Abstract: A numerical procedure is described that will simplify the analysis of the EMP response of structures with dielectric or poorly conducting segments.

Abstract: A numerical simulation of a laboratory experiment involving coupled heat and mass transfer in a horizontal porous medium column with one end subjected to a temperature below 0°C has been carried out. The model is essentially that of Harlan (1973) and is solved numerically by the finite difference method using the Crank-Nicholson scheme. The solution yields temperature, liquid water content, and ice content profiles along the column as a function of time. Comparison of the experimental results and the simulation analysis results shows that Harlan's model, with some modification in the hydraulic conductivity of the frozen medium, can be used successfully to simulate numerically the coupled heat and mass transfer processes when ice lensing does not occur.

Abstract: A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

Abstract: A nonlinear analysis is carried out for the motion of the inviscid, incompressible fluid in a two-dimensional, rigid, open container which is subjected to forced sinusoidal pitching oscillation. Firstly, the problem is defined as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions. Next, the problem is formulated in the form of a pseudo-variational principle, which provides a basis for our discretization. The finite element method and finite difference method are used spacewise and timewise, respectively. Due to the strong nonlinearity of the problem, an incremental method is used for the numerical analysis. Numerical results obtained by the present method are compared with solutions of the linear theory and experimental data. The difference between linear and nonlinear analysis has been clearly indicated.

Abstract: The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.

Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.

Abstract: A finite difference method has been developed to study the inviscid stability of swirling flows to small non-axisymmetric disturbances. We apply the method to Batchelor's trailing line vortex solution [3]. The method appears to be more efficient, and simpler to implement for this class of problem, than previously reported methods.

Abstract: The combined approach of linearisation and finite difference method is used to solve the improved Boussinesq equation. A three-level iterative scheme having second order accuracy and constant coefficients matrix is devised and used in discussing the dynamics of waves having various initial wave packets. The results are in good agreement with the available results.

Abstract: This note will center on the issue of numerical precision as affected by the use of different routing schemes. The investigation is prompted by the observation that in an application of the Kalinin-Miljukov method, accuracy improved when a more refined difference scheme was used in place of the conventional one.

Abstract: A stacked seismic section represents a wave-field recorded at regularly spaced points on the surface. The seismic migration process transforms this recorded data into a reflectivity display. In recent years, Jon F. Claerbout and his co-workers developed migration techniques based on the numerical approximation of the wave equation by finite difference methods. This paper describes an alternative method, termed ASD (for Accurate Space Derivative), and its application to the wave equation migration problem. In this approach to the numerical solution of partial differential equations, partial derivatives are computed by finite Fourier transform methods. This migration method can accommodate media with vertical as well as horizontal velocity variations.

Abstract: Methods of order two and four are developed for the continuous approximation of the solution of a two-point boundary value problem associated with a certain fourth order linear differential equation via quintic and sextic spline functions. Numerical results are summarized for some typical numerical examples and compared with some known finite difference methods of the same order.

Abstract: Convection–diffusion equations are difficult to solve when the convection term dominates because most solution methods give solutions which oscillate in space. Previous criteria based on the one-dimensional convection–diffusion equation have shown that finite difference and Galerkin (linear or quadratic basis functions) will not give oscillatory solutions provided the Peclet number times the mesh size (Pe Δx) is below a critical value. These criteria are based on the solution at the nodes, and ensure that the nodal values are monotone. Similar criteria are developed here for other methods: quadratic Galerkin with upwind weighting, cubic Galerkin, orthogonal collocation on finite elements with quadratic, cubic or quartic polynomials using Lagrangian interpolation, cubic or quartic polynominals using Hermite interpolation, and the method of moments. The nodal values do not oscillate for collocation or moments methods with Hermite cubic polynomials regardless of the value of Pe Δx. A new criterion is developed for all methods based on the monotonicity of the solutions throughout the domain. This criterion is more restrictive than one based only on the nodal values. All methods that are second order (Δx2) or better in truncation error give oscillatory solutions (based on the entire domain) unless Pe Δx is below a critical value. This value ranges from 2 for finite difference methods to 4·6 for Hermite, quartic, collocation methods.

Abstract: Finite-difference and finite-element methods are widely used to solve problems described by sets of partial differential equations. However, the connections between the approximations made in the discretization process and the final solution errors, for a given grid, are often not clearly understood, especially when convection is a dominant factor. These connections are clarified in the present paper and, from the insight gained, a new method of formulating the discrete equations suggests itself. The results of applying two different schemes, that are both based on this method, to an example problem are presented. These are compared with results obtained using two finite-difference schemes. The new approach appears to hold considerable promise for future development.

Abstract: SHAFT79 (Simultaneous Heat And Fluid Transport) is an integrated finite difference program for computing two-phase non-isothermal flow in porous media. The principal application for which SHAFT79 is designed is in geothermal reservoir simulation. SHAFT79 solves the same equations as an earlier version, called SHAFT78, but uses much more efficient mathematical and numerical methods. The present SHAFT79 user's manual gives a brief account of equations and numerical methods and then describes in detail how to set up input decks for running the program. The application of SHAFT79 is illustrated by means of a few sample problems. (MHR)

Abstract: This paper describes a new approach to the solution of two-dimensional boundary value problems which eliminates the disadvantages and combines the advantages of both conformal transformations and numerical methods. The conformal transformations are used to remove potential gradient singularities, and numerical (e.g., finite difference) methods may then be applied to the resulting almost-regular field problems. Boundary value problems previously regarded as very difficult become tractable, and considerable savings in computer time and storage requirements are achieved. The method is applied to the calculation of the even and odd mode capacitances of cylindrical rods between plane parallel ground planes. Excellent agreement with resuIts obtained previously is demonstrated.

Abstract: A two-dimensional mathematical model was developed to calculate the temperature distribution during welding of thin tantalum sheets. The solution of the unsteady heat flow equation was obtained employing a computer program based on the finite difference method. The model does not require any assumptions regarding the shape of the welding pool. Arc parameters, radiative and convective heat losses and the dependence of the relevant physical properties on temperature are taken into account. The shape of the welding pool and isotherms are calculated and compared to experimental results.

Abstract: : A numerical model called VAHM for computing the vertically averaged hydrodynamics of a water body, including salinity effects, has been developed. The model employs the concept of boundary-fitted coordinates to allow for an accurate representation of the boundary of the region being modeled while retaining the simplicity of the finite difference method of solution. Although a general curvilinear coordinate system covers the physical domain, all computations to solve the governing fluid dynamic equations, as well as the computation of the boundary-fitted coordinate system, are performed in a transformed rectangular plane with square grid spacing. A combination implicit- explicit finite difference scheme has been employed to numerically solve the governing equations. With such a scheme, the water surface elevation is computed implicitly using the Accelerated Gauss-Seidel solution technique, whereas the velocity and salinity fields are solved in an explicit manner. The major advantage of such a scheme is that the speed of a surface gravity wave is removed from the stability criteria while many desirable features of an explicit scheme are retained. Although additional work on VAHM remains to be completed before the model can be considered fully operational, results from the three applications of river and ocean boundaries demonstrate that the basic model behaves properly. (Author)

Abstract: A new numerical method called the Finite Analytical Method for solving partial differential equations is introduced. The basic idea of the finite analytic method is the incorporation of the local analytic solution in obtaining the numerical solution of the problem. The finite analytical method first divides the total region of the problem into small subregions in which local analytic solutions are obtained. Then an algebraic equation is derived from the local analytic solution for each subregion relating an interior nodal value at a point P in the subregion to its neighboring nodal values. The assembly of all the local analytic solutions thus provides the finite-analytic numerical solution of the problem. In this paper the finite analytic method is illustrated in solving steady and unsteady heat transfer problems.

Abstract: A set of two-dimensional turbulent boundary-layer equations that describe the three-dimensional fluid flow in a magnetohydrodynamic channel is outlined. The equations are solved by an implicit finite-difference scheme: a method of calculating the pressure gradient is used that avoids the need to iterate over the coefficients of the difference equations. The numerical method is tested by applying it to low- and high-speed flows over a flat plate. It is shown that the governing equations fully satisfy the three-dimensional conservation laws. The importance of accounting for three-dimensional effects is assessed, and an advanced technique for bringing out better the interaction between the boundary layers on sidewalls and electrode walls is discussed.