Showing papers on "Numerical analysis published in 1981"

Lectures on Numerical Methods for Non-Linear Variational Problems

Abstract: Many mechanics and physics problems have variational formulations making them appropriate for numerical treatment by finite element techniques and efficient iterative methods. This book describes the mathematical background and reviews the techniques for solving problems, including those that require large computations such as transonic flows for compressible fluids and the Navier-Stokes equations for incompressible viscous fluids. Finite element approximations and non-linear relaxation, augmented Lagrangians, and nonlinear least square methods are all covered in detail, as are many applications. "Numerical Methods for Nonlinear Variational Problems," originally published in the Springer Series in Computational Physics, is a classic in applied mathematics and computational physics and engineering. This long-awaited softcover re-edition is still a valuable resource for practitioners in industry and physics and for advanced students.

Moving Finite Elements. I

Abstract: We present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or “near-shocks”. These methods allow many nodes automatically to concentrate in the critical regions and move with them. For clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes, with Burgers’ equation as our test equation; but the generalization to much more general spaces and equations (including even certain previous “moving vorticity blobs” of the first author and S. Doss for the Navier–Stokes equations) becomes clear. In this paper we present the theoretical and computational details of our scheme, along with computational trial runs for Burgers’ equation showing that the nodes do concentrate sharply and move with the shocks as desired. The conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a Newton’s method for our implicit stiff ODE solver and were thus limited to ver...

A Numerical Method for Solving the Equations of Compressible Viscous Flow

Abstract: Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.

Numerical Methods Based on Whittaker Cardinal, or Sinc Functions

Abstract: This paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis. Described in this paper are methods of interpolation and approximation of functions and their derivatives, quadrature, the approximate evaluation of transforms (Hilbert, Fourier, Laplace, Hankel and Mellin) and the approximate solution of differential and integral equations. The methods have many advantages over classical methods which use polynomials as bases. In addition, all of the methods converge at an optimal rate, if singularities on the boundary of approximation are ignored.

Properties of numerical algorithms related to computing controllability

Abstract: The numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations. In particular, the numerical rank of a matrix, the numerical stability of algorithms, the sensitivity of problems, and the scaling of problems are discussed. A numerically stable algorithm is given for computing controllability, but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful, and this appears to be an open computational problem.

Numerical Electromagnetics Code (NEC)-Method of Moments. A User-Oriented Computer Code for Analysis of the Electromagnetic Response of Antennas and Other Metal Structures. Part 1: Program Description-Theory. Part 2: Program Description-Code. Volume 1. Revised

Abstract: : The Numerical Electromagnetics Code (NEC-2) is a computer code for analyzing the electromagnetic response of an arbitrary structure consisting of wires and surfaces in free space or over a ground plane. The is accomplished by the numerical solution of integral equations for induced currents. The solution includes Numerical Green's Function for partitioned-matrix solution and a treatment for lossy grounds that is accurate for antennas very close to the ground surface. The excitation may be an incident plane wave or a voltage sour wire, while the output may include current and charge density, electric or magnetic field in the vicinity of structure, and radiated fields. Other options compute the maximum coupling between antennas and facilitate Numerical Electromagnetics Code (NEC-2) Numerical analysis Antenna response Electromagnetic radiation. structure input. Hence the code may be used for antenna analysis, EMP, or scattering studies. Part 1 of the document includes the equations on which the code is based and a discussion of the approximations and numerical methods used in the numerical solution. Some comparisons to demonstrate the range of accuracy of approximations are also included. Details of the coding and a User's Guide are provided as parts 11 and 111, respectively.

Finite dimensional approximation of nonlinear problems

Abstract: In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Karman equations.

Numerical Methods for Harmonic Analysis on the Sphere

Abstract: : This report presents some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere. The data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement 'noise'. The case discussed to greatest length is that of complete, global data sets on regular grids (i.e., lines of latitude and longitude, the latter, at least, separated by constant interval); the case where data are sparsely and irregularly distributed is also considered in some detail. The first section presents some basic properties of spherical harmonics, stressing their relationship to two-dimensional Fourier series. Algorithms for the evaluation of the harmonic coefficients by numerical quadratures are given here, and it is shown that the number of operations is the order of N cubed for equal angular grids, where N is the number of lines of latitude, or 'Nyquist frequency', of the grid. The second section introduces a quadratic measure for the error in the estimation of the coefficients by linear techniques. This is the error measure of least squares collocation, which is a method that can be used for harmonic analysis. Efficient algorithms for implementing collocation on the whole sphere are described. a formal relationship between collocation and least squares adjustment is used to obtain an alternative form of the collocation algorithm that is likely to be stable with dense data sets and, with a minor modification, can be used to implement least squares adjustment as well. The basic principle is that for regular grids the variance-convariance matrix of the data consists of Toeplitz-circulant blocks, so it can be both set up and inverted very efficiently.

Error estimates for some mixed finite element methods for parabolic type problems

Abstract: We consider a class of mixed finit e element methodsfor second order elhptic problems intioduced by Raviart and Thomas and generahze or gwe alternative proofs of previously known error estimâtes for such methods We then extend these results to the corresponding parabohc problems thereby obtaimng estimâtes simüar to those previously known for conventwnal finite element methods for parabohc problems We also obtain corresponding results for a mixed finite element methodfor the stationary and evolutionary Stokes' équations Résumé — On considère une famille de methodes d'éléments finis mixtes pour les problèmes elliptiques du second ordre introduite par Raviart et Thomas, et on presente des généralisations, ou de nouvelles démonstrations, des estimations d'erreur connues auparavant pour ces méthodes On étend ensuite ces résultats aux problèmes paraboliques correspondants, et on obtient de cette façon des estimations semblables à celles déjà connues pour les méthodes d'éléments finis conformes pour les problèmes paraboliques. On obtient aussi des résultats correspondants pour une méthode d'éléments finis mixtes pour les équations de Stokes, dans les cas stationnai}e et d'évolution

Numerical methods for establishing solutions to the inverse problem of electromagnetic induction

Abstract: A previous paper (Parker, 1980) sets out a theory for deciding whether solutions exist to the inverse problem of electromagnetic induction and outlines methods for constructing conductivity profiles when their existence has been demonstrated. The present paper provides practical algorithms to perform the necessary calculations stably and efficiently, concentrating exclusively on the case of imprecise observations. The matter of existence is treated by finding the best fitting solution in a least squares sense; then the size of the misfit is tested statistically to determine the probability that the value would be met or exceeded by chance. We obtain the optimal solution by solving a constrained least squares problem linear in the spectral function of the electric field differential equation. The spectral function is converted into a conductivity profile by transforming its partial fraction representation into a continued fraction, using a stable algorithm due to Rutishauser. In addition to optimal models, which always consist of delta functions, two other types of model are examined. One is composed of a finite stack of uniform layers, constructed so that the product of conductivity and thickness squared is the same in each layer. The numerical techniques developed for the optimal model serve with only minor alteration to find solutions in this class. Models of the second kind are smooth. A special form of the response is chosen so that the kernel functions of the Gel'fand-Levitan integral equation are degenerate, thus allowing very stable and numerically efficient solution. Unlike previously published methods for finding conductivity models, these algorithms can provide solutions with misfits arbitrarily close to the smallest one possible. The methods are applied to magnetotelluric observations made by Larsen in Hawaii.

Modeling of the acoustic wave equation with transform methods

Abstract: Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such er...

Error estimates for the combinedh andp versions of the finite element method

Abstract: In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.

Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems

Abstract: Many of the popular methods for the solution of large matrix equations are surveyed with the hope of finding an efficient method suitable for both electromagnetic scattering and radiation problems and system identification problems.

Operator-theoretic and computational approaches to Ill-posed problems with applications to antenna theory

Abstract: A general framework for regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized: philosophy of resolution, regularization-approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted-norm least-squares solutions of first kind integral equations (and more generally of linear operator equations with nonclosed range). A common problem in all these methods is delineated: each method reduces the problem of resolution to a "nonstandard" minimization problem involving an unknown critical "parameter" whose "optimal" value is crucial to the numerical realization and amenability of the method. The "nonstandardness" results from the fact that one does not have explicitly, or a priori, the function to be minimized; it has to built up using additional information, convergence rate estimates, and robustness conditions, etc. Several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill-posed (identification and pattern synthesis) problems. An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The results will be applied specifically to problems of antenna synthesis and identification. However the thrust of the paper is devoted to the interdisciplinary character of operator-theoretic and numerical methods for ill-posed problems.

Numerical treatment of delay differential equations by Hermite Interpolation

Abstract: A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically. Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.

Fourier-space paths applied to the calculation of diffusion for the Chirikov-Taylor model

Abstract: A compact path-diagram method has been introduced for the calculation of velocity moments of a probability function. This method is complementary to the approach developed earlier by Rechester and White. It is applied to the Chirikov-Taylor model. Analytic expressions for velocity-space diffusion have been derived and compared with numerical computations. A numerical method for path summations has been developed which is more efficient than directly advancing the model equations, and is applicable for small field-amplitude values, where the direct stepping method is impractical.

Analysis of descriptor systems using numerical algorithms

Abstract: In this paper we analyze numerical methods for the solution of the large scale dynamical system E\dot{y}(t)=Ay(t)+g(t),Y(t_{0})=y_{0} , where E and A are matrices, possibly singular. Systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables. Special cases of such systems are algebraic-differential systems. We discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure, detect nonsolvable systems, resolve initial value consistency problems, and are applicable to "stiff" descriptor systems. We also present an algorithm for the control of the local truncation error on only the state variables.

Numerical Analysis of Laminar Natural Convection Between Concentric and Eccentric Cylinders

Abstract: A numerical analysis is carried out to investigate the local and overall heat transfer between concentric and eccentric horizontal cylinders. The numerical procedure, based on Stone's strongly Implicit method, is extended to the 3 × 3 coupled system of the governing partial differential equations describing the conservation of mass, momentum, and energy. This method allows finite-difference solutions of the governing equations without artificial viscosity, and conserves its great stability even for arbitrarily large time steps. The algorithm is written for a numerically generated, body-fitted coordinate system. This procedure allows the solution of the governing equations in arbitrarily shaped physical domains Numerical solutions were obtained for a Raylelgh number In the range 102-103, a Prandtl number of 0.7, and three different eccentric positions of the inner cylinder. The results are discussed in detail and are compared with previous experimental and theoretical results.

A Robust Numerical Technique for Power System State Estimation

Abstract: It is well known in numerical analysis that the least-squares solution via the conventional Gauss' normal equation used in power system state estimation is prone to ill-conditioning problems by its own nature. Under unfavorable circumstances, this may be detrimental to the method's performance. This paper utilizes a numerically more reliable algorithm, known as Golub's method, to solve the least-squares problem as formulated in power system state estimation. Its improved numerical properties stem from the use of orthogonal transformations, which are perfectly conditioned. Details of the algorithm and its implementation are given, as well as results of its application to three different networks, including an actual 121-bus power system.

On a conservation upwind finite element scheme for convective diffusion equations

Abstract: The purpose ofthis paper is to present a new class of upwind finite element schemes for convective diffusion équations and to gwe error analysis These schemes based on an intégral formula have the following advantages (i) They are effective particularly in the case when the convection is dominated^ (n) Solutions obtained by them satisfy a discrete conservation law, (in) Solutions obtained by a scheme with a particuîar choice satisfy a discrete maximum principle {under suitable conditions) We show that the finite element solutions converge to the exact one with rate 0(h) in L(Q, T, H (Q)) and L (0,T,L(a)) Resumé — Le but de cet article est de présenter une classe nouvelle de schémas d'éléments finis conservatîfs et décentres pour des équations de diffusion avec convection, et de donner des estimations d*erreur Les schémas, qui sont basés sur une formule intégrale, ont les avantages suivants (î) Ils sont effectifs surtout dans le cas où la convection est dominante, (n) Des solutions obtenues par eux satisfont a une loi de conservation discrète, (ni) Des solutions obtenues par un schéma particulier satisfont au principe du maximum discret (sous des conditions convenables) On montre que les solutions obtenues par éléments finis convergent vers la solution exacte en 0(h) dans L(0, T, H 1 ^ ) ) et L°°{0, T, L(H)) INTRODUCTION Consider the convective diffusion équation in Q x ( 0 J ) , (0.1) (*) Reçu le 16 novembre 1979 {) Technical System center, Mitsubishi Heavy Industry, Ltd, Kobe, Japan () Department of Mathematics, Kyoto Umversity, Kyoto, Japan R A I R O Analyse numénque/Numencal Analysis, 0399-0516/1981/3/$ 5 00 © Bordas-Dunod 4 K. BABA, M. TABATA where Q is a bounded domain in U. The solution u{x, t) of (0.1) subject to the free boundary condition d^-b.vu = 0 on ÔQ x (0, T) satisfies the mass-conservation law f u(x, t)dx= f M°(X) dx + f A f /(x, 0 j=Q WTM(Q) = { u ; u is measurable in Q, \\ u \\m>PtQ < + oo } , H(Q) = For 0 < a ^ 1 and a non-negative integer m, « L,oc,n = sup { | Dl u(x)\; \ P | = m, x e Q } , m II w i l m a n = E Iwb.oo.n» j=o II « llm+^oca = II u L,oo,n + I u l«+ot,oo,n > C(Q) = { u ; u is continuously differentiable up to order m in Q } , C(Q) = {u;ue C(Q), || W ||m+aj00)n < + oo } . Let X be a Banach space with norm || . ||x. C(0, T ; X) = { u ; u is continuously differentiable up to order m as a function from [0, T] into X } , II u \\C^O,T;X) = £ max { || D/ w(0 | |x ; t e [0, T]}, j=0

The contraction number of a multigrid method for solving the Poisson equation

Abstract: The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.

Numerical methods based on additive splittings for hyperbolic partial differential equations

Abstract: We derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds. These techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude. Our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales, some of which are sometimes treated exactly. The efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator. This is analyzed and a technique is discussed for reducing this error through a simple change of variables. A procedure for generating the appropriate boundary data for the intermediate solutions is also presented.

Finite-Analytic Numerical Solution of Heat Transfer in Two-Dimensional Cavity Flow

Abstract: Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.

High orderP-stable formulae for the numerical integration of periodic initial value problems

Abstract: Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy?=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Stormer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.