Abstract: A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the KreissOliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommentype algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.

Abstract: The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Abstract: It is common practice to partially characterize a filter with a finite portion of its impulse response, with the objective of generating a recursive approximation. This paper discusses the use of mixed first and second information, in the form of a finite portion of the impulse response and autocorrelation sequences. The discussion encompasses a number of techniques and algorithms for this purpose. Two approximation problems are studied: an interpolation problem and a least squares problem. These are shown to be closely related. The linear systems which form the solutions to these problems are shown to be stable. An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test. The close connection between these algorithms suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.

Abstract: A complete reexamination of Geiger9s method in the light of modern numerical analysis indicates that numerical stability can be insured by use of the QR algorithm and the convergence domain considerably enlarged by the introduction of step-length damping. In order to make the maximum use of all data, the method is developed assuming a priori estimates of the statistics of the random errors at each station. Numerical experiments indicate that the bulk of the joint probability density of the location parameters is in the linear region allowing simple estimates of the standard errors of the parameters. The location parameters are found to be distributed as one minus chi squared with m degrees of freedom, where m is the number of parameters, allowing the simple construction of confidence levels. The use of the chi-squared test with n-m degrees of freedom, where n is the number of data, is introduced as a means of qualitatively evaluating the correctness of the earth model.

Abstract: We consider a generalized conjugate gradient method for solving systems of linear equations having nonsymmetric coefficient matrices with positive-definite symmetric part. The method is based on splitting the matrix into its symmetric and skew-symmetric parts, and then accelerating the associated iteration using conjugate gradients, which simplifies in this case, as only one of the two usual parameters is required. The method is most effective for cases in which the symmetric part of the matrix corresponds to an easily solvable system of equations. Convergence properties are discussed, as well as an application to the numerical solution of elliptic partial differential equations.

Abstract: The book represents an introduction to computation in control by an iterative, gradient, numerical method, where linearity is not assumed. The general language and approach used are those of elementary functional analysis. The particular gradient method that is emphasized and used is conjugate gradient descent, a well known method exhibiting quadratic convergence while requiring very little more computation than simple steepest descent. Constraints are not dealt with directly, but rather the approach is to introduce them as penalty terms in the criterion. General conjugate gradient descent methods are developed and applied to problems in control.

Abstract: A new method is developed for accurately predicting resonant frequencies of dielectric resonators used in microwave circuits. By introducing an appropriate approximation in the field distribution outside the resonator, an analytical formulation becomes possible. Two coupled eigenvalue equations thus derived are subsequently solved by a numerical method. The accuracy of the results computed by the present method is demonstrated by comparison with previously published data.

Abstract: A theoretical study is made of the flow behavior of thin Newtonian liquid films being “squeezed” between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations. Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.

Abstract: A new numerical method used to drastically reduce the computation time required to solve the Navier-Stokes equations at flight Reynolds numbers is described. The new method makes it possible and practical to calculate many important three-dimensional, high Reynolds number flow fields on computers.

Abstract: : The book is concerned with the use of mathematical programming techniques for solving ill-conditioned systems of linear equations with various kinds of errors in the right hand side vector. The primary motivation for the work was the spectrum unfolding problem of experimental physics, so the treatment also includes the Fredholm integral equation of the first kind, which can be considered to be an infinite dimensional ill-conditioned system. The basic idea of the new techniques which are developed is the use of priori knowledge about the solution in order to greatly reduce the size of the class of solutions which are consistent with the right hand side errors. The methods are designed to give interval estimates for the solution--the sizes of the intervals being determined by the sizes of the errors in the right hand side, and the constraints imposed on the class of acceptable solutions by the a priori information. The basic a priori constraint which is used is that the solution must be non-negative; but it is shown that many other a priori constraints can be reduced to a simple non-negativity constraint by a suitable transformation of variables. When the non-negativity constraint is taken into account, the problem of estimating lower and upper bounds for the solution can be formulated and solved as a mathematical programming problem. The book treats both the case where the right hand side errors are known absolutely to lie in some bounded region and also the case where the errors are normally distributed. (Author)

Abstract: A simplified method for finding and using discrete small-signal models for switching regulators is presented. With introduction of a new "straight-line" approximation, and application of root locus techniques, it is demonstrated that discrete models may be used accurately to predict wide bandwidth closed-loop behavior with methods simple enough to be useful in the initial design phase of a switching regulator. The principal result is a set of converter transfer functions comparable to the set derived by describing function techniques, but not subject to the low frequency restriction of describing function models. Also presented is a set of pulse-width modulator transfer functions which indicates that the potential small-signal transient behavior of a switching regulator is independent of the choice of modulator.

Abstract: The spin-density-functional (SDF) formalism with the local-spin-density (LSD) approximation is applied to a number of small molecules with the primary aim of testing the approximation for molecular applications. A new numerical method to solve the one-electron wave equation is developed, utilizing the special features of the SDF formalism. Energy curves, dissociation energies, and equilibrium distances for some diatomic molecules [H$sub 2$$sup +$($sup 2$$Sigma$$sup +$/sub g/, $sup 2$$Sigma$$sup +$/sub u/), H$sub 2$($sup 1$$Sigma$$sup +$/sub g/, $sup 3$$Sigma$$sup +$/sub u/), He$sub 2$$sup 2+$($sup 1$$Sigma$$sup +$/sub g/), and He$sub 2$($sup 1$$Sigma$$sup +$/sub g/)] and the vibrational frequencies of H$sub 2$. The deviations from the experimental results are typically $sup 1$/$sub 2$ eV for the energies and less than or equal to 0.1 A for the distances. The LSD approximation is discussed using the concept of an exchange-correlation hole and predictions about the applicability to other molecules are made. The LSD approximation is compared with the Hartree--Fock and multiple-scattering-X$alpha$ methods, and some difficulties in the latter methods are pointed out. It is argued that the SDF formalism within the LSD approximation has physical advantages compared to the Hartree--Fock and X$alpha$ methods and that it should provide a simple and useful method for a broad range of applications. (auth)

Abstract: The instability of the flow induced by a circular cylinder oscillating in an infinite viscous fluid is investigated. The flow is shown to be unstable to a Taylor vortex mode of instability. A series solution of the partial differential system governing the stability of the flow is obtained. The method used has several advantages over the numerical methods used by different authors for related problems. The instability predicted by the theory leads to a flow with no mean velocity component tangential to the cylinders. The disturbance velocity field decays exponentially at the edge of the Stokes layer. The theoretical results are qualitatively confirmed by an experimental investigation of the problem.

Abstract: A numerical method is proposed for solving the balance equations of the steady-state probabilities of a G~/G/c queueing system in a general class The method is based on an iterative calculation of conditional prob­ abilities, instead of absolute probabilities, conditioned by the number of customers in the system By skillfully exploiting a convergence property of the conditional probabilities, it provides a fairly accurate solution of the balance equations with relatively little computational burden In this paper, a numerical method is proposed for solving the balance equa­ tions of the steady-state probabilities of a GI/G/c queueing system in a general class The method is a direct application of the (modified) lumping method introduced in (6) for the stationary distribution of a Markov chain It is based on an iterative calculation of conditional probabilities of the queueing system conditioned by the number of customers in the system By using the conditional probabilities, rather than absolute probabilities, the system of linear equations of the steady-state probabilities is di,'ded into a set of smaller systems of linear equations, and it can be solved with less computational burden by exploiting convergence property of the conditional probabilities Furthermore, errors included in the solution become fairly small The computational time required for solving the balance equations by our method is nearly independent of the value of the utilization factor p Hence, our method is effective even if p is near to l

Abstract: A class of linear algorithms for Volterra Integro-Differential Equations is studied. The concept of the associated canonical fraction is extended to this class and leads to an algebraic criterion forA-stability.

Abstract: Methods are developed for dealing with the various dynamical problems that arise because of convective zones in stars. A system of equations for stellar convection is derived from the full equations of compressible fluid dynamics with the aid of two major approximations. The first of these is the anelastic approximation, which involves both the filtering out of acoustic waves and a suitable linearization of the fluctuating thermodynamic variables. The second one approximates the horizontal structure of convection by expanding the motion in a set of horizontal cellular platforms and severely truncating the expansion. The resulting system of partial differential equations, referred to as the anelastic modal equations, is outlined along with suggested boundary conditions and techniques for solving the equations. Ways of assessing the overall validity of the present treatment are discussed.

Abstract: A modification of the well-known continuation (or homotopy) method for actual computation is worked out. Compared with the classical method, the modification seems to be a more reliable device for supplying useful initial data for shooting techniques. It is shown that computing time may be significantly reduced in the numerical solution of sensitive realistic two-point boundary value problems.

Abstract: The usual successive secant method for solving systems of nonlinear equations suffers from two kinds of instabilities. First the formulas used to update the current approximation to the inverse Jacobian are numerically unstable. Second, the directions of search for a solution may collapse into a proper affine subspace, resulting at best in slowed convergence and at worst in complete failure of the algorithm. In this report it is shown how the numerical instabilities can be avoided by working with factorizations of matrices appearing in the algorithm. Moreover, these factorizations can be used to detect and remedy degeneracies among the directions.

Abstract: In the design of microstrip components, the analytical equations of Wheeler have an advantage in handling-speed and simplicity over alternative quasi-static analyses involving complex computation by numerical methods. Wheeler equations can produce excessive errors, however, unless care is taken particularly in the choice of changeover point between equations for narrow or wide strips. This paper shows that for alumina-type substrates with 8 < er < 12, the Wheeler analysis and synthesis equations produce results within 1% of those given by selected numerical methods, when the changeover points are correctly chosen. Additional new and modified formulae are presented for the direct calculation of microstrip effective permittivity from either W/h or Z0, also to an absolute accuracy of less than 1%. These complete an accurate set of analytical equations for quasi-static analysis or synthesis of microstrip lines.

Abstract: An Eulerian model, describing the dispersion of pollutants in gases and fluids, is developed. The model is based on numerical integration of the dispersion equation, including the effect of advection, diffusion, sinks and of multiple sources. The pseudospectral method is employed for numerical integration of the dispersion equation. The model is not limited to physical problems with periodic boundary conditions, as imposed by the spectral technique. A filtering procedure prevents instabilities caused by aliasing interactions. The emphasis is placed on numerical tests relevant to air pollution studies. Pseudodiffusion is not present in this model. The error in numerical integration is brought down to a few percent of the concentration level of the pollutant. To our knowledge, the model is the most accurate Eulerian model presently available for dispersion calculations. Applications to air pollution studies are discussed. Accuracy of 19 different numerical methods is compared.

Abstract: : A method has been developed and evaluated for merging two dimensional boundary-integral equation and finite element stress analysis programs A variational formulation has been obtained which provides an analytical basis for the study of different merging schemes Extensive numerical results are presented which verify the applicability of the merged analyses to fracture mechanics problems Numerical and experimental results are presented in the area of fatigue life prediction for gas turbine structures with surface or corner cracks to correlate previously developed three-dimensional fracture mechanics analysis methods