the previous distance transformation method is extended into triangular elements both in polar and Cartesian coordinate systems. A new simple and efficient method using an approximate nearly singular point is proposed to deal with the case when the nearly singular point is located outside the element. In general, the results obtained using the polar coordinate system are superior to that in the Cartesian coordinate system when the nearly singular point is located inside the element. Besides, the accuracy of the results is influenced by the locations of the nearly singular point due to the special topology of triangular elements. However, when the nearly singular point is located outside the element, both the polar and Cartesian coordinate systems can get acceptable results. [Copyright &y& Elsevier]

Abstract: This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach. [Copyright &y& Elsevier]

Abstract: Two analytical modifications of the original viscoelastic time-harmonic Lamb’s problem expressions are presented with the aim of improving their numerical integration efficiency. Firstly, a new change of variable in the Lamb’s problem integrands is proposed, which allows a standardization of the integration sampling vector and a complete spatial-frequency field solution after performing only one numerical integration/transformation. Secondly, the Lamb’s problem static integrands are modified and introduced into the original integrands to reduce their spectral content at high wavenumbers and, therefore, the sampling vector lengths needed to avoid aliasing. [Copyright &y& Elsevier]

Abstract: We present the program SecDec 2.0, which contains various new features. First, it allows the numerical evaluation of multi-loop integrals with no restriction on the kinematics. Dimensionally regulated ultraviolet and infrared singularities are isolated via sector decomposition, while threshold singularities are handled by a deformation of the integration contour in the complex plane. As an application, we present numerical results for various massive two-loop four-point diagrams. SecDec 2.0 also contains new useful features for the calculation of more general parameter integrals, related for example to phase space integrals. Program summary: Program title: SecDec 2.0 Catalogue identifier: AEIR_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIR_v2_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 156829 No. of bytes in distributed program, including test data, etc.: 2137907 Distribution format: tar.gz Programming language: Wolfram Mathematica, Perl, Fortran/C++. Computer: From a single PC to a cluster, depending on the problem. Operating system: Unix, Linux. RAM: Depending on the complexity of the problem Classification: 4.4, 5, 11.1. Catalogue identifier of previous version: AEIR_v1_0 Journal reference of previous version: Comput. Phys. Comm. 182(2011)1566 Does the new version supersede the previous version?: Yes Nature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in gauge theories. Numerical integration in the presence of integrable singularities (e.g., kinematic thresholds). Solution method: Algebraic extraction of singularities in dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter , where the coefficients are finite integrals over the unit hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. Reasons for new version: In the previous version the calculation of multi-scale integrals was restricted to the Euclidean region. Now multi-loop integrals with arbitrary physical kinematics can be evaluated. Another major improvement is the possibility of full parallelization. Summary of revisions: [•] No restriction on the kinematics for multi-loop integrals. [•] The integrand can be constructed from the topological cuts of the diagram. [•] Possibility of full parallelization. [•] Numerical integration of multi-loop integrals written in C++ rather than Fortran. [•] Possibility to loop over ranges of parameters. Restrictions: Depending on the complexity of the problem, limited by memory and CPU time. The restriction that multi-scale integrals could only be evaluated at Euclidean points is superseded in version 2.0. Running time: Between a few minutes and several days, depending on the complexity of the problem. Test runs provided take only seconds. [Copyright &y& Elsevier]

This technical note concerns the deterministic sampling points construction strategy for unscented Kalman filter (UKF) and cubature Kalman filter (CKF). From the numerical-integration viewpoint, a new deterministic sampling points set is derived by orthogonal transformation on the cubature points. By embedding these points into the UKF framework, a modified nonlinear filter named transformed unscented Kalman filter (TUKF) is derived. The TUKF can address the nonlocal sampling problem inherent in CKF while maintaining the virtue of numerical stability for high dimensional problems. Moreover, the methodology proposed in this technical note can be used to construct nonlinear filters with improved accuracy for certain problems. The performance of the proposed algorithm is demonstrated through a nonlinear high dimensional problem. [ABSTRACT FROM AUTHOR]

We present two simple designs of matter-wave beam splitters in a trapped Bose-Einstein condensate (BEC). In our scheme, identical pairs of atomic solitons are produced by an adequate control-in time and/or space-of the scattering length. Our analysis is performed by numerical integration of the Gross-Pitaevskii equation and supported by several analytical estimates. Our results show that these devices can be implemented in the frame of current BEC experiments. The system has potential applications for the construction of a soliton interferometer. [ABSTRACT FROM AUTHOR]

Abstract: The threshold for the onset of chaos in asymmetric nonlinear dynamic systems can be determined using an extended Padé method. In this paper, a double-well asymmetric potential system with damping under external periodic excitation is investigated, as well as an asymmetric triple-well potential system under external and parametric excitation. The integrals of Melnikov functions are established to demonstrate that the motion is chaotic. Threshold values are acquired when homoclinic and heteroclinic bifurcations occur. The results of analytical and numerical integration are compared to verify the effectiveness and feasibility of the analytical method. [Copyright &y& Elsevier]

Abstract: This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods. [Copyright &y& Elsevier]

Abstract: In this article, we propose an approach, based on the variation-of-constants formula, for the numerical discretisation over long-time intervals of several stochastic oscillators. Additive and multiplicative noises are treated separately. The proposed schemes permit the use of large step sizes in the presence of a high frequency in the problem and offer various additional properties. These new numerical integrators can be viewed as a stochastic-generalisation of the trigonometric integrators for highly oscillatory deterministic problems. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to present two different approaches, both based on Lie symmetry analysis, allowing to obtain solutions of the Riemann problem with non-classical discontinuous initial data for a system of balance laws. The first method uses the invariance with respect to a Lie group of exact point symmetries, whereas the second one is based on the approximate Lie symmetries. The two methods are applied to a particular 2×2 system of balance laws describing rate-type materials. Moreover, some numerical simulations are performed in order to validate the approximate solutions and check that the exact solution, as a parameter involved in the initial conditions tends to zero, approaches the solution of the classical Riemann problem. [Copyright &y& Elsevier]

Abstract: Computing the centroid of an interval type-2 fuzzy set is an important operation in a type-2 fuzzy logic system, and is usually implemented by Karnik–Mendel (KM) iterative algorithms. By connecting KM algorithms and continuous KM algorithms together, this paper gives theoretical explanations on the initialization methods of KM and Enhanced Karnik–Mendel (EKM) algorithms, proposes exact methods for centroid computation of an interval type-2 fuzzy set, and extends the Enhanced Karnik–Mendel (EKM) algorithms to three different forms of weighted EKM (WEKM) algorithms. It shows that EKM algorithms become a special case of the WEKM algorithms when the weights of the latter are constant value. It also shows that, in general, the weighted EKM algorithms have smaller absolute error and faster convergence speed than the EKM algorithms which make them very attractive for real-time applications of fuzzy logic system. Four numerical examples are used to illustrate and analyze the performance of WEKM algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, two-step extended Runge–Kutta–Nyström-type methods for the numerical integration of perturbed oscillators are presented and studied. The new methods inherit the framework of two-step hybrid methods and are adapted to the special feature of the true flows in both the internal stages and the updates. Based on the EN-trees theory [H.L. Yang, X.Y. Wu, X. You, Y.L. Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794], order conditions for the new methods are derived via the -series defined on the set BT of branches and the -series defined on the subset BWT of BT. The stability and phase properties are analyzed. Numerical experiments show the applicability and efficiency of our new methods in comparison with the well-known high quality methods proposed in the scientific literature. [Copyright &y& Elsevier]

Abstract: The treatment of van der Waals interactions in density functional theory is an important field of ongoing research. Among different approaches developed recently to capture these non-local interactions, the van der Waals density functional (vdW-DF) developed in the groups of Langreth and Lundqvist is becoming increasingly popular. It does not rely on empirical parameters, and has been successfully applied to molecules, surface systems, and weakly-bound solids. As the vdW-DF requires the evaluation of a six-dimensional integral, it scales, however, unfavorably with system size. In this work, we present a numerically efficient implementation based on the Monte-Carlo technique for multi-dimensional integration. It can handle different versions of vdW-DF. Applications range from simple dimers to complex structures such as molecular crystals and organic molecules physisorbed on metal surfaces. [Copyright &y& Elsevier]

Abstract: In this paper, a new approach is presented for the numerical evaluation of arbitrary singular domain integrals based on the radial integration method. The transformation from domain integrals to boundary integrals and the analytical elimination of singularities can be accomplished by expressing the non-singular part of the integration kernels as polynomials of the distance r and using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some numerical examples are provided to verify the correctness and robustness of the presented method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we attack the problem of devising a finite volume method for computational fluid dynamics and related phenomena which can deal with complex geometries while attaining high-orders of accuracy and spectral convergence at a reasonable computational cost. As a first step towards this end, we propose a control volume finite element method for the solution of the advection–diffusion equation. The numerical method and its implementation are carefully tested in the paper where h- and p-convergence are checked by comparing numerical results against analytical solutions in several relevant test-cases. The numerical efficiency of a selected set of operations implemented is estimated by operation counts, ill-conditioning of coefficient matrices is avoided by using an appropriate distribution of interpolation points and control-volume edges. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: The unusual implementation of explicit Falkner methods for solving special second-order initial-value problems increases the order of the method as will be shown by means of numerical examples. In this paper we made an analysis of the propagation of the truncation errors in both the usual and the unusual implementations, thus justifying the numerical results obtained. By that, some error bounds for the global truncation errors on the solution and on the derivative are provided. Some numerical examples confirm that the bounds are realistic. Stability analysis is also addressed, and intervals of stability are presented. [Copyright &y& Elsevier]

Abstract: This paper presents an efficient integrating-factor method for solving a nonlinear Dirac equation (NLD). Starting with the simplest case of one space-variable, this method, unlike other approaches proposed in the bibliography, can be easily extended to problems with more space-variables. Our algorithm is implemented in Matlab© and the numerical experiments performed reveal its effectiveness and reliability. [Copyright &y& Elsevier]

Abstract: A known feature of any mixed interpolation boundary integral equations (BIE)-based methods is that equilibrium is not generally guaranteed in the numerical solution. Here, a complete meshless technique, based on the boundary element-free method (BEFM) with complete equilibrium satisfaction for 2D elastostatic analysis is proposed. The BEFM adopted is a meshless method based on boundary integral equations such as local boundary integral equation (LBIE) method and boundary node method (BNM), differing from them with respect to the integration domain and the approximation scheme. [Copyright &y& Elsevier]

Abstract: The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory. [Copyright &y& Elsevier]

Abstract: In this paper, extended Runge–Kutta–Nyström-type methods for the numerical integration of perturbed oscillators with low frequencies are presented, which inherit the framework of RKN methods and make full use of the special feature of the true flows for both the internal stages and the updates. Following the approach of J. Butcher, E. Hairer and G. Wanner, we develop a new kind of tree set to derive order conditions for the extended Runge–Kutta–Nyström-type methods. The numerical stability and phase properties of the new methods are analyzed. Numerical experiments are accompanied to show the applicability and efficiency of our new methods in comparison with some well-known high quality methods proposed in the scientific literature. [Copyright &y& Elsevier]

In this paper, we show how to compute an over-approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration. We introduce two ways to do so. The first one is a full interval method which handles whole domains for set computation and relies on state-of-the-art validated numerical integration methods. The second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty. Since the derived bracketing systems are piecewise Ch-differentiabIe functions, validated numerical integration methods cannot be used directly. Hence, our contribution resides in the use of hybrid automata to model the bounding systems. We give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set. The computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques. Sufficient conditions are given which ensure the c-practical stability of the enclosures given by our hybrid bounding method. Two examples are also given which show that the performance of our method is very promising. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents a simple but efficient new numerical scheme for the integration of nonlinear constitutive equations. Although it can be used for the integration of a system of algebraic and differential equations in general, the scheme is primarily developed for use with the direct solution methods for solving boundary value problems, e.g. explicit dynamic analysis in ABAQUS/Explicit. In the developed explicit scheme, where no iteration is required, the implementation simplicity of the forward-Euler scheme and the accuracy of the backward-Euler scheme are successfully combined. The properties of the proposed NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, are compared with the properties of the classical forward-Euler scheme and backward-Euler scheme. For this purpose two highly nonlinear examples, with the von Mises and GTN material model considered, have been studied. The accuracy of the new scheme is demonstrated to be at least of the same level as experienced by the backward-Euler scheme, if we compare them on the condition of the same CPU time consumption. Besides, the simplicity of the NICE scheme, which is due to implementation similarity with the classical forward-Euler scheme, is its great Advantage. [Copyright &y& Elsevier]

Abstract: We present a new program performing the sector decomposition and integrating the expression afterwards. The program takes a set of propagators and a set of indices as input and returns the epsilon-expansion of the corresponding integral. Program summary: Program title: FIESTA Catalogue identifier: AECP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECP_v1_0.html Program obtainable from: CPC Program Library, Queen''s University, Belfast, N. Ireland Licensing provisions: GPL v2 No. of lines in distributed program, including test data, etc.: 88 281 No. of bytes in distributed program, including test data, etc.: 6 153 480 Distribution format: tar.gz Programming language: Wolfram Mathematica 6.0 [3] and C Computer: from a desktop PC to supercomputer Operating system: Unix, Linux, Windows RAM: depends on the complexity of the problem Classification: 4.4, 4.12, 5, 6.5 External routines: QLink [1], Vegas [2] Nature of problem: The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression. Solution method: The sector decomposition is based on a new strategy. The sector decomposition, pole resolution and epsilon-expansion are performed in Wolfram Mathematica 6.0 [3]. The data is stored on hard disk via a special program, QLink [1]. The expression for integration is passed to the C-part of the code, that parses the string and performs the integration by the Vegas algorithm [2]. This part of the evaluation is perfectly parallelized on multi-kernel computers. Restrictions: The complexity of the problem is mostly restricted by the CPU time required to perform the evaluation of the integral, however there is currently a limit of maximum 11 positive indices in the integral; this restriction is to be removed in future versions of the code. Running time: Depends on the complexity of the problem. References: [1] http://qlink08.sourceforge.net, open source. [2] G.P. Lepage, The Cornell preprint CLNS-80/447, 1980. [3] http://www.wolfram.com/products/mathematica/index.html2. [Copyright &y& Elsevier]

Abstract: This work proposes an approach for the numerical computation of time and space derivatives of the time-domain solution of scalar wave propagation problems by means of a boundary element method formulation. Here, this formulation employs the so-called convolution quadrature method. Non-homogeneous initial conditions are taken into account by means of a general procedure, known as initial condition pseudo-force procedure, which replaces the initial conditions by equivalent pseudo-forces. The boundary integral equation with initial conditions contribution is differentiated analytically and the quadrature weights of the standard formulation are transformed in order to compute time and space derivatives at interior points. Numerical examples are presented to show the efficiency of the implemented formulation. [Copyright &y& Elsevier]

This paper describes a new algorithm for Monte Carlo integration, based on the Field Estimator for Arbitrary Spaces (FiEstAS). The algorithm is discussed in detail, and its performance is evaluated in the context of Bayesian analysis, with emphasis on multimodal distributions with strong parameter degeneracies. Source code is available upon request. [Copyright &y& Elsevier]

Abstract: This paper addresses the problem of critical point calculations for pairwise comparisons of three normal means. One-sided and two-sided pairwise comparisons are standard multiple comparisons procedures, and while tables of critical points exist for balanced situations with equal sample sizes, only limited tables of critical points exist for unbalanced cases. A new algorithm is developed in this paper using elementary methods which allows the critical points to be found in all situations using only a one-dimensional numerical integration. Programs have been developed to implement the algorithm which will allow experimenters to easily find the required critical points and -values. [Copyright &y& Elsevier]

Abstract: The dual reciprocity boundary element method (DRBEM) is studied thoroughly in the present paper. To the best knowledge of the authors, the DRBEM has never been applied to 3D half-space dynamics previously. In the present paper, the mathematical derivation of the method is presented, with stress placed on peculiarities of the method when applied to transient half-space dynamics. It has been found that semi-infinite domains (half-space) are more difficult to simulate, since the truncation of the discretization of the half-space surface by boundary elements results in excessive spurious wave reflection on the border of the discretized region. Mathematical derivation of the method is followed by its numerical implementation. Wave propagation due to various kinds of loading in the time-domain is studied afterward, aimed at the validation of the method. The main advantage of the DRBEM over its counterparts, used to model infinite and semi-infinite domains (such as classical BEM formulation, integral transformations, thin layer method), is that it produces time- and frequency-independent matrices (mass and stiffness matrix), by preserving a boundary-only discretization (no internal nodes necessary). This makes the method very attractive, since this feature is very close to the common engineering understanding and the final equation of motion has a similar form like the one known from the finite element method. Moreover, the formulation allows for seamless incorporation of non-homogeneous initial conditions, i.e. non-zero initial displacements and velocities and surface tractions can be prescribed. [Copyright &y& Elsevier]

Abstract: Derivation and implementation of a numerical solution of a time-dependent diffusion equation is given in detail, based on the meshless local Petrov–Galerkin method (MLPG). A simple method is proposed that ensures a constant number of support nodes for each point. Numerical integrations are carried out over local square domains. The implicit Crank–Nicolson scheme is used for time discretization. A detailed convergence study was performed experimentally to optimize the number of support nodes, quadrature domain size and other parameters. The accuracy of the MLPG solution is compared with that of standard methods on a unit square and on an irregularly shaped test domain. As expected, the finite difference method on a regular mesh is incompetitive on irregularly shaped domains. MLPG is significantly more accurate when using moving least square shape functions of degree two than with degree one. It is comparable to the finite element method of degree two in the error norm and about two times less accurate in the error norm. [Copyright &y& Elsevier]

Abstract: In this paper, a boundary element formulation for modeling two-dimensional microstructures containing cylindrical voids is presented. Each void is inserted in a standard boundary element mesh in a efficient way using a single special hole element. The traditional discretization in several boundary elements is then avoided. Trigonometric functions are used as base for the element shape functions. A family of hole elements with 3, 4, 5 and 6 nodes is proposed. The numerical integration of the strongly singular kernels found in the element implementation is accomplished by the direct method, resulting in a regularized element. The convergence behavior of the proposed approach is analyzed for several quadrature orders. A static condensation scheme is performed on the system of equation to further improve the formulation. The accuracy of proposed method is illustrated with some examples. [Copyright &y& Elsevier]

Abstract: Numerical solutions of the equal width wave (EW) equation are obtained by using a Galerkin method with quartic B-spline finite elements, a differential quadrature method with cosine expansion basis and a meshless method with radial-basis functions. Solitary wave motion, interaction of two solitary waves and wave undulation are studied to validate the accuracy and efficiency of the proposed methods. Comparisons are made with analytical solutions and those of some earlier papers. The accuracy and efficiency are discussed by computing the numerical conserved laws and , error norms. [Copyright &y& Elsevier]