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.

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.

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.

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)

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.

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

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.

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.

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.

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.

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

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.

Abstract: The rapid increase in electric and magnetic loadings of electrical machines demands improved methods of predicting the end zone field distribution. This paper presents a numerical method for the determination of end zone fields and for the calculation of end-leakage reactance of high-speed alternators. The method takes account of all boundaries, simplified in regard to their geometries and magnetic nature. As a first step, the partial differential equations of the electromagnetic field in the different regions of the end-winding region of homopolar alternators are developed, based on the concept of the magnetic vector potential. These are transformed into finite difference equations, which are retained in a general form allowing for the nonlinearities of iron and the effect of current carrying regions. The theory of the energy method to evaluate the end-leakage reactance is explained. The difficulties of a numerical solution in three dimensions are presented. Then a mathematical model is set up taking the discrete nature of the windings into account. This model makes possible adequate consideration of the effects of the surrounding boundaries and also the effects of the air gap, slots, and of the short-pitched armature winding. An orthogonal lattice system is then fitted to the model. The boundary conditions in differential and difference forms are developed in terms of the magnetic vector potential. An iterative procedure consisting of successive point relaxation of the vector potentials is described.

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.

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.

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

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...

Abstract: element uses as generalized displacements the tangential displacements and their first derivatives, plus the normal displacement and its first and second derivatives at each vertex, a total of 36 in all. The transverse displacement function for the element contains a complete quartic polynomial plus some higher degree terms and allows a cubic variation of normal slope along each edge. The tangential displacement functions are complete cubic polynomials, and it is shown that this formulation leads to a consistent asymptotic strain energy convergence rate of n~6, where n is the number of elements per side of a shell. Results show that this element is exceedingly accurate and far outperforms early lower order elements in predicting stresses as well as displacements. This element may easily be converted into an efficient and useful cylindrical shell element simply by substituting cylindrical shell theory for the shallow shell theory. The purpose of this Note is to present the necessary derivations for doing this and to illustrate the element's usefulness on an example application. The problem of a cylindrical shell with a circular cut-out is analyzed, and the stress concentration results are compared with those from both an approximate analytic analysis2 and a new finite difference variational approach.3 The following presentation is necessarily brief, but more details are available in Ref . 4.

Abstract: Higher‐order finite difference solutions of the Schrodinger equation for the helium atom have been obtained. For the S‐limit equation the ground‐ and first‐excited‐state energy values were found. There was a substantial reduction in the difference error in comparison with the treatment of Winter, Diestler, and McKoy (1968). For the complete (nonrelativistic) Schrodinger equation for He finite difference expressions of error O(h6) gave the ground‐state energy to five significant figures (− 2.9038 hartree). This problem seems to be near the limit of practical solution by the finite difference method.

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.

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)

Abstract: : A numerical discrete-element method of wave motion analysis is summarized and extended for problems involving infinite or semi-infinite solid media in plane and axi-symmetric conditions. Space discretization of a solid medium is accomplished through a lumped-parameter discrete-element model of the medium, whereas the time discretization is embedded within a general numerical integrator. This invariably leads to a system of finite difference equations; thus, the required mathematical conditions for numerical stability can be developed on the basis of available finite difference theory. Explicit stability conditions for plane and axi-symmetric problems are presented. Calculations of wave motions in an infinite or semi-infinite space can be confined to a finite region or interest if the region is terminated by suitable transmitting boundaries such that no significant reflections are generated at these artificial boundaries. Based on the concept of a step-wise transmission of D'Alembert forces, a general transmitting boundary was developed for the discrete-element method of analysis. The boundary was verified extensively through actual calculations of plane strain and axi-symmetric problems, including those with layered half-spaces, elastic-plastic systems, and a problem involving long calculation time. (Author)

Abstract: The conventional central finite difference equations for the plane stress extension of flat plates are derived as a localized Ritz process. A dual differential-variational discretization of this type enables common classification of the finite difference and finite element methods. Also, it provides alternative methods of establishing sufficiency conditions and relative rates of convergence for discrete systems derived from a localized Ritz process, and the existence of solution bounds for discrete systems derived using difference procedures.