Showing papers on "Numerical analysis published in 1970"

Applied numerical analysis

Abstract: The fifth edition of this book continues teaching numerical analysis and techniques. Suitable for students with mathematics and engineering backgrounds, the breadth of topics (partial differential equations, systems of nonlinear equations, and matrix algebra), provide comprehensive and flexible coverage of numerical analysis.

Mathematical Methods of Physics

Abstract: 1. Ordinary Differential Equations. 2. Infinite Series. 3. Evaluation of Integrals. 4. Integral Transforms. 5. Further Applications of Complex Variable. 6. Vectors and Matrices. 7. Special Functions. 8. Partial Differential Equations. 9. Eigenfuctions, Eigenvalues, and Green's Functions. 10. Peturbation Theory. 11. Integral Equations. 12. Calculus of Variations. 13. Numerical Methods. 14. Probability and Statistics. 15. Tensor Analysis and Differential Geometry. 16. Introduction to Groups and Group Representations. Appendix: Some Properties of Functions of a Complex Variable. Bibliography. Index.

Numerical Methods That Work

Abstract: Part I. Fundamental Methods: 1. The calculation of functions 2. Roots of transcendental equations 3. Interpolation - and all that 4. Quadrature 5. Ordinary differential equations - initial conditions 6. Ordinary differential equations - boundary conditions 7. Strategy versus tactics - roots of polynomials 8. Eigenvalues I 9. Fourier series Part II. Double Trouble: 10. Evaluation of integrals 11. Power series, continued fractions, and rational approximations 12. Economization of approximations 13. Eigenvalues II - rotational methods 14. Roots of equations - again 15. The care and treatment of singularities 16. Instability in extrapolation 17. Minimum methods 18. Laplace's equation - an overview 19. Network problems.

Implicit flood routing in natural channels

Abstract: Flood routing in natural channels and many other applications in hydraulic engineering based on the solution of the equations of unsteady flow require fast and accurate numerical methods. Numerical methods which are successful in other applications prove to be inefficient when used for flood routing. An implicit numerical method which is both fast and accurate can be established on the basis of: (1) a centered difference scheme to represent the primary differential equations in finite difference form; and (2) the simultaneous solution of the finite difference equations for each time step. The difference equations constitute a system of nonlinear algebraic equations which can be solved on a digital computer by Newton iteration method. The computational scheme becomes very efficient when advantage is taken of the sparseness of the matrix of coefficients of the linear systems employed in the iteration. Applications of the implicit method show that it can be conveniently used for highly irregular channels.

Numerical Solution of Dielectric Loaded Waveguides: I-Finite-Element Analysis

Abstract: A variational expression of the electromagnetic fields in dielectric loaded waveguides is derived This expression is discretized using the finite-element method and an electromagnetic coupling matrix is derived and evaluated No restriction is placed on the shapes of the triangular elements or the order of the polynomial approximation A general finite-element computer program is described and dispersion curves and field plots of some dielectric loaded waveguides are presented

The Perturbations of Alternating Geomagnetic Fields by Conductivity Anomalies

Abstract: Summary The two-dimensional problems of interest in studying the perturbation of alternating electric current by a sharp discontinuity of conductivity in a conductor are considered, and their applicability to geophysical problems discussed. A numerical method has been developed for solving the appropriate differential equations and boundary conditions. The method has been applied to a vertical discontinuity in conductivity such as at a continental-oceanic interface. The two polarization cases are solved, and the fields and current distributions are determined in detail.

Lumped mass method for Rayleigh waves

Abstract: A simple numerical method is developed for the analysis of generalized Rayleigh waves in multilayered elastic media. The method which completely avoids the use of displacement potentials leads to a simple eigenvalue problem which may be solved by generally available effective computer codes. Numerical results obtained by the method show excellent agreement with previously published solutions obtained by other theories.

Capacitance and Field Distributions for Interdigital Surface-Wave Transducers

Abstract: : A boundary-shift technique used in conjunction with relaxation solutions of Laplace's equation allows the convenient numerical evaluation of the potential in the neighborhood of interdigital comb structures. With this method, the area of computation in the unbounded problem can be restricted to the region of interest near the electrode and interface surfaces. Because of the point-by-point nature of the calculation, a wide range of geometries can be studied with the inclusion of the effects of finger thickness and shape, and of any layers present. Capacitance values for many single interface and layered configurations of surface-wave transducers are presented along with a few representative examples of potential and field maps. (Author)

Numerical Solution of Saint-Venant Equations

Abstract: The general, one-dimensional Saint-Venant equations are presented for a rigid open channel of arbitrary form, not necessarily prismatic, containing a flow that may be spatially varied. The theoretical basis for the method of characteristics is reviewed and used to show that, in the general case, the speed of long-wave disturbances is given by the slope of the characteristic curves. Finite-difference schemes on a rectangular net in the x - t plane and based on the characteristic forms of the Saint-Venant equations, as well as on the direct forms, are given and examined for their stability. The von Neumann technique for stability analysis is presented in detail. Explicit numerical schemes, which are simple, but require small steps in time because of stability problems, are contrasted with implicit schemes that permit numerical solution over large time steps but require the solution of large sets of simultaneous algebraic equations at each step. The double-sweep or progonka method, an exact time- and space-saving technique for solving these (locally linearized) equations, is also given in detail.

The Brightness Temperature of a Vertically Structured Medium

Abstract: Exact equations are derived for calculating the brightness temperature of a medium that is bounded by a plane surface and whose properties (dielectric constant and thermometric temperature) vary only with depth. Although no approximations are made in the development of the principal results, the final equations are presented in a form most convenient for use in the area of microwave radiometry (that is, in the limit where the Rayleigh-Jeans approximation to the Planck blackbody radiation law is valid). Some special cases in which the differential equations arising in the theory can be solved analytically are presented as examples. A practical numerical method for evaluating the exact equations by use of a digital computer is also discussed.

Plane Stress Limit Analysis by Finite Elements

Abstract: A numerical method is presented for the determination of lower bounds on the yield-point load of plane stress problems. In this method, a finite element technique is used to construct a parametric family of piecewise quadratic, equilibrium stress fields. The best lower bound is then found by maximizing the load, subject to the yield constraints, by means of the sequential unconstrained minimization technique. Because all conditions of the Lower Bound theorem are met exactly, the resulting solutions are true bounds. Results are given for square slabs with various cutouts and compared to upper bounds and complete elastic plastic finite element solutions.

Numerical solutions of laminar separated flows

Abstract: Numerical techniques and solutions for compressible and incompressible laminar separated flows using time dependent finite difference equations

Irregularity Of Shape Optimization Problems AndAn Improvement Technique

Abstract: Shape optimization problems of linear elastic bodies, flow fields, magnetic fields, etc. for equilibrium types can be generalized as optimization problems of domains in which elliptic boundary value problems are defined. This paper shows that ordinary domain optimization problems do not have sufficient regularity and proposes a technique to overcome this irregularity. It briefly describes the derivation of the shape gradient functions for a self-adjoint shape optimization problem, and shape identification problems of the Dirichlet type, Neumann type and subdomain gradient assigned type. Using these shape gradient functions, the irregularity of ordinary domain optimization problems is shown through a discussion of the ill-posedness that occurs when the gradient method in Hilbert space is applied directly. To overcome this irregularity, the idea of a smoothing gradient method in Hilbert space is proposed. It is conclusively shown that a numerical method based on this idea coincides with the traction method previously proposed by one of the authors and this conclusion is verified by numerical experiments.

Numerical methods of high-order accuracy for singular nonlinear boundary value problems

Abstract: is a 2n-th order self-adjoint linear differential operator, and it was shown that the Rayleigh-Ritz-Galerkin method is an efficient scheme, both theoretically and numerically, for solving such problems. Our aim here is to extend the results of [2] and [3] to the case of nonsel[adjoint linear differential operators whose coefficients have a singularity at one or both end points of the interval [0, t ]. For ease of exposition, we shall restrict ourselves here to second order operators, as in the particular case of

Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben

Abstract: In this paper, sufficient and necessary maximum conditions are established for a class of mathematical programming problems with an infinite set of restrictions, which is described by a finite number of inequalities. The criteria may be applied to nonlinear approximation problems and to the numerical solution of boundary value problems.

Stepping stone models of finite length

Abstract: The stepping stone model of population structure, of finite length, is analysed with special reference to the variance, and correlation coefficients of gene frequencies. Explicit formulas for these quantities are obtained. The model is also analysed for the genetic variability maintained in the population. In order to check the validity of the analytical results, several numerical computations were carried out using two different methods: iterations and Monte Carlo experiments. The values obtained by these numerical methods agree well with the theoretical values obtained by formulas derived analytically.

Numerical Integration of Navier-Stokes Equations

Abstract: T is extensive literature on the numerical integration of some difference forms of the Navier-Stokes equations. The references quoted herein are those relevant to later discussions and not intended to be complete. There is also extensive literature on the mathematics of difference methods for the solution of partial differential equations for which Ref. 1 is a comprehensive review and contains an extensive bibliography. The present paper is an attempt to develop the implications of a mathematical theorem, which, when put into proper perspective, offers a unified and coherent treatment of various practical aspects of computation. Like approximate differential analyses and physical tests, numerical methods are fallible. Larger and faster computers offer no easy answer to computational difficulties like stability and convergence. Smooth and physically reasonable results of computation are often less accurate than those not so smooth. Since the true asymptotic nature is difficult to establish, both the difference and the differential approximations are nonrigorous; but they are useful, especially with the help of physical experimentation and rigorous mathematical results. We are sympathetic to such heuristic and nonrigorous analysis in favor of obtaining results useful in practice. We shall consider only the difference form of the NavierStokes equations in Eulerian coordinates. Lagrangian coordinates are convenient for flows involving free-surface boundary or active processes associated with fluid elements. It suffers, however, from the serious distortions of the Lagrangian net and from the cummulative errors of the particle paths at large times. Thus various mixed or coupled Eulerian-Lagrangian schemes have been developed even for free boundary problems. Eulerian formulation, even for a single fluid in the absence of a free boundary, has its share of problems, which we shall discuss. We write, for the NavierStokes equations,

Iterative Numerical Methods for Some Integral Equations Arising in Rheology

Abstract: One of the first applications of integral equation inversion techniques in rheology was the exact result of Weissenberg, which enables the shear curve to be obtained from a laminar pipeflow experiment. It is shown that numerical solution of the integral equations occurring for this and other common experiments may be used instead. Numerical inversion programs have been written for the pipeflow and Couette problems. Certain, more difficult inversions, arising with elastic fluid constitutive relations, are also treated; in some cases a simple version of the KBKZ theory seems fairly realistic over a wide range of experiments.