Showing papers on "Finite difference method published in 1970"

Implicit flood routing in natural channels

Abstract: Flood routing in natural channels and many other applications in hydraulic engineering based on the solution of the equations of unsteady flow require fast and accurate numerical methods. Numerical methods which are successful in other applications prove to be inefficient when used for flood routing. An implicit numerical method which is both fast and accurate can be established on the basis of: (1) a centered difference scheme to represent the primary differential equations in finite difference form; and (2) the simultaneous solution of the finite difference equations for each time step. The difference equations constitute a system of nonlinear algebraic equations which can be solved on a digital computer by Newton iteration method. The computational scheme becomes very efficient when advantage is taken of the sparseness of the matrix of coefficients of the linear systems employed in the iteration. Applications of the implicit method show that it can be conveniently used for highly irregular channels.

Numerical Solution of Saint-Venant Equations

Abstract: The general, one-dimensional Saint-Venant equations are presented for a rigid open channel of arbitrary form, not necessarily prismatic, containing a flow that may be spatially varied. The theoretical basis for the method of characteristics is reviewed and used to show that, in the general case, the speed of long-wave disturbances is given by the slope of the characteristic curves. Finite-difference schemes on a rectangular net in the x - t plane and based on the characteristic forms of the Saint-Venant equations, as well as on the direct forms, are given and examined for their stability. The von Neumann technique for stability analysis is presented in detail. Explicit numerical schemes, which are simple, but require small steps in time because of stability problems, are contrasted with implicit schemes that permit numerical solution over large time steps but require the solution of large sets of simultaneous algebraic equations at each step. The double-sweep or progonka method, an exact time- and space-saving technique for solving these (locally linearized) equations, is also given in detail.

Calculation of compressible adiabatic turbulent boundary layers

Abstract: A method of solution of the compressible turbulent boundary-layer equations for twodimensional and axisymmetric flows, with transverse-curvature effects, is presented. The Reynolds shear-stress term is eliminated by an eddy-viscosity concept and the time mean of the product of a fluctuating velocity and temperature term appearing in the energy equation is eliminated by an eddy-conductivity concept. An implicit finite-difference method is used in the solution of both momentum and energy equations after they are linearized. Results are presented for several adiabatic compressible flows, with and without pressure gradients for Mach numbers up to 5. The results show that the method is quite accurate and fast; a typical flow can be calculated in one or two minutes on the IBM 360/65 computer.

Three-dimensional boundary layer near the plane of symmetry of a spheroid at incidence

Abstract: This paper presents incompressible laminar boundary-layer results on both the leeside and windside of a prolate spheroid. The results are obtained by an implicit finite difference method of the Crank–Nicolson type. Particular attention has been given to the determination of separation and of embedded streamwise vortices. No restriction on the angle of attack or the thickness ratio is imposed, nor are there invoked any of the common assumptions such as similarity, conical flow and others. The results suggest an embedded vortex region existing between the regular boundary-layer region and the separated region. At higher angle of attack, the vortex region becomes so thick that it itself may be more appropriately called ‘separated’ also. The latter possibility leads to questions of applicability for existing theories on three-dimensional separation.

Stabilization of finite difference methods of numerical integration

Abstract: To examine the stabilizing effects of a modification of the classical finite difference methods of numerical integration the differential equations of perturbed Keplerian motion are integrated for two examples: an artificial satellite of the Earth, and Hill's variation orbit. The modified methods remove much of the instability that is inherent to the classical methods.

Analytical Treatment of Two-Phase Infiltration

Abstract: An approximate analytical treatment for the problem of one-dimensional infiltration into a homogeneous porous medium is presented. Movement of both the air phase and the water phase is considered. The procedure assumes that capillary pressure can be neglected in the saturation equation, whereas it is retained in an integral equation for the unknown total flow. The two equations are solved in a step-wise manner to yield the saturation profile, and the infiltration rate at any time. Infiltration rate curves are obtained for a number of situations involving different boundary or initial conditions or both. Comparisons are made with results obtained from a finite difference solution.

Numerical Integration of Navier-Stokes Equations

Abstract: T is extensive literature on the numerical integration of some difference forms of the Navier-Stokes equations. The references quoted herein are those relevant to later discussions and not intended to be complete. There is also extensive literature on the mathematics of difference methods for the solution of partial differential equations for which Ref. 1 is a comprehensive review and contains an extensive bibliography. The present paper is an attempt to develop the implications of a mathematical theorem, which, when put into proper perspective, offers a unified and coherent treatment of various practical aspects of computation. Like approximate differential analyses and physical tests, numerical methods are fallible. Larger and faster computers offer no easy answer to computational difficulties like stability and convergence. Smooth and physically reasonable results of computation are often less accurate than those not so smooth. Since the true asymptotic nature is difficult to establish, both the difference and the differential approximations are nonrigorous; but they are useful, especially with the help of physical experimentation and rigorous mathematical results. We are sympathetic to such heuristic and nonrigorous analysis in favor of obtaining results useful in practice. We shall consider only the difference form of the NavierStokes equations in Eulerian coordinates. Lagrangian coordinates are convenient for flows involving free-surface boundary or active processes associated with fluid elements. It suffers, however, from the serious distortions of the Lagrangian net and from the cummulative errors of the particle paths at large times. Thus various mixed or coupled Eulerian-Lagrangian schemes have been developed even for free boundary problems. Eulerian formulation, even for a single fluid in the absence of a free boundary, has its share of problems, which we shall discuss. We write, for the NavierStokes equations,

Computer-aided thermal analysis of power semiconductor devices

Abstract: Thermal analysis of power semiconductor devices is often complicated by odd geometries and the nonlinear properties of materials. It is the type of problem that can best be handled by a computer. Fortunately, numerous general-purpose heat transfer programs have been written that can be applied to power semiconductor deyices. The majority of programs were written for other technologies (aircraft engines, nuclear energy, and space) but they are sufficiently general for electronic applications. These programs are most often based on the method of finite differences. While this method can yield results to any degree of accuracy required, it is not readily apparent just how accurate the results are. In general, a user desires results as accurate as necessary while minimizing the cost of the problem solution. This paper deals with methods of achieving that goal. Descriptions of truncation and convergence errors are given along with methods of estimating their magnitude. Various forms of the finite difference method are discussed. Methods for speeding convergence are shown, including acceleration algorithms and simple guidelines for ordering matrices and selecting node boundaries. Convenient methods of displaying and interpreting the results are also discussed.

A finite difference method for analyzing the compression of poro-viscoelastic media

Abstract: The progress of compression of a water saturated porous medium is derivable from the equation of continuity,Darcy's law, and an appropriate effective stressdilation relationship. Theories of secondary consolidation assume that the effective stress-dilatation relationship is time-dependent. One form of this type of characterization is a system of linear viscoelastic models. The system chosen consists of an elastic element in series with an arbitrary number ofKelvin units. The formulation of this system is a differential-integral equation. The integral portions of the equation are a series of convolution integrals. A finite difference scheme is developed in which the single governing differential-integral equation is broken up into a system of equations of the heat conduction andEuler types. A stability theorem is proved.

Alternating Direction Methods for the Reactor Kinetics Equations

Abstract: A class of finite difference methods known as alternating semi-implicit techniques is presented for the solution of the multigroup diffusion theory reactor kinetics equations in two space dimension...

A Moving Boundary Model of a One‐Dimensional Saturated‐Unsaturated, Transient Porous Flow System

Abstract: A moving boundary model is proposed for one-dimensional transient flow of water through a porous medium of which part is saturated and part is unsaturated. The model is based upon a theory that implies a discontinuous propagation of pore pressure at the saturated-unsaturated interface. The moving boundary model is used to study a gravity drainage problem. Two numerical procedures are developed to solve the problem, an approximate Taylor series method and a finite difference method. The validity of the methods was appraised by comparing the results with experimental data given by Watson [1967]. The Taylor series method is limited in applicability because of a need for accurate determination of derivatives of hydraulic conductivity and of moisture content respecting pressure head. The finite difference solution is very efficient because only changes in the unsaturated region are computed and the need for iteration is obviated.

Finite-Difference Convection Errors

Abstract: Finite-difference methods are making possible more complete mathematical models of aquatic ecosystems than was previously possible with only analytical methods. A basic component of most models of aquatic ecosystems involves the convection or movement of materials by flowing waters. The errors associated with several finite-difference models of convection are investigated. These errors were found to distort results in ways that are not easily detected. The tendency to numerically spread or mix materials was found to be a common error, along with production of oscillation and a tendency to skew distributions. A method of estimating and controlling these errors is developed and investigated.

A one-sweep numerical method for vector-matrix difference equations with two-point boundary conditions

Abstract: A new method is proposed for reducing two-point boundaryvalue problems for vector-matrix systems of linear difference equations to initial-value problems. The method has the advantage that only one sweep is required, and memory requirements are minimal. Applications to potential theory are discussed.

Finite difference methods for surface waveguides

Abstract: Approximate numerical methods based on the use of finite differences are applied to surface waveguides of various types, including those supporting hybrid modes. The ultimate aim is to develope novel waveguides suited to use in guided ground transport systems.

Truncation Error Analysis of Finite Difference Approximations to the Transport Equation

Abstract: The truncation error of difference approximations to the transport equation is examined, and difference equations which are uniformly second-order accurate are derived. Resulting angular quadrature...

Temperature distribution and effectiveness of a two-dimensional radiating and convecting circular fin

Abstract: An analysis is made of the heat-transfer characteristics of a circular fin dissipating heat from its surface by convection and radiation. The temperature is assumed uniform along the base of the fin and constant physical and surface properties are assumed. There is radiant interaction between the fin and its base. Two separate situations are considered. In the first situation heat transfer from the end of the fin is neglected. Solution of the linear conduction equation with nonlinear boundary conditions has been obtained by a least-squares fit method, and also by the finite difference method and the results compared. Results are presented for a wide range of environmental conditions and physical and surface properties of the fin. In the second situation, heat transfer from the end of the fin is also included in the analysis. The solution for the second situation is obtained by a finite-difference procedure only. It is shown that neglecting heat transfer from the end is a good approximation for long fins or for fins of high thermal conductivity material.

Krylov–Bogoljubov Method for Difference Equations

Abstract: An extension to difference equations of the now classical method of Krylov and Bogoljubov is presented. This extension is based primarily (although not entirely) on the analogy between differential and difference equations. A specific example illustrating this method is also presented.