Abstract: The use of 2-point upstream weighting of fluid mobility is proposed as an alternative to the generally employed single-point approximation. Use of the 2-point formula results in the reduction of both numerical dispersion of flood fronts and the sensitivity of predicted areal displacement performance to grid orientation. Stability analysis provides the time step limitation for control of solution oscillations. This, together with limitations for control of overshoot and truncation error, provides a practical basis for the automatic selection of time steps. (18 refs.)

Abstract: The measure of a matrix is used to, first, bound solutions of ordinary differential equations, bound the computer solution by the backward Euler method, and bound the accumulated truncation error; second, to give conditions for the existence and uniqueness of a dc operating point; third, to determine a convergence region for the Newton-Raphson technique and establish its convergence properties. The unifying idea of the paper is the use of the measure of a matrix.

Abstract: A set of equations which transform position and angular orientation of the centroid of the payload platform of a six-degree-of-freedom motion simulator into extensions of the simulator's actuators has been derived and is based on a geometrical representation of the system. An iterative scheme, Newton-Raphson's method, has been successfully used in a real time environment in the calculation of the position and angular orientation of the centroid of the payload platform when the magnitude of the actuator extensions is known. Sufficient accuracy is obtained by using only one Newton-Raphson iteration per integration step of the real time environment.

Abstract: An algorithm is described for numerically evaluating Cauchy principal value (c.p.v.) integrals of the type[Figure not available: see fulltext.]. The remainder is expressed as a contour integral, from which realistic asymptotic estimates are obtained.

Abstract: Graph-theoretic models were developed for the analysis of nonlinear pipe networks. Both symbolic formulation procedures as well as illustrative examples were presented. The topological information contained in the continuity equations together with the component characteristics are used to derive the minimum set of independent equations in a systematic manner. In contrast to conventional methods, the nonlinearities associated with components in the network are treated as an integral part of the formulation procedures and thus they do not require any special treatment. One of the main advantages of the graph-theoretic approach is that the formulation procedure is independent of the numerical technique used to solve the resulting set of nonlinear equations. In other words, once the equations are formulated, a suitable numerical method for solution can be chosen. The graph-theoretic formulation procedures are highly computer worthy.

Abstract: Symmetric laminar incompressible flow past a parabolic cylinder is considered for all Reynolds numbers. In the limit as the Reynolds number based on nose radius of curvature goes to zero, the solution for flow past a semi-infinite flat plate is obtained. All solutions are found by using an implicit alternating direction method to solve the time-dependent Navier-Stokes equations. The solutions found are compared with various other exact and approximate solutions. Results are presented for skin friction, surface pressure, friction drag and pressure drag. The numerical method developed is of particular interest since it combines the alternating direction method with the implicit method for solving the boundary-layer equations. This leads to fast convergence and may be of use in other problems.

Abstract: A numerical solution of Westervelt's inhomogeneous wave equation is obtained to describe the pressure field of sum and difference frequency radiations created by the nonlinear interaction of two monochromatic carriers in the farfield of a circular piston source. Experimental results are presented on the propagation and beam patterns of sum and difference frequency radiations resulting from the interaction of 418‐ and 482‐kHz carriers emitted by a 3‐in.‐diam piston located in a fresh water lake. These results support the numerical methods. Several approximations commonly made to obtain dosed‐form solutions are then evaluated with the numerical solution.

Abstract: Summary General boundary conditions for the problem of electromagnetic induction in a two-dimensional model of a conductor with an arbitrary sub-surface conductivity structure are considered. Program subroutines for both E-polarization and H-polarization cases are given. These boundary condition subroutines can be used to replace the previously presented subroutines and allow the solution of any conductivity configuration within the conducting region by use of the same numerical technique. An example of a particular model with a sub-surface step structure is illustrated. Also, an improved method of calculating the surface values of the tangential component of the H-field (E-case) and the tangential component of the E-field (H-case) at the surface of the conducting region is given for the numerical solution. This new method uses a derivative approximated from the true functional form of the fields instead of a linear approximation and may be applied when a layered or subsurface anomaly is modelled. Some general discussion of the numerical method is given. At present there is considerable interest in the solution of the problem of electromagnetic induction in the Earth and the local perturbations of the fields when a lateral inhomogeneity is encountered. Jones & Price (1970) considered a twodimensional problem with a conducting half-space made up of two quarter-spaces of different conductivity, and Jones & Price (1971) considered a surface or buried region of rectangular cross-section of one conductivity surrounded by a region of different conductivity. Jones & Pascoe (1971) extended this work to consider a region of arbitrary shape and of several conductivities surrounded by a region of different conductivity and gave computer programs for the numerical solution of this problem for both the E-polarization (E parallel to the strike of the structure) and the H-polarization (H parallel to the strike of the structure) cases. The programs given by Jones & Pascoe (1971) may be used to consider long cylinders composed of several conductivities and of arbitrary cross-section embedded in a region of uniform conductivity, but cannot be used to solve the problem in which the surrounding region is not uniform. It is important to be able to solve the more general case in which the surrounding medium is a layered one and is not necessarily the same at great distances from the conductivity inhomogeneities on both sides. In

Abstract: This paper describes a simple numerical method for the solution of the one-dimensional steady-state carrier-transport equations as they apply to semiconductor devices. The proposed method uses an implicit method in its treatment of the recombination-rate term in the continuity equations, rather than the explicit method employed by Gummel and De Mari. The method is particularly suited to the solution of problems involving heavy recombination, and it is shown that, in such cases, it is capable of yielding solutions which are unobtainable by using the Gummel-De Mari method.

Abstract: The problem of transient unconfined seepage under drawdown in riverbanks and dams is solved by using a finite element procedure. An iterative procedure is employed to compute movements of the free surface caused by time wise fluctuations in the external water levels. The finite element solutions are compared with laboratory experiments on a parallel-plate viscous flow model and field observations at a section along the Mississippi River. Correlation between the numerical solutions and observations is found to be good. Consideration is given to such special requirements of numerical techniques as discretization of infinite media and various possible flow situations at discretized end boundaries. An effort is made to obtain the numerical formulations and the computer codes that yield acceptable accuracy with economy. Some projections for use of the method for design analysis are presented.

Abstract: It is shown in this paper that a theoretical method of centers, introduced by Huard, converges linearly. It is also shown, by counter-example, that a modified method of centers due to Huard and a method of feasible direction due to Topkis and Veinot cannot converge linearly even under convexity assumptions. Because of this, a new modified method of centers is introduced which uses a quadratic programming direction finding subroutine. In most uses this new method is not more complicated than Huard's modified method of centers. But it does converge linearly. A method for implementing it without loss of rate of convergence is also discussed.

Abstract: An approximate two-dimensional numerical analysis has been developed for studying double- (or triple-) diffused transistors. The program supplies dc and hf terminal characteristics (e.g., h fe , r bb , f T , I B , V BE ) over a wide range of operating collector currents and voltages for a given set of physical device parameters (mask dimensions, impurity profile, etc.). The approach is based on obtaining a set of differential equations describing current flow in the longitudinal (emitter-collector) direction and a separate differential equation describing current flow in the lateral direction. The assumption is made of space-charge or space-charge-neutral regions with current- and voltage-dependent boundaries. The equations are valid for arbitrary injection levels and automatically include such high-level effects as conductivity modulation, base widening, and emitter current crowding. Both theoretical and experimental results are given for transistors with f T values between 100 MHz and 3 GHz. The validity of the approach is confirmed and some areas requiring further study are outlined. The technique described is felt to be particularly attractive for the design and optimization of high-power microwave devices, due to the small computer execution time and memory requirements.

Abstract: SOME extensions and improvements to the method of successive approximations, and practical means for improving its convergence, are described. A procedure is given in ALGOL 60 for the algorithm of the method. A numerical method of successive approximations for solving optimal control problems was described in [1]. This method was subsequently used to solve a large number of concrete problems. Several types of standard program realizing the algorithm of the method were compiled. The present paper gives some extensions and improvements to the method of [1] and practical devices for improving its convergence. These modifications and devices have been tested and employed in computer evaluations based on the standard programs. The program here described amounts to a procedure written in ALGOL 60, realizing the method of successive approximations together with certain devices for improving its convergence. By using the program, the numerical solutions of quite a wide range of optimal control problems can be obtained automatically.

Abstract: A method to analyze the three-dimensional vibrations of orthotropic cylinders is presented. The field equations are in the form of three coupled, second-order differential equations with nine elastic constants. The vibration problem is solved by means of the Frobenius method, using power series expansion in the radial coordinate, and assuming sinusoidal dependence on the longitudinal and circumferential coordinates and on time. Frequency equation for the free vibration of solid circular cylinders is derived and a numerical example is given. An extensive study of the dynamic behavior of orthotropic cylinders can be made by this analysis.

Abstract: Numerical solutions are obtained to the Navier-Stokes equations for flow past a paraboloid of revolution at zero angle of attack for a full range of Reynolds numbers. The solution approaches the Stokes and Oseen results at low Reynolds number and approaches the boundary-layer solution at high Reynolds number. The numerical method employed is an implicit alternating direction method which converges to the steady-state solution rapidly. Numerical experiments are made to determine the optimal time step for maximum rate of convergence.

Abstract: A numerical-theoretical technique is described for determining the surface current density distribution and subsequently the near- and far-zone fields of an arbitrarily shaped perfectly conducting body excited by an arbitrary primary source. The arbitrary surface is described by dividing it into a number of connected cells which are mathematically described as quadric surfaces. The "arbitrary body" formulation is applied to two configurations; namely the radial dipole above a conducting cylinder of finite length and a quarter-wavelength monopole mounted atop the fuselage of a CH-47 helicopter. The numerical results are compared with those obtained through an experimental program as well as those obtained by alternate numerical means and good agreement is noted.

Abstract: Best possible error estimates are proved for spline semi-discrete approximations to dissipative initial value problems. Error bounds are also established for suitable difference quotients.

Abstract: A method for reducing the bandwidth of matrices is presented. The method was developed for symmetric matrices, but it is also applicable for asymmetric matrices. A detailed flowchart for the algorithm is included. Numerical results obtained by applying this method, as well as several other previously published methods, 10 examples, are tabulated. While no mathematical proof of convergence is attempted, the numerical results indicate that, in general, the method does converge.

Abstract: An analysis is given of the fundamental properties of an iterative numerical method, free of any inherent physical approximations, for evaluating electron transport effects in solids. A formulation applicable to both semiclassical and quantum calculations is used. The method is shown to reduce, in effect, to evaluation of the time development of an electron state, and its application to both steady state and time response calculations is considered. Conditions for iterative convergence are established that are less restrictive than those previously supposed.

Abstract: A numerical method is given for determining the transient flow past a sphere which is impulsively started from rest with constant velocity in a viscous fluid. One of the features of the method is that the calculation of the flow at early times is performed using boundary‐layer variables, which leads to very accurate solutions. The problem is formulated in terms of the stream function and the vorticity. The method used is the semi‐analytical one of series truncation in which the stream function and vorticity are expanded in a series of Legendre functions with argument z = cosθ, where θ is the polar angle. The governing equations are thus reduced to sets of time‐dependent differential equations in the radial variable. In theory the number of these equations is infinite but in practice only a finite number can be solved to give an approximation to the flow. They are solved numerically by the Crank‐Nicolson procedure. Numerical solutions are given for the cases R = 20, 40, 100, 200, 500, 1000, and ∞, where R ...

Abstract: The viscous compressible flow in the vicinity of a right-angle corner, formed by the intersection of two perpendicular flat plates and aligned with the free stream, is investigated. In the absence of viscous-inviscid interactions and imbedded shock waves, a theory is developed that is valid throughout the subsonic and supersonic Mach number range. Within this limitation and the additional assumptions of unit Prandtl number and a linear viscosity-temperature law, a consistent set of governing equations and boundary conditions is derived. The method of matched asymptotic expansions is applied in order to distinguish the relevant regions in the flow field.In the corner region the Crocco integral is shown to apply, even for a three-dimensional flow field. The equations governing the flow in the corner layer consist of four coupled nonlinear elliptic partial differential equations of the Poisson variety. Since they do not lend themselves to analytic solution, numerical methods are employed. Two such methods used here are the Gauss-Seidel explicit technique and the alternating direction implicit method. The merits of both techniques are discussed with regard to convergence rate, accuracy and stability. The calculations show that in cases where the Gauss-Seidel method fails to give converged solutions, owing to instability, the alternating direction implicit method does provide converged solutions. However, in cases where both methods are convergent, there is no appreciable difference in convergence rates. The numerical calculations were done on a CDC 6600 computer.Results of calculations are presented for representative compressible-flow conditions. The extent of the corner disturbance is controlled by the Mach number and wall temperature ratio in a manner analogous to the two-dimensional boundary layer. A swirling motion is noted in the corner layer which is influenced to a great extent by the asymptotic cross-flow profiles. The skin-friction coefficient is shown to increase monotonically from zero at the corner point to its asymptotic two-dimensional value. For cold wall cases, this value is approached more rapidly. The asymptotic analysis indicates that for even colder wall cases, not considered here, an overshoot is possible.

Abstract: A reservoir simulation technique that employs semi- implicit approximations to relative permeabilities exhibits excellent stability and convergence characteristics when applied to water- or gas-coning problems. Recent workers in this area have made a simplifying assumption in order to linearize the flow terms of the semi-implicit finite- difference equations. A method is described of solving efficiently the nonlinear form of the equations and demonstrates that time-step sensitivity is reduced by iterating on the nonlinear terms. In addition, it addresses the problem of allocating a well's production among multiple grid blocks. Example problems include both water-coning and gas-percolation applications.

Abstract: Equations for large coupled flap-lag motion of hingeless elastic helicopter blades are consistently derived. Only torsionally-rigid blades excited by quasi-steady aerodynamic loads are considered. The nonlinear equations of motion in the time and space variables are reduced to a system of coupled nonlinear ordinary differential equations with periodic coefficients, using Galerkin's method for the space variables. The nonlinearities present in the equations are those arising from the inclusion of moderately large deflections in the inertia and aerodynamic loading terms. The resulting system of nonlinear equations has been solved, using an asymptotic expansion procedure in multiple time scales. The stability boundaries, amplitudes of nonlinear response, and conditions for existence of limit cycles are obtained analytically. Thus, the different roles played by the forcing function, parametric excitation, and nonlinear coupling in affecting the solution can be easily identified, and the basic physical mechanism of coupled flap-lag response becomes clear. The effect of forward flight is obtained with the requirement of trimmed flight at fixed values of the thrust coefficient.

Abstract: The application of the Rayleigh-Ritz method for approximating the eigenvalues and eigenfunctions of linear eigenvalue problems in several dimensions is investigated. The object is to improve upon known error estimates for the approximate eigenfunctions. Results for the Galerkin approximation of the eigenfunctions are developed under varying assumptions on the boundary conditions and domain of definition of the eigenvalue problem. These results, coupled with a previous result relating Galerkin and Rayleigh-Ritz approximation of the eigenfunctions, are then used to obtain improved error estimates for the approximate eigenfunctions in theL 2 and uniform norms.