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.

Abstract: A skin-effect equivalent circuit consisting of resistors and inductors is derived from the skin-effect differential equations for simulating the loss of a transmission line. A numerical method is used to analyze the transmission-line differential equations and the skin-effect equivalent circuit, yielding a model which relates the new values of node voltages and line currents to their values at the previous time step. Based on this model, a very simple program was written on a desk-top computer for the transient analysis of lossy trammission lines. Two examples are presented. The first example is an analysis of the step and pulse responses of a 600-m RG-8/U coaxial cable. The computed results show excellent agreement with measured data. The second example studies the current at the end of a 12-in 7-Ω strip line under different loading conditions. Very good agreement has been obtained between the calculated steady-state solution and that obtained by the frequency-domain method.

Abstract: Numerical methods for solving the integrodifferential, integral, and surface-integral forms of the neutron transport equation are reviewed. The solution methods are shown to evolve from only a few ...

Abstract: By the effective index method a two-dimensional field problem is transformed to a problem for a one-dimensional effective waveguide. This method is applied to semiconductor lasers having a gradual lateral variation in the complex permittivity. For the special case of a parabolic variation, analytical formulas for the required gain in the center and the half width of the intensity distribution are derived. The results are compared with a numerical method and very good agreement is found except in some cases where convergence problems occur for the numerical method. This agreement is taken as evidence for the validity of results obtained using the effective index method for analysis of semiconductor laser structures.

Abstract: We describe a sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases. It includes interchanges that, whenever this is possible, avoid the use of any eliminations (with consequent fill-ins) when revising the factorization at an iteration. Test results on some medium scale problems are presented and comparisons made with the algorithm of Forrest and Tomlin.

Abstract: Newton's method plays a central role in the development of numerical techniques for optimization In fact, most of the current practical methods for optimization can be viewed as variations on Newton's method It is therefore important to understand Newton's method as an algorithm in its own right and as a key introduction to the most recent ideas in this area One of the aims of this expository paper is to present and analyze two main approaches to Newton's method for unconstrained minimization: the line search approach and the trust region approach The other aim is to present some of the recent developments in the optimization field which are related to Newton's method In particular, we explore several variations on Newton's method which are appropriate for large scale problems, and we also show how quasi-Newton methods can be derived quite naturally from Newton's method

Abstract: The Osher algorithm for solving the Euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations. A first-order accurate version of the numerical method was derived by Osher circa 1980 for the one-dimensional non-isentropic Euler equations in Cartesian coordinates. In this paper, the extension of the scheme to arbitrary two-dimensional geometries is explained. Results are then presented for several example problems in one and two dimensions. Future work will include extension of the method to second-order accuracy and the development of implicit time differencing for the Osher algorithm.

Abstract: The nonlinear inverse heat conduction problem is the calculation of surface heat fluxes and temperatures by utilizing measured interior temperatures in opaque solids possessing temperature-variable thermal properties. The most widely used numerical method for this problem was developed by Beck. The new sequential procedure presented here reduces the number of computer calculations by a factor of 3 or 4. The general heat conduction model used permits treatment of various one-dimensional geometries (plates, cylinders, and spheres), energy sources, and fin effects. The numerical procedure is illustrated for finite differences, but the basic concepts are also applicable to the finite-element method. Detailed descriptions of the computational algorithms are given and a nonlinear example is provided.

Abstract: The computation of two-dimensional tidal currents using the Alternating Direction Implicit Method (ADI) can be subject to numerical attenuation, parasitic oscillations, and poor reproduction of wave propagation when large time steps are used. The new method described in the paper is designed to overcome these difficulties. It is based on a fractional step method in which momentum advection is calculated using the method of characteristics, horizontal momentum diffusion is calculated using an implicit finite difference scheme, and wave propagation is calculated using an iterative alternating direction implicit algorithm. The resulting method has been incorporated in the CYTHERE-ES1 modelling system, in which tidal flat flooding and drying as well as wind effects and Coriolis acceleration are taken into account. The basic principles of the method, as well as its application to four schematic test cases and two engineering studies, are described.

Abstract: In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.

Abstract: I. Introduction.- 1. Introduction.- 1.1. Optimal Control.- 1.2. System Identification.- 1.3. Optimal Inputs.- 1.4. Computational Preliminaries.- Exercises.- II. Optimal Control and Methods for Numerical Solutions.- 2. Optimal Control.- 2.1. Simplest Problem in the Calculus of Variations.- 2.1.1. Euler-Lagrange Equations.- 2.1.2. Dynamic Programming.- 2.1.3. Hamilton-Jacobi Equations.- 2.2. Several Unknown Functions.- 2.3. Isoperimetric Problems.- 2.4. Differential Equation Auxiliary Conditions.- 2.5. Pontryagin's Maximum Principle.- 2.6. Equilibrium of a Perfectly Flexible Inhomogeneous Suspended Cable.- 2.7. New Approaches to Optimal Control and Filtering.- 2.8. Summary of Commonly Used Equations.- Exercises.- 3. Numerical Solutions for Linear Two-Point Boundary-Value Problems..- 3.1. Numerical Solution Methods.- 3.1.1. Matrix Riccati Equation.- 3.1.2. Method of Complementary Functions.- 3.1.3. Invariant Imbedding.- 3.1.4. Analytical Solution.- 3.2. An Optimal Control Problem for a First-Order System.- 3.2.1. The Euler-Lagrange Equations.- 3.2.2. Pontryagin's Maximum Principle.- 3.2.3. Dynamic Programming.- 3.2.4. Kalaba's Initial-Value Method.- 3.2.5. Analytical Solution.- 3.2.6. Numerical Results.- 3.3. An Optimal Control Problem for a Second-Order System.- 3.3.1. Numerical Methods.- 3.3.2. Analytical Solution.- 3.3.3. Numerical Results and Discussion.- Exercises.- 4. Numerical Solutions for Nonlinear Two-Point Boundary-Value Problems.- 4.1. Numerical Solution Methods.- 4.1.1. Quasilinearization.- 4.1.2. Newton-Raphson Method.- 4.2. Examples of Problems Yielding Nonlinear Two-Point Boundary-Value Problems.- 4.2.1. A First-Order Nonlinear Optimal Control Problem.- 4.2.2. Optimization of Functionals Subject to Integral Constraints.- 4.2.3. Design of Linear Regulators with Energy Constraints.- 4.3. Examples Using Integral Equation and Imbedding Methods.- 4.3.1. Integral Equation Method for Buckling Loads.- 4.3.2. An Imbedding Method for Buckling Loads.- 4.3.3. An Imbedding Method for a Nonlinear Two-Point Boundary-Value Problem.- 4.3.4. Post-Buckling Beam Configurations via an Imbedding Method.- 4.3.5. A Sequential Method for Nonlinear Filtering.- Exercises.- III. System Identification.- 5. Gauss-Newton Method for System Identification.- 5.1. Least-Squares Estimation.- 5.1.1. Scalar Least-Squares Estimation.- 5.1.2. Linear Least-Squares Estimation.- 5.2. Maximum Likelihood Estimation.- 5.3. Cramer-Rao Lower Bound.- 5.4. Gauss-Newton Method.- 5.5. Examples of the Gauss-Newton Method.- 5.5.1. First-Order System with Single Unknown Parameter.- 5.5.2. First-Order System with Unknown Initial Condition and Single Unknown Parameter.- 5.5.3. Second-Order System with Two Unknown Parameters and Vector Measurement.- 5.5.4. Second-Order System with Two Unknown Parameters and Scalar Measurement.- Exercises.- 6. Quasilinearization Method for System Identification.- 6.1. System Identification via Quasilinearization.- 6.2. Examples of the Quasilinearization Method.- 6.2.1. First-Order System with Single Unknown Parameter.- 6.2.2. First-Order System with Unknown Initial Condition and Single Unknown Parameter.- 6.2.3. Second-Order System with Two Unknown Parameters and Vector Measurement.- 6.2.4. Second-Order System with Two Unknown Parameters and Scalar Measurement.- Exercises.- 7. Applications of System Identification.- 7.1. Blood Glucose Regulation Parameter Estimation.- 7.1.1. Introduction.- 7.1.2. Physiological Experiments.- 7.1.3. Computational Methods.- 7.1.4. Numerical Results.- 7.1.5. Discussion and Conclusions.- 7.2. Fitting of Nonlinear Models of Drug Metabolism to Experimental Data.- 7.2.1. Introduction.- 7.2.2. A Model Employing Michaelis and Menten Kinetics for Metabolism.- 7.2.3. An Estimation Problem.- 7.2.4. Quasilinearization.- 7.2.5. Numerical Results.- 7.2.6. Discussion.- Exercises.- IV. Optimal Inputs for System Identification.- 8. Optimal Inputs.- 8.1. Historical Background.- 8.2. Linear Optimal Inputs.- 8.2.1. Optimal Inputs and Sensitivities for Parameter Estimation.- 8.2.2. Sensitivity of Parameter Estimates to Observations.- 8.2.3. Optimal Inputs for a Second-Order Linear System.- 8.2.4. Optimal Inputs Using Mehra's Method.- 8.2.5. Comparison of Optimal Inputs for Homogeneous and Nonhomogeneous Boundary Conditions.- 8.3. Nonlinear Optimal Inputs.- 8.3.1. Optimal Input System Identification for Nonlinear Dynamic Systems.- 8.3.2. General Equations for Optimal Inputs for Nonlinear Process Parameter Estimation.- Exercises.- 9. Additional Topics for Optimal Inputs.- 9.1. An Improved Method for the Numerical Determination of Optimal Inputs.- 9.1.1. Introduction.- 9.1.2. A Nonlinear Example.- 9.1.3. Solution via Newton-Raphson Method.- 9.1.4. Numerical Results and Discussion.- 9.2. Multiparameter Optimal Inputs.- 9.2.1. Optimal Inputs for Vector Parameter Estimation.- 9.2.2. Example of Optimal Inputs for Two-Parameter Estimation.- 9.2.3. Example of Optimal Inputs for a Single-Input, Two-Output System.- 9.2.4. Example of Weighted Optimal Inputs.- 9.3. Observability, Controllability, and Identifiability.- 9.4. Optimal Inputs for Systems with Process Noise.- 9.5. Eigenvalue Problems.- 9.5.1. Convergence of the Gauss-Seidel Method.- 9.5.2. Determining the Eigenvalues of Saaty's Matrices for Fuzzy Sets.- 9.5.3. Comparison of Methods for Determining the Weights of Belonging to Fuzzy Sets.- 9.5.4. Variational Equations for the Eigenvalues and Eigenvectors of Nonsymmetric Matrices.- 9.5.5. Individual Tracking of an Eigenvalue and Eigenvector of a Parametrized Matrix.- 9.5.6. A New Differential Equation Method for Finding the Perron Root of a Positive Matrix.- Exercises.- 10. Applications of Optimal Inputs.- 10.1. Optimal Inputs for Blood Glucose Regulation Parameter Estimation.- 10.1.1. Formulation Using Bolie Parameters for Solution by Linear or Dynamic Programming.- 10.1.2. Formulation Using Bolie Parameters for Solution by Method of Complementary Functions or Riccati Equation Method.- 10.1.3. Improved Method Using Bolie and Bergman Parameters for Numerical Determination of the Optimal Inputs.- 10.2. Optimal Inputs for Aircraft Parameter Estimation.- Exercises.- V. Computer Programs.- 11. Computer Programs for the Solution of Boundary-Value and Identification Problems.- 11.1. Two-Point Boundary-Value Problems.- 11.2. System Identification Problems.- References.- Author Index.

Abstract: A numerical method for calculating the interaction of steep (nonlinear) ocean waves with large fixed or floating structures of arbitrary shape is described. The interaction is treated as a transient problem with known initial conditions corresponding to still water in the vicinity of the structure and a prescribed incident waveform approaching it. The development of the flow, together with the associated fluid forces and structural motions, are obtained by a time-stepping procedure in which the flow at each time step is calculated by an integral-equation method based on Green's theorem. A few results are presented for two reference situations and these serve to illustrate the effects of nonlinearities in the incident waves.

Abstract: In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.

Abstract: : We have studied a number of computational problems in numerical linear algebra. Most of these problems arise in statistical computations. They include the following: (1) Application of the conjugate gradient method to nonorthogonal analysis of variance; (2) Use of orthogonalization procedures in geodetic problems; (3) Algorithms for computing sample variance; (4) Truncated Newton methods; and (5) Imposing Curvature Restrictions on Flexible Functional Forms. (Author).

Abstract: A new numerical method has been developed to investigate three-dimensional, unsteady pipe flows using a new velocity-vector expansion method. Each vector function in the expansion set is divergence-free and satisfies the boundary conditions for viscous flow. Other features of the general technique are as follows: (1) pressure is eliminated from the dynamics; (2) only two unknowns per “mesh point” are required; (3) there is rapid convergence of spectral methods; (4) there is implicit treatment of the viscous term at no extra computational cost; and (5) no fractional time-steps are required. In the present application of the method to flow in a pipe, the behavior of each flow variable near the computational singular point is treated rigorously and expansions in Jacobi polynomials have been shown to be particularly advantageous. The method has been tested on the linear stability problem for Poiseuille flow and has demonstrated rapid convergence of the eigenvalues and eigenfunctions as the number of radial modes is increased.

Abstract: Upwind difference, defect correction and central difference schemes for the solution of the convection-diffusion equation with small viscosity coefficient are compared. It is shown that central difference schemes and hence also standard Galerkin finite element methods are preferable above upwind and defect correction schemes, when Gaussian elimination is used for the solution of the resulting system of equations.When iterative solution methods are employed good results can be achieved by a defect-correction method, whereas upwind difference schemes are generally inaccurate.

Abstract: A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn > 39, and storage is proportional ton rather thann2.

Abstract: The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL 2,H 1 andL ? , provided that certain relationships hold between the frequency, mesh size and outer radius.

Abstract: A numerical method for the unconstrained minimization of a convex nonsmooth function of several variables is presented. It is closely related to the ‘bundle type’ approach and to the conjugate subgradient method. A way is suggested to reduce the amount of information to be stored during the computational procedure. Global convergence of the method to the minimum is proved.

Abstract: This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or ‘staircase’, structure. The present paper looks at ‘inversion’ routines within the simplex method, particularly those for sparse triangular factorization of a basis by Gaussian elimination and for solution of triangular linear systems. The succeeding paper examines ‘pricing’ routines. Both papers describe extensive (though preliminary) computational experience, and can point to some quite promising results.

Abstract: We propose an approximate method, of general applicability, for analyzing the discontinuities in slab dielectric waveguides. The method is based on replacing the unbounded configuration by a corresponding periodic multilayer structure. Hence the entire spectrum becomes discrete, and this makes the problem easier. The characteristic equation for the modal solution in a periodical multilayer dielectric waveguide is expressed in a matrix form which can be readily solved with a computer. In order to solve the discontinuity problems of slab waveguides, we expressed the reflected and transmitted fields by truncated modal expansions. The unknown amplitudes of the reflected and transmitted modes are determined by the boundary conditions. Numerical results are graphically shown including the cases which have been difficult to analyze by other methods. The accuracy of our results is checked by evaluating the relative errors and comparing with other available results.