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...

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

Abstract: The paper describes a proposed modification to the transmission-line-matrix method of numerical analysis (TLM). The conventional formulation of this technique uses squares and cubes for space quantisation in two and three dimensions, respectively. This restriction, which is due to the necessity of time synchronism, severely affects the efficiency of the method. The paper introduces a technique in which the propagation space can be irregularly graded according to the nature of the problem under investigation. A new formulation for TLM in the rθ-plane of polar coordinates is also introduced; here the space elements are non-uniform by nature.

Abstract: This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.

Abstract: A numerical method is presented to compute one unknown constitutive parameter of an inhomogeoeous lossy dielectric slab from the reflected field in the time domain. The method is based upon a space-time discretization of the integral equation for the reflected field. In the inversion, especially those space-time points where the numerical computation of the electric-field strength in the slab is most accurate are taken into account. This is achieved by computing the unknown parameter iteratively. Alternately solving equations for an approximate direct-scattering problem and an approximate inverse-scattering problem yields successive approximations for the electric field in the slab and the unknown constitutive coefficient. Both problems lead to an infinite system of linear equations from which a finite subsystem is selected. General criteria for this selection are presented. Various profiles have been reconstructed numerically from the reflected field due to a sine-squared incident pulse.

Abstract: Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of "P-series" is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nystrom methods fory?=f(y,y?), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].

Abstract: An analytical method for determining eigenvalues and eigenfunctions of continuous beams with arbitrary boundary conditions is developed by using a general solution of a differential equation for the lateral vibration of the beam. This method can be applied to nonuniform cross section beams and calculated to higher eigenvalues very accurately. The dynamic response of a continuous beam traversed by a moving load with constant velocity is studied. The analysis is conducted by the method of modal analysis. Numerical examples are presented to illustrate the applicability of the analysis and to investigate the dynamic characteristics of continuous beams.

Abstract: The pseudospectral-Chebyshev methods are shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems. The analysis demonstrates that the rate of convergence is greater for finite difference methods or the finite element method. For a single first-order hyperbolic equation, the method is seen as remaining stable even when the coefficient changes sign, although in this case it is specified that care must be taken to have adequate spatial resolution. It is noted that this fact, combined with the fact that collocation methods are easy to apply in the nonlinear case, shows that the pseudospectral method is in general preferable to the Galerkin or Tau methods.

Abstract: We present an efficient numerical method for computing electromagnetic (EM) scattering of arbitrary three‐dimensional (3-D) local inhomogeneities buried in a uniform or two‐layered earth. In this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite‐element mesh and whose boundary is only a slight distance away from the inhomogeneity. The scheme uses two sets of independent equations. The first is a set of finite‐element equations derived from a variational integral, and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary. The Green’s function is used to derive the second set of equations. An iterative algorithm has been developed to solve these two sets of equations. The solutions are the electric fields at nodes inside the finite‐element mesh. The scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region. The scheme is used for modeling 3-D inho...