Abstract: In this paper, the Alternating Group Explicit (AGE) method is developed and applied to derive the solution of a 2 point boundary value problem. The analysis clearly shows the method to be analogous to the A.D.I. method. The extension of the method to ultidimensional problems and techniques for improving the convergence rate and attaining higher order accuracy are also given.

Abstract: The finite‐difference method is a powerful technique for studying the propagation of elastic waves in boreholes. Even for the simple case of an open borehole with vertical homogeneity, the snapshot format of the method displays clearly the interaction between the borehole and the rock, and the origin and evolution of phases. We present an outline of the finite‐difference method applied to the acoustic logging problem, including a boundary condition formulation for liquid‐solid cylindrical interfaces which is correct to second order in the space increments. Absorbing boundaries based on the formulations of Reynolds (1978) and Clayton and Engquist (1977) were used to reduce reflections from the grid boundaries. Results for a vertically homogeneous sharp interface model are compared with the discrete‐wavenumber method and excellent agreement is obtained. The technique is also demonstrated by considering sharp and continuous transitions (damaged zones) at the borehole wall and by considering the effects of wa...

Abstract: This paper presents are view of the numerical methods for the analysis of the homogeneous and inhomogeneons,isotropic and anisotropic, microwave and optical dielectric waveguides with arbitrarily-shaped cross sections.The characteristics of various methods are compared,and a set of qualittative criteria to guide the selection of an appropriate method for a given problem is proposed. The main approaches discussed are those of point matching, integral equations, finite difference, and finite element.

Abstract: In this paper we present a finite‐difference method to numerically determine the normal modes for the sound propagation in a stratified ocean resting on a stratified elastic bottom. The compound matrix method is used for computing an impedance condition at the ocean–elastic bottom interface. The impedance condition is then incorporated as a boundary condition into the finite difference equations in the ocean, yielding an algebraic eigenvalue problem. For each fixed mesh size this eigenvalue problem is solved by a combination of efficient numerical methods. The Richardson mesh extrapolation procedure is then used to substantially increase the accuracy of the computation. Two applications are given to demonstrate the speed, accuracy, and efficiency of the method.

Abstract: A numerical technique based on Patankar's "SIMPLER" algorithm is developed to determine the flow characteristics and performance of a two-dimensional vertical axis wind turbine. The conservation of mass and momentum equations are solved using a finite difference procedure without the necessity of introducing an irrotationality constraint. The computational domain is subdivided into control volumes in cylindrical coordinates and the turbine blades are modeled as a porous cylindrical shell of one control volume thickness. The characteristics of the turbine are computed and compared with previous investigations. The results show a very good agreement.

Abstract: This paper examines iterative methods for solving the semiconductor device equations. The emphasis is on fully coupled methods, because of the failure of decoupled methods for on-state devices. Using the PISCES-II device simulator as a vehicle, incomplete factorization and operator decomposition iterative methods are presented for solving the Newton equations. The dependencies of these methods on factors such as choice of variables, bias condition and initial guess are analyzed. The results are compared with sparse Gaussian elimination.

Abstract: A numerical simulation of a pressure swing adsorption process is presented for a system in which a small concentration of an adsorbable component is separated from an inert carrier. Linear equilibrium and a linear rate expression are assumed. The model equations were solved by orthogonal collocation and by finite difference methods with consistent results. The theory is shown to provide a good representation of the experimental data of Mitchell and Shendalman (1973) for the system CO2-He-silica gel.

Abstract: SUMMARY A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function-vorticity form of the governing equations was approximated by upwind or central finite-differences. Both types of finite-difference approximations were solved by a multi-grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0 1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.

Abstract: The error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by Numerov's method with mesh lengthh isO(k6h4). We show that a simple correction technique of Paine, de Hoog and Anderssen reduces the error to one ofO(k3h4). Numerical examples demonstrate the usefulness of this correction even for low values ofk.

Abstract: The Chebyshev spectral collocation method for the Euler gas-dynamic equations is described. It is used with shock fitting to compute several two-dimensional, gas-dynamic flows. Examples include a shock-acoustic wave interaction, a shock/vortex interaction, and the classical blunt body problem. With shock fitting, the spectral method has a clear advantage over second order finite differences in that equivalent accuracy can be obtained with far fewer grid points.

Abstract: This paper reports a new explicit and conditionally stable finite difference equation for heat conduction. It predicts results with an accuracy comparable with or better than that obtainable by other methods. Stability of operation can be extended to any desired degree by subdividing the basic time step and increasing the number of nodes. Some existing difference equations are special cases of the new equation reported in this paper. The new solution has been tested by calculating the response of a slab to transient and progressive waves whose analytical solutions are known.

Abstract: A comparative study of seven discretization schemes for the equations describing convection-diffusion transport phenomena is presented. The (differencing) schemes considered are the conventional central- and upwind-difference schemes, together with the Leonard,1 Leonard upwind1 and Leonard super upwind difference1 schemes. Also tested are the so called locally exact difference scheme2 and the quadratic-upstream difference scheme.3,4 In multidimensional problems errors arise from ‘false-diffusion’ and function approximations. It is asserted that false diffusion is essentially a multidimensional source of error. No mesh constraints are associated with errors in function approximation and discretization. Hence errors associated with discretization only may be investigated via one-dimensional problems. Thus, although the above schemes have been tested for one- and two-dimensional flows with sources, only the former are presented here. For 1D flows, the Leonard super upwind difference scheme and the locally exact scheme are shown to be far superior in accuracy to the others at all Peclet numbers and for most source distributions, for the test cases considered. Furthermore, the latter is shown to be considerably cheaper in computational terms than the former. The stability of the schemes and their CPU time requirements are also discussed.

Abstract: A three-dimensional finite-difference scheme is used to solve Maxwell's equations in closed microwave systems. The calculation of the sinusoidal steady-state amplitude and phase of the EM-field, and the derivation of the power density are discussed. The method is tested on some canonical examples which can be solved analytically. Finally, some nontrivial three-dimensional examples are given.

Abstract: The computation of flowing well bottom-hole pressure from the pressure of the block containing the well, or the computation of well flow rate when the flowing bottom-hole well pressure is specified, are important considerations in reservoir simulation. While this problem has been addressed by several authors, some important aspects of the problem ae not adequately treated in the literature. An analytical method is presented for computing the well block factors (constants of the productivity index) for a well coated anywhere in a square or rectangular block. Equations for well geometric factors and well fraction constants are given for grid blocks of various types, containing a single well, encountered in reservoir simulation studies. The equations given can be used for both block-centered and point-distributed grids in 5- and 9-point (2-dimensional) finite difference formulations. However, the radial flow assumption utilized in deriving the equations is not always strictly valid. For most practical situations it provides an adequate approximation for near well flow. 12 ref.).

Abstract: A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.

Abstract: A coupled strongly implicit procedure (CSIP) is presented for application in solving the algebraic equation system that results from discrete modeling of incompressible fluid flow problems. The proposed procedure uses a derived pressure equation, representing conservation of mass, that is obtained through the substitution of discrete momentum equations into the discrete mass conservation equation. If a conservative formulation is used, the conservation property is not altered by the use of this pressure equation. The characteristics of the new procedure are examined through application to three test problems having significant inertial influences, in addition to the diffusion of momentum. It is observed that the proposed procedure is robust and stable over a wide range of its parameters. Cost comparisons indicate that while not optimized from a coding viewpoint, the proposed procedure is marginally more expensive than a fully optimized SIMPLE procedure. However, the significant benefits available with the...

Abstract: The stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves. Wave propagation ideas are applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Various known instability phenomena are accounted for in a unified way. Results show: (1) dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface; (2) algebraic GKS instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present; and (3) GKS-stability and P-stability can be established in certain problems by showing that all numerical reflection coefficients have modulus less than 1.

Abstract: We present a numerical method for solving a three-dimensional radiation field by decomposing the radiation field into transverse modes that satisfy the free space wave equation and the boundary conditions. The formalism includes betatron oscillations in a realizable wiggler, finite emittance, and energy spread in an amplifier or oscillator configuration. This formalism can be generalized to calculate sidebands resulting from the synchrotron oscillations. The transverse mode spectral method is compared to the finite difference method, spectral method, and transform spectral method. Examples from a computer code using Gaussian-Hermite expansion are given. We show self-focusing of the radiation beam due to gain and refraction of the FEL.