Abstract: Techniques from numerical analysis and crystallographic refinement have been combined to produce a variant of the Truncated Newton nonlinear optimization procedure. The new algorithm shows particular promise for potential energy minimization of large molecular systems. Usual implementations of Newton's method require storage space proportional to the number of atoms squared (i.e., O(N2)) and computer time of O(N3). Our suggested implementation of the Truncated Newton technique requires storage of less than O(N1.5) and CPU time of less than O(N2) for structures containing several hundred to a few thousand atoms. The algorithm exhibits quadratic convergence near the minimum and is also very tolerant of poor initial structures. A comparison with existing optimization procedures is detailed for cyclohexane, arachidonic acid, and the small protein crambin. In particular, a structure for crambin (662 atoms) has been refined to an RMS gradient of 3.6 × 10−6 kcal/mol/A per atom on the MM2 potential energy surface. Several suggestions are made which may lead to further improvement of the new method.

Abstract: A temperature control system for individual rooms within a home is disclosed having a pair of thermostatically controlled switches settable to "occupied" and "unoccupied" temperatures respectively with a clock controlled switching arrangement for determining which thermostatically controlled switch is effective during predetermined hours of the day. The clock selected thermostatically controlled switch is coupled by way of a relay to control a forced air duct opening and closing louver arrangement so that the duct may be opened for either heating or cooling during unoccupied periods of time a lesser percentage than it is opened during periods of the day when the room is typically occupied.

Abstract: We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number ( N = O (1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M , and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness ( ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

Abstract: We present a new numerical method for studying the evolution of free and bound waves on the nonlinear ocean surface. The technique, based on a representation due to Watson and West (1975), uses a slope expansion of the velocity potential at the free surface and not an expansion about a reference surface. The numerical scheme is applied to a number of wave and wave train configurations including longwave-shortwave interactions and the three-dimensional instability of waves with finite slope. The results are consistent with those obtained in other studies. One strength of the technique is that it can be applied to a variety of wave train and spectral configurations without modifying the code.

Abstract: In this stochastic approach to global optimization, clustering techniques are applied to identify local minima of a real valued objective function that are potentially global. Three different methods of this type are described; their accuracy and efficiency are analyzed in detail.

Abstract: 1. Preliminary material 2. Numerical methods on the real line 3. Numerical methods on an arc 'Gamma' 4. The Sinc-Galerkin method 5. Steady problems 6. Time-dependent problems Appendix A. Linear algebra References.

Abstract: Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Abstract: We construct explicit Runge–Kutta (–Nystrom) methods for the integration of first (and second) order differential equations having an oscillatory solution. Special attention is paid to the phase errors (or dispersion) of the dominant components in the numerical oscillations when these methods are applied to a linear, homogeneous test model. RK(N) methods are constructed which are dispersive of orders up to 10, whereas the (algebraic) order of accuracy is only 2 or 3. Application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced.

Abstract: In this work we review the present status of numerical methods for partial differential equations on vector and parallel computers. A discussion of the relevant aspects of these computers and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial-boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. A brief discussion of application areas utilizing these computers is included.

Abstract: A numerical method for solving the complete Navier-Stokes equations for incompressible flows is introduced that is applicable for investigating three-dimensional transition phenomena in a spatially growing boundary layer. Results are discussed for a test case with small three-dimensional disturbances for which detailed comparison to linear stability theory is possible. The validity of our numerical model for investigating nonlinear transition phenomena is demonstrated by realistic spatial simulations of the experiments by Kachanov and Levchenko1 for a subharmonic resonance breakdown and of the experiments of Klebanoff et al.2 for a fundamental resonance breakdown.

Abstract: I: Initial Value Problems for Ode's and Parabolic Pde's.- 1. Algorithms for Semiconductor Device Simulation.- 2. Hierarchical Bases in the Numerical Solution of Parabolic Problems.- 3. Extrapolation Integrators for Quasilinear Implicit ODE's.- 4. Numerical Problems Arising from the Simulation of Combustion Phenomena.- 5. Numerical Computation of Stiff Systems for Nonequilibrium.- 6. Finite Element Simulation of Saturated-Unsaturated Flow Through Porous Media.- II: Boundary Value Problems for ODE's and Elliptic PDE's.- 7. Numerical Pathfollowing Beyond Critical Points in ODE Models.- 8. Computing Bifurcation Diagrams for Large Nonlinear Variational Problems.- 9. Extinction Limits for Premixed Laminar Flames in a Stagnation Point Flow.- 10. A Numerical Method for Calculating Complete Theoretical Seismograms in Vertically Varying Media.- 11. On a New Boundary Element Spectral Method.- III: Hyperbolic PDE's.- 12. A High Order Non-Oscillatory Shock Capturing Method.- 13. Vortex Dynamics Studied by Large-Scale Solutions to the Euler Equations.- IV: Inverse Problems.- 14. Numerical Backprojection in the Inverse 3D Radon Transform.- 15. A Direct Algebraic Algorithm in Computerized Tomography.- 16. A Two-Grid Approach to Identification and Control Problems for Partial Differential Equations.- V: Optimization and Optimal Control Problems.- 17. Solving Large-Scale Integer Optimization Problems.- 18. Numerical Treatment of State & Control Constraints in the Computation of Feedback Laws for Nonlinear Control Problems.- 19. Optimal Production Scheme for the Gosau Hydro Power Plant System.- VI: Algorithm Adaptation on Supercomputers.- 20. The Use of Vector and Parallel Computers in the Solution of Large Sparse Linear Equations.- 21. Local Uniform Mesh Refinement on Vector and Parallel Processors.- 22. Using Supercomputer to Model Heat Transfer in Biomedical Applications.- Speakers.

Abstract: The fundamental principles of adaptive grid generation for the numerical analysis of physical phenomena described by systems of partial differential equations are examined in an analytical review. Topics addressed include weight functions, equidistribution in one dimension, the specification of coefficients in the linear weight, the attraction to a given grid on a curve, evolutionary forces, and metric notation. Consideration is given to curve-by-curve methods, finite-volume methods, variational methods, and temporal aspects.

Abstract: Many optimization models of neural networks need constraints to restrict the space of outputs to a subspace which satisfies external criteria Optimizations using energy methods yield "forces" which act upon the state of the neural network The penalty method, in which quadratic energy constraints are added to an existing optimization energy, has become popular recently, but is not guaranteed to satisfy the constraint conditions when there are other forces on the neural model or when there are multiple constraints In this paper, we present the basic differential multiplier method (BDMM), which satisfies constraints exactly; we create forces which gradually apply the constraints over time, using "neurons" that estimate Lagrange multipliers The basic differential multiplier method is a differential version of the method of multipliers from Numerical Analysis We prove that the differential equations locally converge to a constrained minimum Examples of applications of the differential method of multipliers include enforcing permutation codewords in the analog decoding problem and enforcing valid tours in the traveling salesman problem

Abstract: I. Continued Fractions. Since these play an important role, the first chapter introduces their basic properties, evaluation algorithms and convergence theorems. From the section dealing with convergence it can be seen that in certain situations nonlinear approximations are more powerful than linear approximations. The recent notion of branched continued fraction is introduced in the multivariate section and is later used for the construction of multivariate rational interpolants. II. Pade Approximants. A survey of the theory of these local rational approximants for a given function is presented, including the problems of existence, unicity and computation. Also considered are the convergence of sequences of Pade approximants and the continuity of the Pade operator which associates with a function its Pade approximant of a certain order. A special section is devoted to the multivariate case. III. Rational Interpolants. These rational functions fit a given function at some given points. Many results of the previous chapter remain valid for this more general case where the interpolation conditions are spread over several points. In between the rational interpolation case and Pade approximation case lies the theory of rational Hermite interpolation where each interpolation point can be assigned more than one interpolation condition. Some results on the convergence of sequences of rational Hermite interpolants are mentioned and multivariate rational interpolants are introduced in two different ways. IV. Applications. The previous types of rational approximants are used here to develop several numerical methods for the solution of classical problems such as convergence acceleration, nonlinear equations, ordinary differential equations, numerical quadrature, partial differential equations and integral equations. Many numerical examples illustrate the different techniques, and it is seen that nonlinear methods are very useful in situations involving singularities. Subject Index.

Abstract: The relationship of the axisymmetric flow between large but finite coaxial rotating disks to the von Karman similarity solution is studied. By means of a combined asymptotic – numerical analysis, the flow between finite disks of arbitrarily large aspect ratio, where the aspect ratio is defined as the ratio of the disk radii to the gap width separating the disks, is examined for two different end conditions: a ‘closed’ end (shrouded disks) and an ‘open’ end (unshrouded or free disks). Complete velocity and pressure fields in the flow domain between the finite rotating disks, subject to both end conditions, are determined for Reynolds number (based on gap width) up to 500 and disk rotation ratios between 0 and – 1. It is shown that the finite-disk and similarity solutions generally coincide over increasingly smaller portions of the flow domain with increasing Reynolds number for both end conditions. In some parameter ranges, the finite-disk solution may not be of similarity form even near the axis of rotation. It is also seen that the type of end condition may determine which of the multiple similarity solutions the finite-disk flow resembles, and that temporally unstable similarity solutions may qualitatively describe steady finite-disk flows over a portion of the flow domain. The asymptotic – numerical method employed has potential application to related rotating-disk problems as well as to a broad class of problems involving flow in regions of large aspect ratio.

Abstract: Mesh Refinement and Redistribution Efficient Techniques for the Analysis of Pollutant Migration Direct General Finite Difference Techniques for Elliptic Problems Defined in Bounded and Unbounded Two-Dimensional Domains Solution Strategies for Elastic and Inelastic Contact Problems of Solids Recent Developments in Finite Difference Methods for the Computation of Transient Flows Numerical Analysis of Rain Effects on an Airfoil Solution Techniques for Boundary Integral Matrices Some Transient and Coupled Problems - A State-of-the- Art Review The Prediction of Instabilities using Bifurcation Theory Long Time Calculations and Non-linear Maps Modelling of Coupled Thermo-elastoplastic-hydraulic Response of Clays Subjected to Nuclear Waste Heat Numerical Modelling of Free- Surface Flows Transient Algorithms and Fluid-Structure Interaction - An Overview Nonlinear Transient Dynamic Analysis of Reinforced Concrete Structures using a Three-Dimensional Approach.

Abstract: A forward error analysis is presented for the Bjorck-Pereyra algorithms used for solving Vandermonde systems of equations. This analysis applies to the case where the points defining the Vandermonde matrix are nonnegative and are arranged in increasing order. It is shown that for a particular class of Vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only, being independent of the condition number of the coefficient matrix. By comparing appropriate condition numbers for the Vandermonde problem with the forward error bounds it is shown that the Bjorck-Pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right-hand side vector on the computer. A technique for computing “running” a posteriori error bounds is derived. Several numerical experiments are presented, and it is observed that the ordering of the points can greatly affect the solution accuracy.

Abstract: with partial differential equations. The governing equations are developed in the context of a physical model and both analytical and numerical methods to solve the problem are explained. The first two chapters serve as introductory material for the subject treated in the book. In Chapter 1 the different difference approximations for derivatives are introduced. The important concepts in finite difference methods, such as error analysis and stability are treated well. Efforts have been made to distinguish between implicit and explicit methods in solving problems. In Chapter 2 the author discusses methods of solution for the first order partial differential equations. The method of characteristics is introduced and the applications of difference methods are reviewed. Chapter 3 forms a major part of the book, gives a thorough treatment to the development of second order hyperbolic partial differential equations and various analytical methods for solving them. The concepts of domains of influence and dependence are explained well. The author has also dealt with conservation laws and propagation of discontinuities in the context of the wave equation. The latter part of this chapter contains several examples dealing with the application of difference methods in solving both linear and nonlinear wave propagation problems. In Chapter 4 parabolic equations have received only a cursory treatment in comparison to the hyperbolic system in the previous chapter. In the context of heat conduction both implicit and explicit difference methods for solving the problems are illustrated. Stability analysis has been carried out for the difference schemes discussed in Chapters 3 and 4. Chapter 5 deals with analytical treatment of the governing equations in fluid dynamics. Though the subject has been dealt with in some detail, it does not contain any description of difference methods for dealing with shocks or other fluid dynamics problems. The exercises at the end of the chapters places emphasis on analytical methods. The inclusion of a list of references for further study would have been useful to motivated readers to further their knowledge in this field. A nice feature of this book is that it gives a good blend of analytical and numerical methods for solving problems. This book could serve as an introductory text to students in engineering sciences and as a guide to practicing engineers to acquaint them with difference methods for solving transient problems.

Abstract: The frequency-dependent resistances and inductances of cables can either be found from analyt ical formulas, or with numerical methods based on finite elements or subdivision of conductors. While analytical formulas are limited to coaxial configurations, numerical methods can be used for non-concentric configurations as well. This paper discusses the method of subdivision into subconductors of circular, square or elemental shape, and compares the results for the case of a coaxial cable, where exact solutions are available from analytical formulas. The inclusion of ground return impedances is discussed next. The method is then applied to the calculation of impedances of pipe-type cables with magnetic pipe material, and of internal impedances of stranded conductors in the power line carrier frequency range.

Abstract: This paper presents a numerical method for predicting the behavior of an elastic membrane wing under aerodynamic loading. A method for finding the pressure distribution generated by flow over a given threedimensional surface is combined with another for finding the shape of a given membrane under a given pressure distribution. The pressure is calculated using a vortex lattice simulation of potential flow, and the shape is determined using a finite element representation of the membrane. An iterative scheme is employed to solve the resulting nonlinear equations which relate the shape and loading to the displacements of the surface. A simple example is given, in which the lift and stress distribution are calculated for a membrane wing with the shape and boundary constraints of an idealized hang glider. The method is equally applicable to yacht sails.

Abstract: We solve the Helmholtz equation in an exterior domain in the plane. A perfect absorption condition is introduced on a circle which contains the obstacle. This boundary condition is given explicitly by Bessel functions. We use a finite element method in the bounded domain. An explicit formula is used to compute the solution out of the circle. We give an error estimate and we present relevant numerical results.

Abstract: A numerical implementation of a discretization scheme of the hydrodynamic model for submicron devices is described and applied to a one‐dimensional ballistic diode. The performance of the numerical method and the physical results of the simulation for different biases and lattice temperatures, and a brief comparison to Monte Carlo simulations, are also given.

Abstract: There are spurious phenomena in the numerical approximation of the hyperbolic equations of fluid dynamics that may be investigated by invoking concepts which originate from wave propagation theory. Many of the significant results which have been obtained by pursuing this kind of analysis are reviewed in this paper by using as an illustration a family of implicit approximations of the simple linear advection equation. Included in this family of algorithms are the common six-point implicit finite difference scheme, the linear finite element/Galerkin scheme and the ‘box’ method. The phase and group velocities of sinusoidal solutions are brought into the analysis of the accuracy and of the spurious reflection or scattering phenomena which are created at computational boundaries and in non-uniform grids. General properties become apparent in this Fourier/wave propagation approach to the analysis. One of these is in the form of an analogy with quantum mechanics. Another shows that certain energy norms of the errors are independent of time discretization, i.e. depend on space discretization alone.