Showing papers on "Finite difference method published in 1971"

Quantitative Evaluation of Numerical Diffusion (Truncation Error)

Abstract: Truncation error limits the use of numerical finite difference approximations to solve partial differential equations. In the solution of convection-diffusion equations such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing 2-phase fluid flow can also be rearranged into a convection-diffusion form. Miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs and probably all numerical analysts recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this study is to give the user more than just a qualitative feel for the importance of truncation error. Insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. (14 refs.)

On finite-difference methods for the Korteweg-de Vries equation

Abstract: The purpose of this paper is to set up and analyse difference schemes for solving the initial-value problem for the socalled Korteweg-de Vries equation. After the discussion of a difference scheme which is correctly centered in both space and time, the construction of difference schemes which implicitly contain the effect of dissipation is described.

A Finite Difference Method for Unsteady Flow in Variably Saturated Porous Media: Application to a Single Pumping Well

Abstract: Finite difference equations were derived by using the divergence theorem to convert the nonlinear partial differential equation (which approximately describes liquid flow in a variably saturated, elastic porous medium) to an integral equation, and then to integrate around individual mesh volume elements. Original nonlinearity of the differential equation was preserved by keeping saturations and relative conductivities current with hydraulic heads during the iterative matrix solution method. The problem of axisymmetric flow to a water well that penetrates one or more elastic rock units, the upper one of which is unconfmed, provides a convenient vehicle for analysis of the procedural and theoretical study of unconfined and semiconfined flow. Of the three methods tried to solve the matrix equation that resulted from the finite difference equations (which included a form of the direct alternating direction implicit method, the iterative alternating direction implicit method, and the line successive overrelaxation method), the line successive overrelaxation method was the fastest and was selected for use in a general computer program. A comparison with analytical solutions that use Boulton's convolution integral as a velocity boundary condition at the water table for a single aquifer and an aquifer-aquitard system demonstrates close correspondence of the numerical and analytical solutions, even for a case where the water table is lowered appreciably.

Analysis of Turboalternator Magnetic Fields by Finite Elements

Abstract: The nonlinear quasi-Poisson equation that describes static magnetic fields in saturable iron is solved approximately by minimizing the corresponding nonlinear energy functional. The minimization is performed by means of the method of finite elements, using firstorder elements and a quadratically convergent iterative solution method. The method is applied to a turboalternator and used to predict all the normal shop-floor test results. Excellent agreement is found between experimental and computed values. Computing times are found to be extremely fast, and it is concluded that this method is capable of producing results comparable to those obtained by finite difference methods, but at very much reduced cost.

Solving the Biharmonic Equation as Coupled Finite Difference Equations

Abstract: A technique is proposed for solving the finite difference biharmonic equation as a coupled pair of harmonic difference equations. Essentially, the method is a general block SOR method with convergence rate $O(h^{{1 / 2}} )$ on a square, where h is mesh size.

On the structure of error estimates for finite-difference methods

Abstract: In this paper we study in an abstract setting the structure of estimates for the global (accumulated) error in semilinear finite-difference methods. We derive error estimates, which are the most refined ones (in a sense specified precisely in this paper) that are possible for the difference methods considered. Applications and (numerical) examples are presented in the following fields: 1. Numerical solution of ordinary as well as partial differential equations with prescribed initial or boundary values. 2. Accumulation of local round-off error as well as of local discretization error. 3. The problem of fixing which methods out of a given class of finite-difference methods are "most stable". 4. The construction of finite-difference methods which are convergent but not consistent with respect to a given differential equation.

Minimum Weight Design of Stiffened Fiber Composite Cylinders

Abstract: : A structural synthesis capability for stiffened fiber composite cylindrical shells has been developed. The design variables are the configuration and material parameters. The material parameters considered are the fiber volume content and the ply orientations in the skin. Both longitudinal and circumferential stiffeners are assumed to have hat cross-sections. The instability loads of the heterogeneous anisotropic cylinder under combined axial, radial, and torsional loading are calculated with smeared stiffener theory. In the synthesis scheme, multiple load conditions on the cylinder are permissible. The optimal design problem is cast into a model of a nonlinear mathematical programming problem which is solved with the penalty function technique of Fiacco-McCormick. Since weight is independent of the ply orientations there exist alternative optima. Special modifications on the iteration procedure are made, so that the ply angles tend to move in such a way that the overall strength of the cylinder is improved. Numerical examples are discussed. (Author-PL)

Elastic-wave propagation in two evenly-welded quarter-spaces

Abstract: This paper deals with elastic-wave propagation in two evenly-welded quarter-spaces. A compressional line source can be located at any point within either medium. The numerical solutions to this problem have been obtained by using the finite difference method. A computer program has been written to obtain synthetic seismograms of the horizontal and vertical displacements at all nodes of the superimposed grid, for the following cases: (a) elastic-wave propagation in a quarter-space, and (b) elastic-wave propagation in two quarter-spaces. Reflected, converted, transmitted, and diffracted phases are identified and interpreted. Surface and interface waves, originated at the corner by diffraction of the source pulse, are investigated as a function of the rigidity contrast and the velocity contrast between the two media and of the position of the source. Two-dimensional seismic modeling techniques have been used to provide a qualitative experimental verification of the numerical results.

Calculation by a finite difference method of supersonic turbulent boundary layers with tangential slot injection

Abstract: Finite difference calculation based on eddy diffusivity and mixing length flow theory to characterize supersonic turbulent boundary layer with tangential slot injection

Computation of nonequilibrium merged stagnation shock layers by successive accelerated replacement

Abstract: Results are presented for fully merged shock layer computations in the stagnation region of a hypersonic body. Numerical solutions are obtained with a new technique which avoids several troublesome aspects of previous methods. The effects of nonequilibrium chemistry are studied for a multicomponent air mixture and the resulting electron concentrations are calculated. The analysis assumes a continuum approach and employs the well known "locally similar" flow model. The governing system of equations constitutes a two-point boundary value problem which is solved using a simple finite difference method called successive accelerated replacement, or SAR. Special attention is given to a singularity that appears in the continuity equation, and the importance of an acceleration factor on the convergence of the solution is discussed. Solutions are obtained for a Reynolds number range of 814 to 4000 and for flight speeds from 15,000 to 26,000/fps. Predicted results for the electron concentration are compared with experimental data and good agreement is obtained.

De Saint-Venant Equations Experimentally Verified

Abstract: The unsteady spatially varied flow equations (De Saint-Venant equations) are being solved by implicit finite differences with explicit description at the boundaries. Imposition of improper boundary conditions which violate the physics of the problem resulted into either violation of continuity or numerical instability problems or meaningless results. The magnitude of the spatial increment used in this implicit solution scheme was critical on steep slopes (2.0%). The temporal distribution of flow and time to equilibrium was altered considerably depending on the specified value of Δ x . Hydrographs on milder slopes (0.5% and 1.0%) were affected to a progressively lesser extent as the channel slope was decreased. The time to equilibrium flow starting from a dry bed was nonlinear with respect to change of channel length, channel slope, and rate of lateral inflow. For a given channel length and slope, time to equilibrium approached a constant for high rates of later inflow.

Integral Method for Flow Between Corotating Disks

Abstract: An integral method is developed for the three-dimensional, nonboundary-layer flow which occurs for laminar, radially inward through-flow of an incompressible Newtonian fluid between parallel corotating disks. The method is a forward-stepping procedure which forces satisfaction of integrals of the governing differential equations, plus boundary conditions, plus the governing differential equations at every radius. The velocity components are represented by polynomials of order N; the method is extendable with extraordinary ease to any value of N. It is reported that, with N = 8, the results agree very closely with results earlier obtained by a conventional finite-difference method and which agree with experiment. It is pointed out that the method presented is extremely conservative of computational time and might be adapted to many other problems.

Local Estimation of the Global Discretization Error

Abstract: The lowest order term of the global discretization error of the numerical solution to a system of ordinary differential equations satisfies a well-known differential equation. It is observed that the integration of this differential equation and thus the estimation of the discretization error becomes almost trivial if it is an exact differential equation. It is shown that there exist both RK-methods and multistep methods, the error equation of which is exact.

Magnetic stars with an external non-linear force-free field

Abstract: The possible existence of strong magnetic fields in stars is discussed and a method of constructing highly distorted models of magnetic, rotating stars developed. For stars with both poloidal and toroidal fields at the surface a force-free outer boundary condition is necessary. Non-linear solutions of the force-free equations must be used. The force-free equations and the structure equations for a white dwarf are solved simultaneously by a finite difference method.

The compressible turbulent boundary layer on a blunt swept slab with and without leading- edge blowing

Abstract: Finite difference method for solving equations for compressible turbulent boundary layers on swept infinite cylinders

Difference methods for the solution of the time-dependent semiconductor flow equations

Abstract: The letter describes the solution of the instationary semiconductor transport equations by means of finite-difference methods. This new approach is based on the formal similarity with the equations of incompressible flow. Implicit schemes have proven useful for the computation of both transient and steady-state solutions. Comparisons with experimental results are extremely favourable for f.e.t.s.

Analysis of Space Truss as Equivalent Plate

Abstract: An alternative approach to the direct method analysis of a two-layer rectangular-mesh space truss is considered. A reduction in dimensionality of the problem is sought by deriving the differential equation for a plate equivalent in behavior to the space truss. The differential equation is found from the nodal equations of equilibrium of the truss in which the nodal deflections are expanded in Taylor series form. The direct solutions of a uniformly loaded square space truss with three alternative boundary conditions are compared with finite difference solutions for the equivalent plate under the same boundary conditions. The comparison of examples shows that the equivalent plate is satisfactory for estimating chord forces of a space truss except over concentrated supports. As the influence of boundary conditions is seen to be significant, care is required in interpreting deflections drived from an equivalent plate.

Difference schemes with best possible truncation error

Abstract: In this paper we present a variety of schemes for solving the initial value problem for a class of hyperbolic systems of partial differential equations. These schemes arise as solutions of constrained minimization problems for a quadratic form. The form is an expression for the local truncation error for a certain class of difference schemes.

Computer Analysis of Free Surface Well Flow

Abstract: A mathematical analog of immiscible multiphase well flow, considering three compressible fluids—two liquids and one gas—is solved with fully implicit finite differences. A Newton iteration scheme is utilized to solve the system of nonlinear difference equations. The method is applied to free surface gravity well flow, including the effect of partial penetration. The importance of capillarity, of air dissolved in water, of water compressibility, as well as the effect of the multiphase flow approach upon the shape of the “free surface,” are analyzed.

A Nonlinear Dynamic Optimization Technique for Controlling Xenon Induced Oscillations in Large Nuclear Reactors

Abstract: The problem of controlling the spatial power distribution in a reactor core under changing load conditions is formulated as a nonlinear optimization problem. The one-dimensional distributed core model is approximated by using finite differences to obtain a set of nonlinear ordinary differential equations. Linear programming is then used in an iterative scheme to determine the optimum control rod strategy.

PETROS 3: A Finite-Difference Method and Program for the Calculation of Large Elastic-Plastic Dynamically-Induced Deformations of Multilayer Variable-Thickness Shells.

Abstract: : The governing equations for the arbitrarily large-deformation elastic-plastic transient responses of variable-thickness, hard-bonded, multilayer, multimaterial, thin, Kirchhoff shells of any initial shape are formulated and solved by the finite difference technique. The material is assumed to be initially isotropic and to exhibit elastic, strain-hardening, strain-rate-sensitive, and temperature dependent behavior. the structure may be subjected to a variety of initial velocity distributions, transient mechanical loads, and/or transient thermal loads. These capabilities and features are contained in a computer program PETROS 3 which has been applied to a variety of example problems. Included herein is a FORTRAN IV listing and a description of PETROS 3 together with the data input and solution output for several example problems. (Author)

A bound for the optimum relaxation factor for the successive overrelaxation method

Abstract: An upper bound on the optimum relaxation factor for use with the successive overrelaxation method is derived for a class of linear systems arising from the numerical solution by finite difference methods of a boundary value problem involving the self-adjoint differential equation

On the Convergence of Approximating Solutions for Linear Distributed Parameter Optimal Control Problems

Abstract: Optimal control problems for distributed parameter systems, particularly systems described by partial differential equations, are often treated using mathematical function space techniques. As a result, the equations which define the optimal control are frequently obtained in abstract terms. Numerical solutions are obtained by approximating the abstract operations in a computationally feasible manner. In obtaining approximating solutions, the finite difference method is widely used.After an approximate optimal control has been found, the question arises whether a sequence of these approximating optimal controls converges to an optimal control of the original system.A condition of convergence of a sequence of approximating solutions of initial value problems by the finite difference method was given by H. F. Trotter. This theorem is concerned with a homogeneous system and does not give the condition of convergence of approximating optimal controls for a distributed parameter system.In this paper the condit...