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An effort has been undertaken at the NASA Langley Research Center to assess the capabilities of available computational methods for use in propulsion integration design studies of transonic transport aircraft, particularly of pylon/nacelle combinations which exhibit essentially no interference drag. The three computer codes selected represent state-of-the-art computational methods for analyzing complex configurations at subsonic and transonic flight conditions. These are: EULER, a finitie volume solution of the Euler equation; VSAERO, a panel solution of the Laplace equation; and PPW, a finite difference solution of the small disturbance transonic equations. In general, all three codes have certain capabilities that allow them to be of some value in predicting the flows about transport configurations, but all have limitations. Until more accurate methods are available, careful application and interpretation of the results of these codes are needed.

We find estimates for the error in replacing an integral $\int f d\mu$ with respect to a fractal measure $\mu$ with a discrete sum $\sum_{x \in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $\Delta f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpi\'{n}ski gasket, both for the standard self-similar measure and energy measures, and for other fractals.

Loop integration results have been obtained using numerical integration and extrapolation. An extrapolation to the limit is performed with respect to a parameter in the integrand which tends to zero. Results are given for a non-scalar four-point diagram. Extensions to accommodate loop integration by existing integration packages are also discussed. These include: using previously generated partitions of the domain and roundoff error guards.

The integration of the equations of motion in gravitational dynamical systems -- either in our Solar System or for extra-solar planetary system -- being non integrable in the global case, is usually performed by means of numerical integration. Among the different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie series has shown some advantages. In its original form (Hanslmeier 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the method by deriving an expression of the Lie-terms when other major forces are considered. As a matter of fact, previous studies had been made but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations: relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration of the equation of motions for typical Near-Earth objects and planet Mercury.

Higher order digital nets have recently been recognized as one of the most promising branches of quasi-Monte Carlo methods. The notable feature of higher order digital nets is that they can exploit the smoothness of a function for numerical integration and achieve an improved convergence rate of the integration error for smooth functions. One prominent construction of such nets is based on digitally interlacing the components of the classical digital net whose number of components is a multiple $d$ of the dimension. In this study we consider the weighted unanchored Sobolev spaces of smoothness $\alpha \ge 2$, and derive an upper bound on the mean square worst-case error for digitally shifted higher order digital nets. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be made efficient use of to obtain good polynomial lattice point sets that are used for interlaced components. Through this approach we are able to get some tractability results under certain conditions on the weights. Fast construction using the fast Fourier transform requires the construction cost of order $O(sdN \log N)$ operations using O(N) memory, where $N$ is the number of points and $s$ is the dimension, which is a significant reduction in the construction cost as compared to higher order polynomial lattice rules. Numerical experiments confirm that the performance of our constructed point sets often outperforms in terms of our introduced quality criterion the performances of higher order digital nets with Sobol' sequences and Niederreiter-Xing sequences used as interlaced components, indicating the usefulness of our algorithm.

Hybrid shell elements have long been regarded with reserve by the commercial finite element developers despite the high degree of reliability and accuracy associated with such formulations. The fundamental reason is the inherent higher computational cost of the hybrid approach as compared to the displacement-based formulations. However, a noteworthy factor in favor of hybrid elements is that numerical integration to generate element matrices can be entirely avoided by the use of symbolic integration. In this paper, the use of the symbolic computational approach is presented for an assumed-stress hybrid shell element with drilling degrees of freedom and the significant time savings achieved is demonstrated through an example.

We examine the effect of numerical integration on the convergence of high order pyramidal finite element methods. Rational functions are indispensable to the construction of pyramidal interpolants so the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include rational functions and show that despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.

We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta, and whose potential is a function of the generalized positions. We apply the various techniques to the well-known H\'enon-Heiles system, and use the Smaller Alignment Index (SALI) method of chaos detection to evaluate the percentage of its chaotic orbits. The accuracy and the speed of the integration schemes in evaluating this percentage are used to investigate the numerical efficiency of the various techniques.

The use of two Class 2/fixed mesh/fixed order/multistep integration packages of the PECE type for the numerical integration of the second order, nonlinear, ordinary differential equation of the satellite orbit problem. These two methods are referred to as the general and the second sum formulations. The derivation of the basic equations which characterize each formulation and the role of the basic equations in the PECE algorithm are discussed. Possible starting procedures are examined which may be used to supply the initial set of values required by the fixed mesh/multistep integrators. The results of the general and second sum integrators are compared to the results of various fixed step and variable step integrators.

Gauss quadrature formula for approximate integration by computer - numerical integration

Numerical integration methods for computing flow of gas in chemical nonequilibrium behind normal shock wave

Journal of Research of the National Bureau of Standards

Many numerical codes now under development to solve Einstein's equations of general relativity in 3+1 dimensional spacetimes employ the standard ADM form of the field equations. This form involves evolution equations for the raw spatial metric and extrinsic curvature tensors. Following Shibata and Nakamura, we modify these equations by factoring out the conformal factor and introducing three ``connection functions''. The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. We evolve small amplitude gravitational waves and make a direct comparison of the numerical performance of the modified equations with the standard ADM equations. We find that the modified form exhibits much improved stability.

Error analyses for numerically integrating first order ordinary differential equations

Comparison of fourth order Runge-Kutta numerical integration technique with two numerical integration formulas in mathematical models used for computing rigid body trajectories

Numerical error comparisons for integration of near earth orbits in various coordinate systems

Mellin-Barnes (MB) techniques applied to integrals emerging in particle physics perturbative calculations are summarized. New versions of AMBRE packages which construct planar and nonplanar MB representations are shortly discussed. The numerical package MBnumerics.m is presented for the first time which is able to calculate with a high precision multidimensional MB integrals in Minkowskian regions. Examples are given for massive vertex integrals which include threshold effects and several scale parameters.

In this paper we show two examples of numerical orbital integrations (Planar Circular Restricted Three Body Problem) in which even though the conservation of Jacobi's constant is near to 1 part in 10e8, the integration proves to be wrong. That is, for some particular cases the Jacobi's constant is insensitive to very important qualitative (not quantitative) changes in the integrated orbit which are produced by numerical errors. We provide specific recomendations to test and ensure that a numerical code is working properly, regardless the numerical method employed.

Bell System Technical Journal, 47: 4. April 1968 pp 511-527. Numerical Integration of Systems of Stiff Nonlinear Differential Equations. (Sandberg, I.W. ; Shichman, H.)

Bell System Technical Journal, 47: 4. April 1968 pp 511-527. Numerical Integration of Systems of Stiff Nonlinear Differential Equations. (Sandberg, I.W. ; Shichman, H.)

Bell System Technical Journal, 47: 4. April 1968 pp 511-527. Numerical Integration of Systems of Stiff Nonlinear Differential Equations. (Sandberg, I.W. ; Shichman, H.)

viii, pages 157-372

viii, pages 157-372

A set of nonlinear and coupled equations governing the thermomechanical deformations of a viscoplastic body undergoing simple shearing deformations is integrated in time by using the Forward-Difference-Galerkin-Finite-Element (FDGFE) method and the Crank-Nicolson-Galerkin-Finite-Element (CNGFE) method. In the latter scheme the number of unknown functions is increased so that the governing equations involve only first order spatial derivatives. It is shown that the solutions obtained by the two methods agree qualitatively but the CNGFE method seems to introduce considerable damping into the system. However, the time increment needed to obtain a stable solution by the CNGFE method is 200 times that required by the FDGFE method. Keywords: Shear bands, Finite element method, Crank nicolson method.

Operational unification of finite difference methods for numerical integration of ordinary differential equations

A new class of linear multistep methods for numerical integration of differential equations is reported that permits satellite computation solutions to be corrected at certain points in the past as the integration advances in time. Algorithms have been developed for the solution of both first- and second-order differential equations. The back correction method appears to be more efficient than classical methods when dominant and perturbing forces can be separated.

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.

LONG-TERM GOALS: 1. Application of multidimensional Fourier analysis for the extremely fast numerical integration of the partial differential equations of surface water waves is the long-term goal of this work. The approach is a generalization of linear Fourier analysis and therefore is a hyperfast extension of the fast Fourier transform (FFT) to the nonlinear Euler water wave equations. 2. The present report discusses progress made in both the shallow and deep-water environments, anticipating the marriage of the two in the coming year for a complete, higher order, hyperfast integration to Euler. An important application of the approach is possibility of the on-board computation of wave properties as derived from shipboard direct wave and/or radar measurements of the sea surface, leading to prediction of sea state conditions, including the presence of rogue waves in real time.

We developed Micro SAINT computational networks for numerical integration and solving initial value problems for linear and nonlinear first- and second-order ordinary differential equations, as well as for systems of differential equations. These Micro SAINT computer programs are written with a user friendly approach where the user will be required to supply the input information and the functional form(s) of the function(s) in the function library section of Micro SAINT without any changes in the main programs. These computational modules could be used as subnetworks in modeling psychophysiological and biomedical problems of interest in naval aerospace medical research. For example, Micro SAINT developed models can be used by staff medical officers to predict psychophysiological performance of naval aircrew personnel under sustained operational work schedules. (KR)

In many computer-implemented signal-processing systems, it is necessary to perform numerical integrations. If the function to be integrated is a band-limited time signal represented by a set of samples, serious distortion can occur if the wrong integration technique is used. In particular, if the integration is used to calculate a Fourier transform, large error side- lobes will be introduced if Simpson's rule or other higher-order rules are used. It is shown that the rectangular rule introduces no distortion and gives a result which is consistent with the number of samples.

All the original objectives of the project have been addressed. Lacunae have been identified and studied theoretically in the solutions of the Maxwell's equations that govern the propagation of electromagnetic waves in vacuum. The case of acoustics has also been analyzed along the same lines. A lacunae-based numerical algorithm has been built for the long-term integration of the Maxwell's equations. Its key property is the grid convergence that is uniform in time; in other words, there is no error build-up during the integration over arbitrary long intervals. Lacunae-based artificial boundary conditions (ABCs) have also been constructed and tested; their actual performance fully meets the theoretical expectations. The corresponding results have been presented at scientific conferences and published in peer reviewed journals. New theoretical and numerical issues have also been discovered in the course of the project that were not known at the time of its inception. They have warranted the extension of the project by six months beyond its original expiration date. It turns out that implementation of the lacunae-based ABCs for the Maxwell's equations necessitates that special auxiliary field sources be used in the form of solenoidal currents and zero charges. Particular types of such sources have been analyzed and employed previously. A general procedure for their construction has been introduced and implemented at the most recent stage of the project. The results of this implementation will be reported in a forthcoming paper.

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.

We consider the radiation transfer problem in the discrete-ordinate, plane-parallel approach. We introduce two benchmark problems with exact known solutions and show that for strongly non-homogeneous media the homogeneous layers approximation can lead to errors of 10% in the estimation of the intensity. We propose and validate a general purpose numerical method that transforming the two-boundary problem into an initial boundary problem, using an adaptative step integration and an interpolation of the local optical properties, can improve the accuracy of the solution up to two orders of magnitude. This is furthermore of interest for practical applications, such as atmospheric radiation transfer, where the scattering and absorbing properties of the media vary strongly with height and are only known, based on measurements or models, at certain discrete points.

One-loop two-, three- and four-point scalar functions are analytically integrated directly such that they are expressed in terms of Lauricella's hypergeometric function $F_D$. For two- and three-point functions, exact expressions are obtained with arbitrary combination of kinematic and mass parameters in arbitrary space-time dimension. Four-point function is expressed in terms of $F_D$ up to the finite part in the expansion around 4-dimensional space-time with arbitrary combination of kinematic and mass parameters. Since the location of the possible singularities of $F_D$ is known, information about the stabilities in the numerical calculation is obtained. We have developed a numerical library calculating $F_D$ around 4-dimensional space-time. The numerical values for IR divergent cases of four-point functions in massless QCD are calculated and agreed with golem95 package.

The field method is developed for arbitrary open branch domains subjected to general linear boundary conditions. Although closed branches are within the scope of the method, they are not treated here. The numerical feasibility of the method has been demonstrated by implementing it in a computer program for the linear static analysis of open branch shells of revolution under asymmetric loads. For such problems the field method eliminates the well-known numerical problem of long subintervals associated with the rapid growth of extraneous solutions. Also, the method appears to execute significantly faster than other numerical integration methods.

For abstract, see N76-33552.

The term 'trajectory problem' is taken to include problems that can arise, for instance, in connection with contour plotting, or in the application of continuation methods, or during phase-plane analysis. Geometrical techniques are used to construct difference methods for these problems to produce in turn explicit and implicit circularly exact formulae. Based on these formulae, a predictor-corrector method is derived which, when compared with a closely related standard method, shows improved performance. It is found that this latter method produces spurious limit cycles, and this behavior is partly analyzed. Finally, a simple variable-step algorithm is constructed and tested.

Report discusses several methods of performing numerical integration with computer. When data can be expressed as state vector that is dependent variable in a differential equation, self-starting integrators can be used to predict future data.

Due to the stringent design requirement for aerospace or nuclear structural components, considerable research interests have been generated on the development of constitutive models for representing the inelastic behavior of metals at elevated temperatures. In particular, a class of unified theories (or viscoplastic constitutive models) have been proposed to simulate material responses such as cyclic plasticity, rate sensitivity, creep deformations, strain hardening or softening, etc. This approach differs from the conventional creep and plasticity theory in that both the creep and plastic deformations are treated as unified time-dependent quantities. Although most of viscoplastic models give better material behavior representation, the associated constitutive differential equations have stiff regimes which present numerical difficulties in time-dependent analysis. In this connection, appropriate solution algorithm must be developed for viscoplastic analysis via finite element method.

We consider two strongly hyperbolic Hamiltonian formulations of general relativity and their numerical integration with a free and a partially constrained symplectic integrator. In those formulations we use hyperbolic drivers for the shift and in one case also for the densitized lapse. A system where the densitized lapse is an external field allows to enforce the momentum constraints in a holonomically constrained Hamiltonian system and to turn the Hamilton constraint function from a weak to a strong invariant. These schemes are tested in a perturbed Minkowski and the Schwarzschild space-time. In those examples we find advantages of the strongly hyperbolic formulations over the ADM system presented in [arXiv:0807.0734]. Furthermore we observe stabilizing effects of the partially constrained evolution in Schwarzschild space-time as long as the momentum constraints are enforced.

The Air Force Institute of Techology (AFIT) is developing a simulation framework to model a wide variety of existing and proposed Quantum Key Distribution (QKD) systems. This research investigates using graphical processing unit (GPU) technology to more efficiently integrate optical pulses modeled within this framework. The goal is to reduce the simulation execution time. A GPU algorithm is presented for performing numerical integration of optical pulses described by Gaussian curves to improve pulse energy and power calculations. In order to measure the performance of the algorithm a optimal timing method is needed. A timer using Comute Unified Device Architecture (CUDA) events is selected over a Windows system application programming interface (API) timer. The problem sizes studied produce speedups greater than 60x on the NVIDIA Tesla C2075 compared to the Intel i7-3610QM CPU.

B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge-Kutta methods. Connections to renormalization theory in perturbative quantum field theory have been established in recent years. The algebraic structure of classical Runge-Kutta methods is described by the Connes-Kreimer Hopf algebra. Lie-Butcher theory is a generalization of B-series aimed at studying Lie-group integrators for differential equations evolving on manifolds. Lie-group integrators are based on general Lie group actions on a manifold, and classical Runge-Kutta integrators appear in this setting as the special case of R^n acting upon itself by translations. Lie--Butcher theory combines classical B-series on R^n with Lie-series on manifolds. The underlying Hopf algebra combines the Connes-Kreimer Hopf algebra with the shuffle Hopf algebra of free Lie algebras. We give an introduction to Hopf algebraic structures and their relationship to structures appearing in numerical analysis, aimed at a general mathematical audience. In particular we explore the close connection between Lie series, time-dependent Lie series and Lie--Butcher series for diffeomorphisms on manifolds. The role of the Euler and Dynkin idempotents in numerical analysis is discussed. A non-commutative version of a Faa di Bruno bialgebra is introduced, and the relation to non-commutative Bell polynomials is explored.

A numerical algorithm suitable for rapid numerical solution of advanced turbulence-model equations on elliptic regions has been devised. The algorithm is particularly useful for obtaining accurate solutions when (a) one of the dependent variables is singular approaching a solid boundary, and (b) the number of mesh points available to resolve the flowfield is small. The new algorithm is used in conjunction with the MacCormack semi-implicit scheme for solving the time-averaged Navier-Stokes equations to obtain solutions for a laminar Mach 2 boundary layer and for a turbulent Mach 3 boundary layer. In both cases, numerical accuracy is shown to be excellent for meshes having less than 20 points normal to the surfaces. As a corollary result of this study we have shown that, using the new algorithm and the MacCormack scheme, time-averaged Navier-Stokes solutions are now feasible using a microcomputer. (Author)

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to the non-integrable case.

We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence which gives the conditions and the appropriate choice of the grid sizes. The method is applied to Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to the non-integrable case.

Algorithm for use in estimating accumulated numerical integration errors

Cowell numerical integration and satellite orbit calculation

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.

It is noted that there are some integrals which cannot be evaluated by determining an antiderivative, and these integrals must be subjected to other techniques. Numerical integration is one such method; it provides a sum that is an approximate value for some integral types. This module's purpose is to introduce methods of numerical integration and to describe related computer programing techniques for electronically carrying out the calculations. The material is seen to require very little background or knowledge about computers and their use. Exercises are provided at various points. It is expected that users work each of these problems, as they are viewed as essential to a genuine understanding of the ideas developed. (MP)