Abstract: The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Abstract: Various finite difference and finite element methods for solving the one-dimensional convective-dispersive equation are investigated. A Fourier series analysis is performed to determine the dissipative and dispersive characteristics of these numerical methods. The analysis indicates that the commonly observed phenomenon of overshoot of a concentration pulse is due to the inability of the numerical schemes to propagate the small wavelengths which are important to the description of the front. Furthermore, the numerical smearing of a sharp front is due to dissipation of these small wavelengths. The finite element method was found to be superior to finite difference methods for solution of the convective-dispersive equation.

Abstract: A finite difference formulation is developed for computing the frequency domain electromagnetic fields due to a point source in the presence of two‐dimensional conductivity structures. Computing costs are minimized by reducing the full three‐dimensional problem to a series of two‐dimensional problems. This is accomplished by Fourier transforming the problem into the x-wavenumber (kx) domain; here the x-direction is parallel to the structural strike. In the kx domain, two coupled partial differential equations for H⁁x(kx,y,z) and E⁁x(kx,y,z) are obtained. These equations resemble those of two coupled transmission sheets. For a requisite number of kx values these equations are solved by the finite difference method on a rectangular grid on the y-z plane. Application of the inverse Fourier transform to the solutions thus obtained gives the electric and magnetic fields in the space domain. The formulation is general; complex two‐dimensional structures containing either magnetic or electric dipole sources can ...

Abstract: A new method of solving radiative transfer problems is described including a comparison of its speed with that of the doubling method, and a discussion of its accuracy and suitability for computations involving variable optical properties. The method uses a discretization in angle to produce a coupled set of first-order differential equations which are integrated between discrete depth points to produce a set of recursion relations for symmetric and anti-symmetric angular sums of the radiation field at alternate depth points. The formulation given here includes depth-dependent anisotropic scattering, absorption, and internal sources, and allows arbitrary combinations of specular and non-Lambertian diffuse reflection at either or both boundaries. Numerical tests of the method show that it can return accurate emergent intensities even for large optical depths. The method is also shown to conserve flux to machine accuracy in conservative atmospheres

Abstract: A general method of characteristics for solving the multigroup transport equations is developed. This is combined with an adaptive difference scheme, called the modified diamond scheme, and is then applied to the finite difference form of the equation. This formulation is obtained from the discrete ordinates equation, which in turn derives from the multigroup equation, both on the basis of consistency arguments. In this connection two forms of the multigroup equation are used, and the diffusion and other important limits also have a bearing on the final difference equation. The new approaches resolve a number of theoretical and practical difficulties with S/sub n/-type transport calculations, in particular in curved and multidimensional geometries. They lead to a firmer basis for discrete ordinates quadrature sets and to better control, mesh cell by mesh cell, over flux extrapolation, including methods to smooth out unwanted flux oscillations. The total effect is a more consistent treatment of the transport equation together with improved accuracy, fewer breakdowns, and more speed in the calculations, while keeping close to the physics of the problem and retaining the basic simplicity of the difference approach.

Abstract: A numerical technique was 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, making point as well as area discretization, of the subsurface. Potential distribution 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 were obtained with significantly less attendant computational costs than with the relaxation, finite-element or network solution techniques, for models of comparable dimensions.

Abstract: Analytical and empirical studies of a finite difference method for the solution of the transonic flow about an harmonically oscillating wing are presented along with a discussion of the development of a pilot program for three-dimensional flow. In addition, some two- and three-dimensional examples are presented.

Abstract: This study has been undertaken to evaluate an uncertainty in a finite difference method for two-dimensional neutron diffusion calculations and to provide a simple method to eliminate the uncertaint...

Abstract: Relaxation procedures for solving steady irrotational transonic flow have become well established. After their initial success of treating small disturbance theory [1] with a uniform Cartesian mesh, they have progressed to dealing with the full transonic potential equation using conformally mapped body coordinates along with a rotated iterative difference scheme to carry out the solution [2]. These methods now solve two- or three-dimensional flows with either a subsonic or supersonic free stream very effectively. Two features common to all these methods are: (1) the numerical differencing technique changes its character, depending on whether the flow is locally subsonic or supersonic, with supersonic differencing generally being first order; and (2) shock waves form automatically and are spread over a few mesh widths. All of these techniques share two essential assumptions: the flow must be steady and irrotational; and the latter restricts flows to M < 1. 3.

Abstract: Various techniques for smoothing numerical forecast integrations are compared for both finite difference and finite element Galerkin methods. The difference in the accuracy of finite element and finite difference methods is analysed to illustrate the removal of ‘aliasing’ by the Galerkin approach. This is then related to the appropriate choice of smoothing technique. The extra accuracy of finite elements allows the use of more selective methods. The use of filtering is shown to be suspect when significant energy is present in high wavenumbers and the Sadourny gravity-wave smoothing is also discussed.

Abstract: This paper presents an efficient two-point finite-difference method for solving the compressible laminar and turbulent boundary-layer equations for a given external velocity distribution (standard problem) as well as an efficient method for solving the same equations for a prescribed positive wall shear or displacement thickness (inverse problem). In the equations the Reynolds stress terms are modeled by using the eddy-diffusivity formulas developed by Cebeci and Smith. The accuracy of the method is investigated for both incompressible and compressible turbulent flows. A cf C f f g h H K L M P Pr Pr< Re u,v UT

Abstract: The utility of the finite-element Galerkin technique in advection-diffusion flow problems is examined by comparison with several finite-difference schemes in one dimension The calculations show that for relatively coarse grids, finite-element solutions are either comparable to or significantly better than those obtained from the finite-difference schemes considered For advection-dominated flows, the superiority of the finite-element technique is attributed to spatial coupling of time-derivative terms inherent in the Galerkin discretization This procedure, absent from conventional finite-difference schemes, leads to very accurate phase properties for the approximate solution even when coarse grids are used A two-dimensional analogue of the advection-diffusion problem further illustrates the advantages and accuracy of the finite-element method in conjunction with the use of isoparametric elements

Abstract: The local properties of non-linear differential-difference equations are investigated by considering the location of the roots of the eigen-equation derived from the lineraised approximation of the original model. A general linear system incorporating one time delay is considered and local stability results are obtained for cases in which the coefficient matrices satisfy certain assumptions. The results have applications to recent Biological and Economic models incorporating time lags.

Abstract: The finite element approximation of the convective diffusive transport equation can be expressed in terms of well-known finite difference notation. Examination of the relationship obtained reveals that in contrast to finite difference schemes, which approximate the differential equation at a point, the finite element method can be interpreted as an approximation of the integrated form of this equation.

Abstract: This paper describes a general method for solving the three-dimensional laminar and turbulent boundary-layer equations in orthogonal curvilinear coordinates. As in the earlier two papers, the Reynolds shear-stress terms are modeled by an eddy viscosity formulation developed by Cebeci and the governing equations are solved by a very efficient two-point finite-difference method. The accuracy of the method is investigated for turbulent flows.

Abstract: A computational system is presented for the calculation of long waves in harbors and coastal seas. The numerical method used is a three level implicite finite difference scheme, using the fractional steps technique. Tests on harbors under gale conditions with waves of one minute period, show that nonlinear instabilities develop, but they can be dissipated by iterating for equation coefficients.

Abstract: Integral equation methods are described. It is shown that the recent developments in these methods applied to two dimensional aerofoils give comparable results to the more advanced finite difference methods in shorter computing times.

Abstract: The finite‐difference boundary value method is applied to the calculation of Born–Oppenheimer vibrational energies and expectation values of R−2 for an excited state of H2. We estimate the accuracy attainable by this method, point out a systematic error in the previous calculations of Tobin and Hinze, and correct several unjustified statements in the literature. Finally we point out that there is a large uncertainty in the final results due to choice of interpolation scheme.

Abstract: A finite difference approximation to the equations of motion without the field acceleration terms, and the salt conservation equation with an isotropic diffusion coefficient, are used to model the circulation of a rectangular basin having the gross dimensional characteristics of a bar-built estuary. The recursion relations include the dynamic effects of horizontal salinity variations. Two cases are examined; in the first case the basin has a constant depth and in the second case the bottom slope varies linearly across the basin. The resulting patterns of circulation are qualitatively analyzed using a vorticity equation. The analysis indicates that the dynamic effects of horizontal salinity gradients can be significant in controlling the circulation in localized sections of the basin.