Abstract: Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal order error estimates in $L^2 $ and $W^{1,2} $ are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems $(b \gg a)$, these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

Abstract: Introduction to Groundwater Modeling presents an overview of the fundamental concepts and applications of computerized groundwater modeling.

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: 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: An approach is presented for the direct modeling of electromagnetic penetration problems which involves a hybrid technique combining the frequency-domain method of moments (MM) and the finite-difference time-domain (FD-TD) method. The hybriding is based upon a novel use of a field equivalence theorem due to Schelkunoff, which permits a field penetration problem to be analyzed in steps by treating the relatively simple external region and the relatively complex internal region separately. The method involves first, determination of an equivalent short-circuit current excitation in the aperture regions of the structure using MM for a given external illumination. This equivalent current excitation over the aperture is next used to excite the complex loaded interior region, and the penetrating fields and induced currents are computed by the FD-TD method. A significant advantage of this frequency domain/time domain hybriding is that no Green's function need be calculated for the interior region. This hybrid approach takes advantage of the ability of MM to solve exterior problems using patch models and also takes advantage of the ability of FD-TD to model in great detail localized space regions containing metal structures, dielectrics, permeable media, anisotropic or nonlinear media, as well as wires.

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: A fast computer code COSAL for transition prediction in three dimensional boundary layers using compressible stability analysis is described. The compressible stability eigenvalue problem is solved using a finite difference method, and the code is a black box in the sense that no guess of the eigenvalue is required from the user. Several optimization procedures were incorporated into COSAL to calculate integrated growth rates (N factor) for transition correlation for swept and tapered laminar flow control wings using the well known e to the Nth power method. A user's guide to the program is provided.

Abstract: The forward problem in electrophysiology?the computation of the potential distribution due to a known electrical source in a known volume conductor?is discussed. Three methods of solution are considered: 1) the finite difference method 2) a discretized integral equation method 3) the analytic method.

Abstract: In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a “one-off” formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.

Abstract: The miscible displacement of one incompressible fluid by another in a porous medium is described by a system of two nonlinear equation, one elliptic in form for the pressure and the other parabolic in form for the concentration. The pressure and the fluid velocity will be approximated by a mixed finite element method, and the concentration by a finite difference method based on the use of a modified method of characteristic procedure. A convergence analysis is given for the method.

Abstract: A two-dimensional self consistent MOS transistor model accounting for the avalanche effect is described. The classical semiconductor equations—Poisson's equation and the two carrier equations—are solved with the finite difference method. The pair production rate is evaluated at any mesh point and dominates the inhomogeneity term of the carrier continuity equations in case of avalanche. Calculated and measured current-voltage characteristics are in good agreement and thus support our model. For a 3 μm device the electrical potential, the carrier densities, and the generation rates are shown in quasi three-dimensional plots from which the avalanche generation in the pinch-off region becomes apparent. Furthermore, hole storage close to the interface is seen to take place in the channel up to the vicinity of the source region. The corresponding barrier lowering leads to increased electron injection from the source and enhanced avalanche. The barrier lowering is supported by the influence of the parasitic bulk resistance.

Abstract: A mathematical model of the three-dimensional two-phase reacting flows in gas turbine combustors has been developed which takes into account the mass, momentum, and energy coupling between the phases. The fundamental equations of motion of the droplets are solved numerically in a Lagrangian frame of reference, using a finite-difference solution of the governing equations of the gas. Well-known relations are used to model the heat and mass transfer processes and the initial droplet heat-up is allowed for. The entire fuel spray is constructed using a finite number of size ranges obeying a two parameter droplet size distribution. The results are found to be in close agreement with experimental data. An important feature of this analytical technique is that it permits the rational selection or specification of fuel nozzle design.

Abstract: Analyses are made to compare the two approaches of using the transient method to determine permeability of rocks: the simplified version and the numerical version. The simplified version assumes that when the sample volume is much smaller than the reservoir volume, the fluid storage in a rock sample can be neglected and the pressure decay in the upstream reservoir can be approximated by an exponential function of time. The numerical version uses finite difference method to solve the differential equation of pressure decay and matches the observed pressure decay with the calculated decay curves. The pressure decay calculated by the numerical version is not generally a simple exponential function of time, as suggested by the simplified version. Our analyses show that the numerical version fits to the observed data very well. The permeability value determined by the simplified version tends to be greater than that determined by the numerical version. The difference in apparent permeability between the two depends on many factors such as rock properties (porosity and compressibility), sample size, reservoir volumes, etc. A relative fluid storage is used to illustrate the difference of these two approaches. For a relative fluid storage value of greater than 0.03, the difference in permeability between the two is more than 30%. For a system with a fluid storage of less than 0.01, these two versions agree with each other well.

Abstract: A method of local mesh refinement has been developed in which additional nodes are used only in regions where they are necessary. The extra mesh lines are not extended across the entire simulator as in the usual current practice. Refinement of a two-dimensional element, herein defined as the rectangle formed by mesh lines with nodes at the vertices, consists of bisecting it in each coordinate direction, thus dividing the original element into four smaller, but similar ones, and creating five nodes. The resulting smaller elements may themselves be refined by the same procedure. An element must be refined if two adjacent elements of the same size have been refined or if one smaller adjacent element has been refined. Three configurations of five nodes each which arise in current refinement methods also appear in the new procedure. In addition, a configuration of six nodes also occurs. Finite difference analogs for all configurations were developed. An extensive study of local mesh refinement near a well was made for the repeated five spot in a homogeneous reservoir with unit mobility ratio. Results of the finite difference solutions were compared with the analytic solution. Generally, it was found there should be at least twomore » consecutive increments of the same size between changes in increment size. When this practice is followed, a local minimum in truncation error occurs at each node where there is a change in increment size. The more accurate solution obtained in the vicinity of the well by use of smaller increments there results in a more accurate solution also in the coarser unrefined mesh.« less

Abstract: In this report the theory and the computational methods used to perform numerical field calculations on reverse biased two-dimensional (or three-dimensional with circular symmetry) structures are fully described and completely justified. Finite difference methods are used to approximate Poisson's equation and, together with a depletion region logic, solutions to semiconductor field problems are obtained without the need to solve the complete set of device equations. Two unique aspects of the methods are the depletion logic and the approach taken to handle dielectric interfaces.

Abstract: A compressible stability analysis computer code is developed. The code uses a matrix finite difference method for local eigenvalue solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used initial value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed which may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black box stability code. Sample stability calculations are presented for the boundary layer profiles of a Laminar Flow Control (LFC) swept wing.

Abstract: The artificial compressibility method is briefly reviewed and the effect of different forms of the artificial viscosity are studied. Successful finite element solutions of transonic airfoil problems are presented. Iterative procedures including VLSOR, Zebroid, fast solver, firstand second-degree methods, and variable acceleration parameters such as preconditioned steepest descent and conjugate gradient are discussed and necessary modifications for transonic flow computations by finite elements are implemented leading to fast, reliable, and efficient calculations.

Abstract: Direct iterative methods for solving the linear system $AX = Y$ split A into a difference $M - N$. By viewing N as a weak multiplication operator, we determine the convergence rates of block direct iterative methods for elliptic and parabolic difference equations. The difference equations may arise from very general partial differential equations on general domains in m space dimensions.

Abstract: The methodology used to implement structural sensitivity calculations into a major, general-purpose finite-element analysis system (SPAR) is described. This implementation includes a generalized method for specifying element cross-sectional dimensions as design variables that can be used in analytically calculating derivatives of output quantities from static stress, vibration, and buckling analyses for both membrane and bending elements. Limited sample results for static displacements and stresses are presented to indicate the advantages of analytically calculating response derivatives compared to finite difference methods. Continuing developments to implement these procedures into an enhanced version of SPAR are also discussed.