Abstract: A numerical technique is developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations by using point- as well as area-discretization of the subsurface. Potential distributions at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct explicit matrix inversion techniques. An empirical mixed boundary condition is used at the “infinitely distant” edges of the lower half-space. Accurate solutions using area-discretization method are obtained with significantly less attendant computational costs than with the relaxation, finite-element, or network solution techniques for models of comparable dimensions.

Abstract: We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y (2n)+f(x,y)=0,y (2j)(a)=A 2j ,y (2j)(b)=B 2j ,j=0(1)n−1,n≧2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.

Abstract: When finite-difference methods are used to solve the benchmark problem of natural convection in a square cavity, a very fine grid is required to obtain predictions that are accurate to 1-2%. The derivation of the finite-difference equations requires the introduction of many approximations; this study systematically evaluates these approximations to establish which are mainly responsible for the fine-grid requirement. The poorest approximations are then improved one by one, resulting in a scheme that yields highly accurate predictions using a relatively coarse grid. The method of evaluating the accuracy of the approximations, the improved approximations themselves, and the solution method used all contain novel features. Storage and computing time requirements for a new sparse matrix solver, which was used in the current study to simultaneously solve for stream function and vorticity, are presented.

Abstract: The existence of discrete shock profiles for difference schemes approximating a system of conservation laws is the major topic studied in this paper. The basic theorem established here applies to first-order accurate difference schemes; for weak shocks, this theorem provides necessary and sufficient conditions involving the truncation error of the linearized scheme which guarantee entropy satisfying or entropy violating discrete shock profiles. Several explicit difference schemes are used as examples illustrating the interplay between the entropy condition, monotonicity, and linearized stability. Entropy violating stationary shocks for second-order accurate Lax-Wendroff schemes approximating systems are also constructed. The only tools used in the proofs are local analysis and the center manifold theorem.

Abstract: A general class of finite difference methods for solving nonlinear two point boundary value problems is considered. These methods can also be interpreted as collocation methods. A convergence analysis on uniform meshes is given. This analysis is based upon a theorem of H. B. Kelley and a previous paper by the author. A specific example is given in detail and results of some numerical computations are included.

Abstract: Reactions of isolated ion pairs in solution have been modelled using the Debye–Smoluchowski equation for diffusion and conduction An activation step was incorporated using a partially reflecting boundary condition The method of matched expansions and the Abelian theorem of Laplace transforms was used to give an approximate solution of the Debye–Smoluchowski equation Numerical integrations based on the finite‐difference method confirmed these approximate analytic formulas

Abstract: New explicit finite difference methods are developed for approximating the discontinuous time dependent solutions of nonlinear hyperbolic conservation laws. The analysis is based on the method of lines approach of decoupling the space and time discretizations and analyzing each independently before combining them into a composite method. Particular attention is given analyzing to high order spatial differences, artificial dissipation and the accurate approximation of boundary conditions. Both a third order iterated leap-frog predictor-corrector and a second order iterated Runge--Kutta method are shown to have excellent stability and accuracy properties for the time integration. These methods are A-stable when iterated to convergence and have the special property of allowing for local improvements in the stability and accuracy of the computed solution. The paper is designed to aid a scientist or engineer construct a numerical method specially tailored to a specific problem. The analysis requires an elementary knowledge of the numerical solution of ordinary differential equations, finite difference theory and gas dynamics.

Abstract: An efficient numerical scheme to compute steep gravity waves in water of shallow uniform depth is described. The problem is formulated as a system of integrodifferential equations for the free surface. A numerical procedure based on Newton’s iterations is devised to solve these equations. Solutions of high accuracy for depth as small as 1/120 of a wavelength are presented. Numerical confirmation is obtained for the existence of maxima of the potential and kinetic energies of the waves as functions of the steepness.

Abstract: The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.

Abstract: In this paper, an evaluation of the methods of finite elements and finite differences, as applied to nonlinear magnetic field problems in electrical machines, is presented. The evaluation covers the aspects of effectiveness, numerical accuracy, modeling implementation considerations as well as computer storage and execution time requirements of the two methods. The evaluation includes static as well as sinusoidally time varying fields. The method of finite elements is found to be superior in improved accuracy, computer time and storage requirements, as well as programming implementation aspects. A crucial finding of this investigation is that results derived from the finite element analysis tend to converge asymptotically to corresponding experimental test data, as the discretization mesh fineness is increased. This is not the case for finite differences, where the results strongly indicate that there are lower bounds beyond which inherent numerical error cannot be decreased by an increase in the degree of fineness of the corresponding discretization mesh. Details of the analysis, on which these findings stand, are presented here.

Abstract: The effect of layered structure of rock formation on free convection in a geothermal reservoir is investigated in this work. The model examined is that of a rectangular reservoir comprised of three horizontal permeable layers with different permeabilities. The reservoir is considered to be bounded by impermeable surfaces on the sides and at the bottom. The upper boundary of the aquifer is permeable, which permits the recharge and discharge of water to and from the aquifer. A transient two-dimensional convective flow is developed when the impermeable boundaries are raised suddenly to high temperatures. The governing nonlinear partial differential equations with appropriate boundary and initial conditions are solved numerically by finite difference methods. Application of the direct method for solving Poisson’s equation for stream function made it possible to carry out the solution for a much longer time than possible with iterative techniques. Numerical results are obtained for various parameters and configurations of the geothermal reservoir. The influence of a less permeable middle layer on the flow and heat transfer characteristics in the aquifer is discussed. The computed vertical temperature profiles are similar in shape to the complex temperature profiles observed at the HGP-A well.

Abstract: The formulation of the problem of a rapidly propagating crack in a double cantilever beam specimen is re-examined using Reissner's variational principle. The governing equations are first solved to obtain the static compliance which is in good agreement with measured values. The equations of motion in conjunction with the energy balance criterion for a running crack are solved using a finite difference method. Predicted crack growth versus time, crack speed versus crack length and dynamic stress intensity factor versus crack length are all found to be in very good agreement with their measured counterparts for a polymer.

Abstract: A method is described for the calculation of transonic flow based on an integral equation formulation. The integral equation is descretized in a way analogous to that used in the well-established panel methods for incompressible flow. The resulting system of non-linear equations is solved by means of a quasi-Newton method. By the introduction of artificial viscosity and directional bias shock waves are captured in a way similar to that of current finite difference methods for transonic flow. Results of calculations, using the transonic small distrubance equation, are presented for a non-lifting 10 % parabolic arc aerofoil. Paper (No. 79-1459) presented at AIAA 4th Computational Fluid Dynamics Conference, Williamsburg, Virginia, USA, July 23-24, 1979.

Abstract: In this paper a short survey of the available numerical techniques for solving electrostatic problems is given. It is determined that techniques based on integral equations have several advantages over other available techniques which are used to solve Laplace's equation. A pair of integral equations is derived which can be used to solve Laplace's equation in regions containing conductor-dielectric and dielectric- dielectric boundaries. A computer program to solve these equations in the case for geometries with axial symmetry is described. Results based on this program are given.

Abstract: This paper presents a number of observations made and techniques developed in the process of constructing an accurate finite-difference method for the calculation of transonic flow through a cascade. These techniques can be applied to related problems such as flows over airfoils and inlets. A simple method is given to account for wind-tunnel sidewall effects in transonic flow calculations. A method is then presented which can be utilized to produce more rapid convergence for coupled viscous-inviscid transonic flow relaxation calculations. Finally, a stable set of fully second-order-accurate difference equations for the full-transonic potential flow equation is presented, along with some principles governing its construction and use. Calculations are presented to demonstrate the importance and effectiveness of these new methods.

Abstract: A method is described for the calculation of transonic flow based on an integral equation formulation. The integral equation is descretized in a way analogous to that used in the well-established panel methods for incompressible flow. The resulting system of non-linear equations is solved by means of a quasi-Newton method. By the introduction of artificial viscosity and directional bias shock waves are captured in a way similar to that of current finite difference methods for transonic flow. Results of calculations, using the transonic small distrubance equation, are presented for a non-lifting 10 % parabolic arc aerofoil. Paper (No. 79-1459) presented at AIAA 4th Computational Fluid Dynamics Conference, Williamsburg, Virginia, USA, July 23-24, 1979.

Abstract: The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure. This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.

Abstract: In this paper, we consider the convective instability of a heat conducting micropolar fluid layer between rigid boundaries. The eigenvalue problem governing the perturbations is solved, numerically, using finite difference method and Wilkinson's iteration technique. The heat induced by microrotation leads to the onset of instability not only due to adverse temperature gradient but also for positive temperature gradient. In the case of rigid boundaries, the critical Rayleigh number is seen to be higher than that of free boundaries. Here we notice that the cell pattern for positive temperature gradient are highly elongated when compared with that of negative temperature gradient.

Abstract: A method is presented for determining the vibration characteristics of a rotating blade whose cross-sectional dimensions or mechanical properties may vary sharply or even discontinuously along its length. The coupled flapwise bending, chordwise bending, and torsional vibration of the blade is analyzed by the method of the new quotient which is based on a variational statement proposed by Nemat-Nasser. In this approach, the nonuniform blade properties may be approximated by step (piecewise constant) functions. Two illustrative examples are given, and the results are compared with available experimental data and other numerical solutions. The comparison shows that the method of the new quotient yields very good results.

Abstract: : A new energy based finite difference analytical technique is introduced. The method incorporates certain energy concepts and the ability to use arbitrary, irregular meshes within the framework of the Finite Difference Method. This formulation reduces any governing partial differential equations to a set of difference equations containing partial derivatives up to and including the second order. Further certain strong simularities with the popular Finite Element Method are shown and the ability to solve problems with irregular boundaries is discussed. To demonstrate the Finite Difference Energy Method several plate bending problems are solved. (Author)

Abstract: Marine riser response calculations are usually based on finite element or finite difference methods. This paper outlines the modal analysis method as an alternate approach for calculating marine riser time-dependent stresses. An example problem shows that five natural vibration modes give acceptable convergence and engineering accuracy. Dynamic response calculations are, therefore, greatly simplified as only a limited number of eigenvalues and eigenfunctions are needed. In addition, the time-dependent part of the solution can be determined from elementary single-degree-of-freedom-type equations. 9 references, 5 figures, 2 tables.