Showing papers on "Finite difference method published in 1985"

Generalized discrete variable approximation in quantum mechanics

Abstract: The formal definition of the generalized discrete variable representation (DVR) for quantum mechanics and its connection to the usual variational basis representation (VBR) is given. Using the one dimensional Morse oscillator example, we compare the ‘‘Gaussian quadrature’’ DVR, more general DVR’s, and other ‘‘pointwise’’ representations such as the finite difference method and a Simpson’s rule quadrature. The DVR is shown to be accurate in itself, and an efficient representation for optimizing basis set parameters. Extensions to multidimensional problems are discussed.

A practical method for modeling fluid and heat flow in fractured porous media

Abstract: A Multiple Interacting Continua method (MINC) is presented which is applicable for numerical simulation of heat and multi-phase fluid flow in multidimensional, fractured porous media. This method is a generalization of the double-porosity concept. The partitioning of the flow domain into computational volume elements is based on the criterion of approximate thermodynamic equilibrium at all times within each element. The thermodynamic conditions in the rock matrix are assumed to be primarily controlled by the distance from the fractures, which leads to the use of nested grid blocks. The MINC concept is implemented through the Integral Finite Difference (IFD) method. No analytical approximations are made for the coupling between the fracture and matrix continua. Instead, the transient flow of fluid and heat between matrix and fractures is treated by a numerical method. The geometric parameters needed in a simulation are preprocessed from a specification of fracture spacings and apertures, and the geometry of the matrix blocks. The MINC method is verified by comparison with the analytical solution of Warren and Root. Illustrative applications are given for several geothermal reservoir engineering problems.

Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method

Abstract: A numerical boundary integral method, relating slip and traction on a plane in an elastic medium by convolution with a discretized Green function, can be linked to a slip-dependent friction law on the fault plane. Such a method is developed here in two-dimensional plane-strain geometry. The method is more efficient for a planar source than a finite difference method, and it does not suffer from dispersion of short wavelength components. The solution for a crack growing at constant velocity agrees closely with the analytic solution, and the energy absorbed at the smeared-out crack tip in the numerical calculation agrees with energy absorbed at the analytic singularity. Spontaneous plane-strain shear ruptures can make a transition from sub-Rayleigh to near- P propagation velocity. Results from the boundary integral method agree with earlier results from a finite difference method on the location of this transition in parameter space. The methods differ in their prediction of rupture velocity following the transition. The trailing edge of the cohesive zone propagates at the P -wave velocity after the transition in the boundary integral calculations.

Numerical simulation of three‐dimensional flow in a cavity

Abstract: SUMMARY Previous three-dimensional simulations of the lid-driven cavity flow have reproduced only the most general features of the flow. Improvements to a finite difference code, REBUFFS, have made possible the first completely successful simulation of the three-dimensional lid-driven cavity flow. The principal improvement to the code was the incorporation of a modified QUICK scheme, a higher-order upwind finite difference formulation. Results for a cavity flow at a Reynolds number of 3200 have reproduced experimentally observed Taylor-Gortler-like vortices and other three-dimensional effects heretofore not simulated. Experimental results obtained from a unique experimental cavity facility validate the calculated results.

Group explicit iterative methods for solving large linear systems

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.

Finite Difference Synthetic Acoustic Logs

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...

Review of Numerical Methods for the Analysis of Arbitrarily-Shaped Microwave and Optical Dielectric Waveguides

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.

A numerical method for bottom interacting ocean acoustic normal modes

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.

Finite difference model for vertical axis wind turbines

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.

Iterative methods in semiconductor device simulation

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.

An efficient algorithm for finite-difference analyses of heat transfer with melting and solidification

Abstract: A simple algorithm incorporating the equivalent heat capacity model is described for the finite-difference heat transfer analysis involving melting and solidification. The latent heat of fusion is ...

Correction of Numerov's eigenvalue estimates

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.

Numerical and experimental study of driven flow in a polar cavity

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.

Numerical simulation of a PSA system. Part I: Isothermal trace component system with linear equilibrium and finite mass transfer resistance

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.

Spectral methods for the Euler equations. II - Chebyshev methods and shock fitting

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.

An explicit conditionally stable finite difference equation for‐heat conduction problems

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.

A critical evaluation of seven discretization schemes for convection–diffusion equations

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.

Field and power-density calculations in closed microwave systems by three-dimensional finite differences

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.

A comparative study of finite element and finite difference methods for cauchy-riemann type equations*

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.

Spectral methods for the Euler equations. I - Fourier methods and shock capturing

Abstract: Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one dimensional, periodic astrophysical ""nozzle'' problem.

A coupled strongly implicit procedure for velocity and pressure computation in fluid flow problems

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...