Abstract: Wave equation migration is known to be simpler in principle when the horizontal coordinate or coordinates are replaced by their Fourier conjugates. Two practical migration schemes utilizing this concept are developed in this paper. One scheme extends the Claerbout finite difference method, greatly reducing dispersion problems usually associated with this method at higher dips and frequencies. The second scheme effects a Fourier transform in both space and time; by using the full scalar wave equation in the conjugate space, the method eliminates (up to the aliasing frequency) dispersion altogether. The second method in particular appears adaptable to three‐dimensional migration and migration before stack.

Abstract: Several competitive numerical integration techniques for nonlinear dynamic analysis of structures by the finite element method are examined and compared for a plane stress problem. Both material and geometric nonlinearities are included in the finite element formulation. Three explicit methods are investigated: (1)Central difference predictor; (2)two-cycle iteration with the trapezoidal rule; and (3)fourth-order Runge-Kutta method. A nodewise solution technique is generated at any state in the analysis. Three implicit methods also are studied: Newmark-Beta method, Houbolt method, and Park's stiffly-stable method. Among the three explicit methods, it is concluded that the central difference predictor is the best, whereas the performances of the other two methods are about equal. The three implicit approaches are nearly equal, but Park's stiffly-stable method is somewhat better than the Newmark-Beta method, and Houbolt's procedure must be rated third.

Abstract: A flexible finite difference method is described that gives approximate solutions of linear elliptic partial differential equations, Lu = G, subject to general linear boundary conditions. The method gives high-order accuracy. The values of the unknown approximation function U are determined at mesh points by solving a system of finite difference equations LhU = IhG. LhU is a linear combination of values of U at points of a standard stencil (9-point for two-dimensional problems, 27-point for three-dimensional) and IhG is a linear combination of values of the given function G at mesh points as well as at other points. A local calculation is carried out to determine the coefficients of the operators Lh and Ih so that the approximation is exact on a specific linear space of functions. Having the coefficients of each difference equation, one solves the resulting system by standard techniques to obtain U at all interior mesh points. Special cases generalize the well-known 0(h6) approximation of smooth solutions of the Poisson equation to 0(h6) approximation for the variable coefficient equation -div(p grad[u]) + Fu = G. The method can be applied to other than elliptic problems.

Abstract: A mathematical model describing the physical behavior of hot-water geothermal systems is presented. The model consists of a set of coupled partial differential equations for heat and mass transfer in porous media and an equation of state relating fluid density to temperature and pressure. The equations are solved numerically using an integrated finite difference method which can treat arbitrary nodal configurations in one, two, or three dimensions. The model is used to analyze cellular convection in permeable rock layers heated from below. Results for cases with constant fluid and rock properties are in good agreement with numerical and experimental results from other authors.

Abstract: A prediction method for three-dimensional reacting flows is described. It comprises a numerical solution technique for the time-averaged governing partial differential equations and physical modeling for the turbulence, combustion, and thermal radiation. The requirement of computational economy is strongly emphasized by the method, which employs an implicit numerical technique of the finite-difference kind to solve the governing equations iteratively. The turbulence model is of the "two-equation" variety, while the combustion model is based on a "fast kinetics" statistical approach. A newly developed flux model is employed for the thermal radiation. Comparisons of predictions and data are presented for industrial furnaces, but the method is applicable to all forms of combustion chambers including gas turbine cans. Nomenclature apyan — coefficient in finite-differ ence equation A b = area of cell boundary A i, A 2, A 3 = coefficients of the Taylor series for radiation intensity BlfB2,B3 = coefficients of the Taylor series for radiation intensity B* = defined by (B2 + B22 + B/) i/2

Abstract: Tables of coefficients for high order accurate, compact approximations to the first ten derivatives on and at the midpoints of uniform nets are presented. The exact rational weights are generated and tested by means of symbolic manipulation implemented through MACSYMA. These weights are required in the application of deferred corrections to new methods for solving higher order two point boundary value problems.

Abstract: A variable mesh finite difference scheme for a class of parabolic differential equations which exhibit shock-like structures is developed. It is shown that a properly chosen variable mesh will yield results comparable in accuracy to one using a much finer uniform mesh. Computable criteria and schemes for generating such variable meshes are given. A scheme is then applied to the Burgers' and modified Burgers’ equations with a small viscosity. Excellent agreement is obtained with known exact solutions.

Abstract: Publisher Summary This chapter describes the Hodie method and its performance for solving elliptic partial differential equations. It discusses a new flexible, high-accuracy finite difference approximation to the elliptic partial differential equation. A rectangular mesh is put and at each mesh point an estimate is obtained as the solution of a finite difference equation. For simplicity of exposition, the chapter presents an assumption that the mesh is uniform with mesh spacing and this assumption is not essential to the method though it improves its efficiency in some cases. It is found that after the auxiliary points are chosen, the coefficients of the difference equation are determined to make the approximation exact on a given linear space of functions. The chapter presents a general discussion of the method's computational properties and potential applicability along with a comparative performance evaluation using the ELLPACK system. The usual difference equation for a second order problem can be derived by making the scheme exact on quadratic polynomials and it is automatically exact on cubic polynomials. It is found that the order of the discretization error is the same as the order of the truncation error.

Abstract: The two-dimensional diffusion equation with nonperiodic driving term is solved by a finite-difference method. Moving parts, inductance-limited eddy currents, and strong saturation of the ferromagnetic materials are taken into account. The numerical procedure is based on time stepping and iterative solution by successive line overrelaxation. The method is used for the calculation of the dynamic behaviour of high-speed electromagnets.

Abstract: The boundary integral equation method (BIEM) is shown to be an efficient and accurate numerical technique for solving problems of Darcy flow in porous media. The BIEM is combined with conformal transformation to the complex potential plane to solve free surface problems without iteration and with relatively few nodal points. Since the BIEM reduces the effective dimensions of the problem by one, the computer time varies approximately as the inverse square of the point spacing, whereas in finite element or finite difference methods the time varies approximately as the inverse of the fourth power of the point spacing. Two examples of the BIEM are presented herein. They are flow through an underdrained dam and a seawater intrusion problem.

Abstract: A mathematical model is evaluated to describe cesium reaction and transport in a soil. The implicit finite difference method using quasilinearization technique is applied to solve the parabolic partial differential equation. This equation is a combination of the dispersion, convective transp

Abstract: The behavior of a hot, magnetized plasma brought into contact with a cold wall is studied numerically in one and two dimensions. A fully nonlinear, time‐dependent magnetohydrodynamic plasma model which includes thermal conduction, resistive diffusion, radiation, and ionization is used. The model is solved numerically with an Eulerian computer code which employs implicit finite difference methods. One‐dimensional calculations for cylindrical geometry examine the effect of the electrical properties of the wall on the plasma. Two‐dimensional calculations for cylindrical geometry show the formation of a wall‐induced instability which enhances thermal conduction losses from the plasma; the re‐emergence of short wavelengths, a new feature of unstable behavior, is evident in the calculations. Two‐dimensional calculations for toroidal geometry show that heat losses to a cold wall lead to double‐vortex convection flow of the plasma with no evidence of the formation of smaller scale convective cells.

Abstract: A moving boundary value problem is considered to investigate the one-dimensional propagation of fire fronts in forests. The problem is solved by the finite difference method with real physical parameters of forests. The theoretical and experimental results are compared to determine the sensitivity of the model for forecasting and control of forest fires.

Abstract: Highly efficient finite‐difference resistivity modeling algorithms which yield accurate results are put forward. The given medium is discretized and divided into rectangular blocks by using a very coarse system of vertical and horizontal grid lines, whose distance from the source(s) increases logarithmically. Expressions are derived to compute the longitudinal conductance and transverse resistance associated with each of these blocks for a parallel‐layer medium followed by a generalized treatment to accommodate arbitrarily shaped structures. Since the values of Dar Zarrouk parameters are derived from the exact resistivity distribution of the given medium, fine features such as a thin but anomalously resistive bed which ordinarily would be missed entirely in coarse discretization can be taken into account. Further reduction in the size of the model is achieved by making use of a symmetry wherever possible. In most cases the computation of the potential field which involves the inversion of a small sparse m...

Abstract: The numerical method presented treats the primitive-variable form of the Navier-Stokes equations. It is shown how to treat the generalised orthogonal coordinate form of the equations in order to retain the numerical stability of the linearised equations when these are approximated by finite differences. A property analogous to diagonal dominance in more simple systems is shown to exist for the complete set of difference approximations to the flow equations so that the matrix of the finite-difference equations has all of its eigen values in the left-hand half-plane. It follows that the linearized equations are unconditionally stable. An entirely new difference scheme for the continuity equation is derived and shown to be superior to the more commonly used “central-difference” approximations for the high-Reynolds-number flow considered. The total “package” is tested against experiment on a shear flow through a 90° rectangular bend. The experimental measurements are of total-pressure distributions, and these indicate the presence of a strong secondary flow. The computed results give a close agreement to the experimental results.

Abstract: Solutions of Boussinesq's equation for groundwater seepage from a ditcn with vertical sides extending in depth to a horizontal impermeable floor were obtained numerically by using the finite difference and finite element methods for the case when the seepage rate from the ditch into the soil is constant with time Both solutions agreed satisfactorily with experimental results from a Hele-Shaw analog It was found, however, that the execution time in the computer for the finite difference method was an order shorter than that for the finite element method, and thus the finite difference method is to be preferred The finite difference method was also used to obtain a numerical solution for the reverse situation when water seeps out of the soil into the ditch at a constant rate