Abstract: The method first replaces the differential equations of ground-water flow by finite-difference approximations that include unknown sink/source terms. The resulting system of algebraic linear equations has a rectangular matrix of coefficients. This system, together with linear inequalities relating sink/source terms, heads or both, and together with an objective function, forms a linear programming (LP) model. The method is applied to small-scale models of confined and unconfined saturated flow for steady-state and transient cases. The steady-state LP models are solved using available computer codes. For the transient confined model, the Crank-Nicolson scheme is used, and a single LP problem is solved covering all of the time steps. For the transient unconfined model, a predictor technique is used, and a LP problem is solved at each corrector step. The optimal solutions are consistent with the results of traditional analyses.

Abstract: The finite element method, using smooth splines as basis functions, applied to the model problem $u_t = cu_x $ with periodic data generates a differential-difference equation whose phase error is closely estimated and compared with the phase error of both explicit and high order implicit centered differencing. We also compute and compare the minimum work required to obtain a fixed error for several fully discrete schemes.

Abstract: An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program

Abstract: The orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors. There are two ways to view applications of the orthogonal collocation method. In one view it is a numerical method for which the convergence to the exact answer can be seen as the approximation is refined in successive calculations by using more collocation points, which are similar to grid points in a finite difference method. Another viewpoint considers only the first approximation, which can often be found analytically, and which gives valuable insight to the qualitative behavior of the solution. The answers, however, are of uncertain accuracy, so that the calculation must be refined to obtain useful numbers. However, with experience and appropriate caution, the first approximation is often sufficient and is easy to obtain. Thus it is very often useful in engineering work, where valid approximations are accepted. We present both viewpoints here: we use the first a...

Abstract: A finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented. The partial differential equation for the unsteady transonic flow was linearized by dividing the flow into separate steady and unsteady perturbation velocity potentials and by assuming small amplitudes of harmonic oscillation. The resulting linear differential equation is of mixed type, being elliptic or hyperbolic whereever the steady flow equation is elliptic or hyperbolic. Central differences were used for all derivatives except at supersonic points where backward differencing was used for the streamwise direction. Detailed formulas and procedures are described in sufficient detail for programming on high speed computers. To test the method, the problem of the oscillating flap on a NACA 64A006 airfoil was programmed. The numerical procedure was found to be stable and convergent even in regions of local supersonic flow with shocks.

Abstract: A numerical procedure that uses an explicit finite difference method to solve the wave equation is described. This technique results in a propagation algorithm that can accurately propagate an arbitrary electric field through a uniform medium or a medium that is nonuniform, transversely flowing, saturable, and contains index inhomogeneities. By using the propagation algorithm to propagate an arbitrary field back and forth between two resonator mirrors, the three-dimensional transverse mode and the output beam characteristics for a laser resonator can be determined. The advantage of the finite difference method is that unlike integral techniques the computational accuracy and efficiency improve as the resonator Fresnel number increases. The computational techniques are explained, and results for several specific empty cavity confocal unstable resonators are presented and compared to results obtained using an established calculation technique. The application of the finite difference method to inhomogeneous laser media is described, and computational results for an existing CO2 gas dynamic laser are presented and compared to measured data. The medium kinetics and shock wave models used in the calculations are described.

Abstract: Whereas considerable effort has been expended in generating approximations to the spatial derivatives encountered in porous media flow, the time derivative has received relatively little attention. In spite of the fact that sophisticated finite element formulations have been developed for the spatial derivatives, finite difference methods are generally applied to the time derivative. The Galerkin method of approximation permits a high-order approximation in time as well as in space. The resulting approximate equations have been successfully solved by using a prismatic element with triangular cross section. The time axis runs the length of the prism and is subdivided into elements that may be linear, quadratic, or cubic. Because this formulation requires in general the solution for several time levels simultaneously, there is a resulting increase in computer time required to solve the larger matrix. Numerical experiments indicate that the selection of an optimum numerical scheme is dependent not only on the particular problem considered but also on the sequence of time steps used.

Abstract: 0. Summary. Three subjects are considered in this paper. First, the notion of the resolving power of approximation methods, i.e. the number of intervals (or function values) per wavelength necessary to attain a preassigned error when approximating a given frequency, permits the evaluation of the relative efficiency of three types of spatial differencing used in setting up differential-difference equations for the 1-periodic parabolic problem ut = uxx. The three types of high-order spatial schemes considered are (explicit) centered differencing, the super-convergent smooth spline-Galerkin schemes discovered by Thomee, and the very high-order implicit schemes (Mehrstellenverfahren) which generalize Numerov's method. Seven time discretizations are introduced, namely Euler, backward-Euler, duFort-Frankel, 2nd order explicit, trapezoidal, Calahan-Zlamal, and 4th order Pade. The computational work necessary to solve each full discretization is minimized, for given error requirements, by balancing the number of intervals per wavelength against the number of time intervals per eth-life. The resulting data is used to compare the relative efficiency of these finite element and finite difference methods. Secondly, some corresponding results for the hyperbolic problem ut = ux are briefly reviewed. Finally, as Numerov-trapezoidal differencing turns out to be almost optimal for the heat equation, a tridiagonal implicit difference formula which extends Numerov differencing to the general 2nd order linear differential operator in one space dimension is presented. The technique used in deriving this scheme inspires certain difference analogs of some finite element schemes. It also leads to a curious modification of the diamond difference scheme for ut = ux. This “alternating kite” scheme has O(Δ2 + Δx3) truncation error with no more computational work than the diamond scheme itself.

Abstract: It is apparent that the finite difference approximation to the Laplacian obtained by Williamson, Hewlett & Tammemagi (1974) differs from that given by Jones R: Pascoe (1971). The difference is related to the manner in which first derivatives are approximated in the two cases. Jones & Pascoe assunzed a central diflerence form for first derivatives from the outset, whereas the method of Williamson et al. implies a linear cornhination of backward and forward differences for first derivatives. It is instructive to examine the two approaches in greater detail. Following Jones & Pascoe, Taylor's theorem may bc used to obtain the field values around the point '0' as: for the real part of the field values and similar equations for the imaginary part (a misplaced minus sign which appeared in the earlier paper has been correctly located above). Combining the equations for f1 and f3 gives:

Abstract: The problem of steady state free convection in an unconfined aquifer bounded by ocean on the sides with geothermal heating from below is investigated in this paper. The governing nonlinear partial differential equations with nonlinear boundary conditions are approximated by a set of linear subproblems on the basis of perturbation method. The equations for the zero- and first-order approximations are of the elliptic type that can be solved numerically by the finite difference method. Numerical results, accurate to the first-order approximations, are obtained for temperature, pressure, and stream function as well as for the shape of the water table. The influence of the location and the size of the heat source as well as various parameters on heat transfer and fluid flow characteristics in a rectangular geothermal aquifer is discussed.

Abstract: An incremental finite element method of the large elastic-plasic deformation of non-axisymmetric metal diaphragms which are edge-clamped and bulged by hydrostatic pressure has been formulated. The diaphragm is divided into a number of flat triangular elements and the behavior of a typical element is described in terms of the displacements of its nodes. The authors derive stiffness matrices, taking account of effects of shape change on the equilibrium equations, from a total differential of the equivalent nodal forces. In order to check the theory, numerical calculations were carried out for the bulging of a circular diaphragm under hydrostatic pressure. Theoretical results were in reasonable agreement with experiments for aluminium sheet and also with numerical solutions by means of a finite difference method.

Abstract: The Colorado method for the solution of the non-linear form of Laplace's, Poisson's, and the diffusion partial differential equations is explained. Various boundary conditions can be satisfied. The transformation of the partial differential equation into a large set of finite difference equations is given. The discretization is based on a grid system consisting of two sets of orthogonal grid lines. The resulting meshes are nonuniform. Successive line overrelaxation method is used for the solution of the nonlinear equations in two steps. For the improvement of convergence, two methods of acceleration of convergence are described.

Abstract: In this thesis finite-difference approximations to the three boundary value problems for Poisson’s equation are given, with discretization errors of O(H^3) for the mixed boundary value problem, O(H^3 |ln(h)| for the Neumann problem and O(H^4)for the Dirichlet problem respectively . First an operator is constructed, which approximates the nor-ma1 derivative with a truncation errors of O(H^3). The derivation by which this result is obtained contains an improvement upon the one used for a similar operator by Bramble and Hubbard; it became thus possible to make their results valid under more general conditions. For points in a square net where the nine-point approximation to the Laplace operator cannot be used, because of their position near the boundary of the region under consideration, several approximations to the Laplace operator are given, dependent on the particular point configuration, which are all O(H^2). The above-mentioned operators are then used to formulate finite-difference problems with solutions approximating the corresponding continuous problems with the desired accuracy. These error bounds are an improvement of O(H) upon the most accurate approximations known for the Neumann and Robin problems, while the approximation given for the Dirichlet problem has as an advantage over a similar approximation given by Bramble and Hubbard, which is of the same order, that its res-ulting coefficient matrix is of positive type. All our approximations share this important property. For all three problems simple numerical examples are given. It is also pointed out that certain results of Bramble and Hubbard, with respect to the error in the numerical partial derivatives of the solutions, are valid for our approximations.

Abstract: Finite difference method for the numerical solution of electromagnetic waveguide discontinuity problems is presented. The method of boundary relaxation is applied, using finite difference techniques in the nonuniform section of the waveguide and using a modal representation of the field in the uniform sections of the waveguide. To illustrate the process some two-dimensional diffraction problems in an electromagnetic waveguide with rectangular cross section are solved.

Abstract: The compressible laminar and turbulent boundary-layer equations representing the fiow of cesium seeded argon plasma over successive segments of the cathode wall of an MHD channel were investigated and solved for both finite and intinite rates of reaction. The equations solved include the electron energy and the electron continuity equations as well as the usual global boundary-layer equations. The effects of finite rates of reaction and ambipolar diffusion are considered, and the inner boundary conditions for both the electron energy and the electron continuity equations are obtained by investigating a sheath analysis. The governing equations are solved using a finite-difference scheme without introducing a local similarity assumption. The finite rate results are considerably different from the infinite rate ones. The main reason is the ambipolar diffusion which depletes the electrons near the wall. No special problems have been found in the solution of the turbulent flow equations. This is due to the high mobility of electrons. (auth)

Abstract: THE application of finite-difference schemes to the solution of axisymmetric problems for the kinetic equations of the theory of rarefied gases are discussed. A numerical procedure without storage of the distribution function in the computer memory is proposed for an equation with a model collision operator. Two axisymmetric problems are solved as examples: the flow of a uniform stream of a rarefied gas past a circular disk and the outflow of a gas into a vacuum through a circular orifice in a flat wall.

Abstract: The momentum integral method of Klineberg is extended to allow the study of two-dimension al laminar viscous-in viscid interactions over a continuous range of wall cooling ratio. By redefining the Klineberg profile quantities, it is shown that the functions appearing in the governing differential equations become, to a good approximation, independent of wall cooling ratio. A considerable simplification of the method is thus affected and flows under nonisothermal wall conditions can be studied. Experimental comparisons are made for a wide class of body shapes which show that the method provides a good description of the major features of viscousinviscid interactions in both supersonic and hypersonic flow and is superior to other existing methods. In the light of these comparisons, some further improvements to the method are considered.

Abstract: Integrations of the shallow water equations on the sphere using a second order finite difference method and a spectral method are compared. By increasing the resolution, a good estimate of the exact solution may be made, thus allowing an estimate of the accuracy of each integration. The particular initial conditions used are a Rossby wave of zonal wavenumber 4 which moves with little change in shape or amplitude and a Rossby wave of zonal wavenumber 8 which undergoes large changes within 5 days. All the models perform reasonably for the wavenumber 4 integration. A 5° times; 3° grid for the finite difference simulation is insufficient to resolve the breakdown in wavenumber 8 despite there being 9 points per zonal wavelength. A spectral model with a truncation at wavenumber 16 uses less storage and takes the same computing time as the grid-point model. However, it is able to predict quite accurately the breakdown.

Abstract: A new implicit finite-difference scheme for viscous flows is presented. The scheme is based on Simpson's rule and two-point Hermite interpolation, is uniformly accurate to fourth order in time and space, and is unconditionally stable according to a Fourier stability analysis. Numerical solutions of Burger's equation are presented to illustrate the order and accuracy of the scheme. 1. Introduction. The invention of the electronic digital computer stimulated the intensive development of numerical methods for the solution of fluid flow problems. The majority of the methods first developed were explicit and of low order because their utility on the small slow early computers derived from their simplicity. This trend has continued to the present, despite the accuracy of high order schemes and the unconditional stability of implicit schemes. The new parallel and pipeline com- puters, however, have spurred interest in complex difference methods by making feasible the large calculations required by such schemes. Recently Rusanov (5) and Burstein and Mirin (2) have studied explicit third order schemes for hyperbolic systems, while Zwas and Abarbanel (6) have developed an explicit fourth order method for special hyperbolic systems. We present in this paper a new unconditionally stable implicit scheme for viscous flows which is uniformly accurate to fourth order in time and space. We first describe the method and examine the local truncation error. We then present a linearized stability analysis of the scheme. Finally, we present several examples to illustrate the accuracy and stability of the method.