Showing papers on "Finite difference method published in 1982"

Adaptive mesh refinement for hyperbolic partial differential equations

Abstract: We present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Our approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, data structures and grid generation procedure, and conclude with numerical examples in one and two space dimensions.

Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures*

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.

On the transport-diffusion algorithm and its applications to the Navier-Stokes equations

Abstract: This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (h+Δt+h×h/Δt in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).

Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods

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

A Numerical Method for Solving the Equations of Compressible Viscous Flow

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.

A Review of Neutron Transport Approximations

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 ...

A hybrid moment method/finite-difference time-domain approach to electromagnetic coupling and aperture penetration into complex geometries

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.

New Method for Tidal Current Computation

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.

COSAL: A black-box compressible stability analysis code for transition prediction in three-dimensional boundary layers

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.

Finite difference methods and their convergence for a class of singular two point boundary value problems

Abstract: We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0

Solution Methods of Electrical Field Problems in Physiology

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.

Solution of Burgers' equation with a large Reynolds number

Abstract: Numerical difficulties arise in the solution of Burgers' equation for the case of a large Reynolds number. The aim of this paper is to extend the finite element method proposed by Caldwell et al. to the general case of n elements. The method is tested out on Burgers' equation for two different initial conditions and proves to be much more accurate than finite difference methods for a large Reynolds number.

High order methods for the numerical solution of two-point boundary value problems

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.

High order methods for the numerical solution of two-point boundary value problems

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.

Simulation of miscible displacement in porous media by a modified method of characteristic procedure

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.

Three-Dimensional Model of Spray Combustion in Gas Turbine Combustors

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.

Finite-difference methods for eigenvalues

Abstract: Some useful ways of improving the speed and accuracy of finite-difference methods for eigenvalue calculations are proposed, and are tested successfully on several problems, including one for which the potential is highly singular at the origin.

Parametric analyses of the transient method of measuring permeability

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.

Theoretical basis for field calculations on multi-dimensional reverse biased semiconductor devices

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.

Finite Element Solutions of Transonic Flow Problems

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.

Finite-difference solution of the compressible stability eigenvalue problem

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.