Abstract: A variable order variable step finite difference algorithm for approximately solving m-dimensional systems of the form y'' = f(t,y), t $\in$ [a,b] subject to the nonlinear boundary conditions g(y(a),y(b)) = 0 is presented. A program, PASVAR, implementing these ideas has been written and the results on several test runs are presented together with comparisons with other methods. The main features of the new procedure are: a) Its ability to produce very precise global error estimates, which in turn allow a very fine control between desired tolerance and actual output precision. b) Non-uniform meshes allow an economical and accurate treatment of boundary layers and other sharp changes in the solutions. c) The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.

Abstract: This paper investigates the phenomenon of ‘noise’ which is common in most time-dependent problems. The emphasis is on the achievable accuracy that is obtained with various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level. A series of experiments are made where the space domain is discretized using the finite element method and the variation with time is approximated by several finite difference methods. The conclusion is reached that the Crank–Nicolson scheme with a simple averaging process is superior to the other methods investigated.

Abstract: The book presents the theory and applications of the analytical techniques used in finding stresses in highway and other bridge decks. Current trends in bridge design and construction are discussed and are followed by the various analytical methods. The plate method is dealt with, initially by the basic derivation and solution of the plate equation. A chapter is devoted to the determination of the equivalent plate rigidities of various representative types of bridge deck. An approximate method of determination of bending moments for initial design is described. Various special applications of orthotropic plate theory are covered and the finite difference method for plates is described, including a summary of the dynamic relaxation method. The last four chapters deal with the stiffness method and its application: grillage and space frame analysis, the folded plate method, the finite element method, and the finite strip method. The book is intended for use by bridge designers and students with a particular interest in bridge engineering. /TRRL/

Abstract: : Problems relating to the computation of viscous compressible flows based on numerical solutions of the Navier-Stokes equations are reviewed. A general introduction to the Navier-Stokes equations and a discussion of their interest in aerodynamic problems are first presented. Then the following aspects of numerical methods are considered: limitation of the computational domain and boundary conditions on the outer boundary; various approaches in finite difference methods and description of some representative schemes; treatment of boundary conditions at a solid wall; treatment of shock waves, and general considerations on accuracy and computing times. Finally reported computations of two-dimensional or three-dimensional flows are presented in table form with summary indications on the problems treated and the methods used.

Abstract: A high accuracy difference method (hermitian method) for the solution of evolution equations of parabolic type is presented. Its most original feature is to use several unknowns (the value of the solution and its spatial derivatives) at every nodal point of the computational grid. It is shown that this method has better computational performance than classical schemes on non-uniform and coarse meshes.

Abstract: A numerical model based on the equations of unsteady flow in open channels is used to compute unsteady flows in rivers and reservoirs. The cross sections of the waterways range from uniform to highly irregular, the type of flow ranges from slowly varied to abrupt changes in discharge, and nearly all combinations of boundary conditions are encountered. The model uses an implicit finite difference method. The versatility, accuracy, stability, and efficiency of the method is demonstrated by field measurements.

Abstract: A mathematical model describing the physical behavior of hot-water geothermal systems is presented. The model consists of a set of coupled partial differential equations for heat and mass transfer in porous media and an equation of state relating fluid density to temperature and pressure. The equations are solved numerically using an integrated finite difference method which can treat arbitrary nodal configurations in one, two, or three dimensions. The model is used to analyze cellular convection in permeable rock layers heated from below. Results for cases with constant fluid and rock properties are in good agreement with numerical and experimental results from other authors.

Abstract: A certain fourth-order differential equation is solved numerically by the method of finite differences. Conditions on the original differential equation are given which are sufficient to quarantee that the matrix thus produced is monotone so that a straightforward error analysis is possible. This error analysis is given in detail. Examples are given which demonstrate the validity of this error analysis.

Abstract: A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference methods and computational algorithms are described. Results are compared with other calculation methods and experiments.

Abstract: This paper considers the Dirichlet problem for the two-dimensional biharmonic equation in a bounded region consisting of a finite sum of rectangles. The biharmonic equation is first split into two Poisson equations and two classes of finite difference schemes are defined for obtaining the numerical solution. These classes correspond to the type of difference approximation defined for the missing boundary condition. Discretization error for the difference schemes in these two classes is shown to be of order $h^{{3 / 2}} $ and $h^2 $, respectively, as the mesh size $h \to 0$. The effect on discretization error of the different approximations within a class is also examined.

Abstract: The theoretical study of time-varying two-phase flow problems in several space dimensions introduces such a complicated set of coupled nonlinear partial differential equations that numerical solution procedures for a high-speed computer are required in almost all but the simplest examples Efficient attainment of realistic solutions for practical problems requires a finite difference formulation that is simultaneously implicit in the treatment of mass convection, equations-of-state, and the momentum coupling between phases We describe such a method, discuss the equations on which it is based, and illustrate its properties by means of examples In particular, we emphasize the capability for calculating physical instabilities and other time-varying dynamics, at the same time avoiding numerical instability The computer code is applicable to problems in reactor safety analysis, the dynamics of fluidized dust beds, raindrops or aerosol transport, and a variety of similar circumstances, including the effects of phase transitions and the release of latent heat or chemical energy

Abstract: In order to complete the existing theory of beams subjected to follower forces, an exact solution for the case of triangularly distributed loads is presented. The results obtained may be compared with those reported in the literature and which were based on approximate calculations by means of the Method of Galerkin and the Finite Difference Method. The conclusion is that there is good agreement between all these results, thus confirming the dependability of the approximate methods.

Abstract: The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and element methods. The finite difference method requires a large number of nodes and is slow to converge as permeability is variable. The finite element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient conditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.

Abstract: Conclusion Compared to the rigorous procedures the solution to the previously stated problem, given by Eqs (4) and (5) is approximate, but avoids the cumbersome calculations involved in the former In this connection, the stochastic analysis of a single degree of freedom system subjected to random wind and seismic excitations to study the response characteristics was undertaken by the authors The exciting force was assumed to be nonstationary in character, and was represented by the product of a deterministic shape function and a stationary random process characterized by its power spectral density The choice of deterministic function and power spectral density was based on certain characteristics observed in a large number of past records of excitation process The application of Eqs (4) and (5) to study the peak response characteristics of the system revealed that the probability estimates for various appropriate values of X are about 05% below those obtained by an exact procedure

Abstract: Equation systems describing one-dimensional, transient, two-phase flow with separate continuity, momentum, and energy equations for each phase are classified by use of the method of characteristics. Little attempt is made to justify the physics of these equations. Many of the equation systems possess complex-valued characteristics and hence, according to well-known mathematical theorems, are not well-posed as initial-value problems (IVPs). Real-valued characteristics are necessary but not sufficient to insure well-posedness. In the absence of lower order source or sink terms (potential type flows), which can affect the well-posedness of IVPs, the complex characteristics associated with these two-phase flow equations imply unbounded exponential growth for disturbances of all wavelengths. Analytical and numerical examples show that the ill-posedness of IVPs for the two-phase flow partial differential equations which possess complex characteristics produce unstable numerical schemes. These unstable numerical schemes can produce apparently stable and even accurate results if the growth rate resulting from the complex characteristics remains small throughout the time span of the numerical experiment or if sufficient numerical damping is present for the increment size used. Other examples show that clearly nonphysical numerical instabilities resulting from the complex characteristics can be produced. These latter types of numerical instabilities are shown to be removed by the addition of physically motivated differential terms which eliminate the complex characteristics. (auth)

Abstract: Response matrix equations in two-dimensional geometry have been derived in the form of a set of coupled integral equations of the Fredholm type that have been solved by the moments method. The set of Legendre polynomials defined at the material interfaces has been chosen as the base for representing the partial interface currents and the response matrices.The method has been applied to the solution of the one-group diffusion equation and its convergence has been investigated in a series of numerical experiments, involving expansions of up to order 14. It turned out that the P1 approximation should be adequate for the majority of the two-dimensional problems occurring in power reactor design. Furthermore, the response method has a substantially higher computer efficiency than the finite difference method, both in processor time and in storage locations. As a by-product, the nature of the singularities around edges and corners of material interfaces has been analyzed by numerical experimentation.

Abstract: In the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non-dimensional form and also the results have been obtained in non-dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.

Abstract: Numerical and experimental studies of non-Darcy flow in porous media are examined. Laboratory experiments using a screened gravel include radial flow to a simulated well using a sector and two-dimensional flow through a bank with vertical sides in a flume. Permeameter tests on the gravel were used to estimate coefficients in the nonlinear relation between head loss and velocity. Results are presented of analyses performed using a finite difference solution of the appropriate partial differential equation boundary value problem. The flow nets and discharges obtained are compared with the experimental results and the corresponding solutions for Darcy flow.

Abstract: The application of spatial discretization (discrete ordinate method) to a class of integro-differential equations is discussed. It is shown that consistency in the approximation of the operators implies convergence of the approximate solution to the true solution.

Abstract: The use of Lagrangian finite element methods for solving a Poisson problem produces systems of linear equations, the global stiffness equations. The components of the vectors which are the solutions of these systems are approximations to the exact solution of the problem at nodal points in the region of definition. There is thus associated with each nodal point an equation which can be thought of as a difference equation. Difference equations resulting from the use of polynomial trial functions of various orders on regular meshes of square and isosceles right triangular elements are derived. The rival merits of this technique of setting up a standard difference equation, as distinct from the more usual practice with finite elements of the repeated use of local stiffness matrices, are considered.