Abstract: Conventional finite-difference operators for numerical differentiation become progressively inaccurate at higher frequencies and therefore require very fine computational grids. This problem is avoided when the derivatives are computed by multiplication in the Fourier domain. However, because matrix transpositions are involved, efficient application of this method is restricted to computational environments where the complete data volume required by each computational step can be kept in random access memory. To circumvent these problems a generalized numerical dispersion analysis for wave equation computations is developed. Operators for spatial differentiation can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band. For specified levels of maximum relative error in group velocity ranging from 0.03% to 3%, differentiators have been designed that have the largest possible bandwidth for a given operator length. The relation between operator length and the required number of grid points per shortest wavelength, for a required accuracy, provides a useful starting point for the design of cost-effective numerical schemes. To illustrate this, different alternatives for numerical simulation of the time evolution of acoustic waves in three-dimensional inhomogeneous media are investigated. It is demonstrated that algorithms can be implemented that require fewer arithmetic and I/O operations by orders of magnitude compared to conventional second-order finite-difference schemes to yield results with a specified minimum accuracy.

Abstract: Conjugate heat transfer for two-dimensional, developing flow over an array of rectangular blocks, representing finite heat sources on parallel plates, is considered. Incompressible flow over multiple blocks is modeled using the fully elliptic form of the Navier-Stokes equations. A control-volume-based finite difference procedure with appropriate averaging for the diffusion coefficients is used to solve the coupling between the solid and fluid regions. The heat transfer characteristics resulting from recirculating zones around the blocks are presented. The analysis is extended to study the optimum spacing between heat sources for a fixed heat input and a desired maximum temperature at the heat source.

Abstract: Although there are acceptable methods for calculating whole body electromagnetic absorption, no completely acceptable method for calculating the local specific absorption rate (SAR) at points within the body has been developed. Frequency domain methods, such as the method of moments (MoM) have achieved some success; however, MoM requires computer storage on the order of (3N) 2 and computation time on the order of (3N) 3 where N is the number of cells. The finite-difference time-domain (FDTD) method has been employed extensively in calculating the scattering of metallic objects, and recently is seeing some use in calculating the interaction of EM fields with complex, lossy dielectric bodies. Since the FDTD method has storage and time requirements proportional to N, it presents an attractive alternative to calculating SAR distribution in large bodies. This paper describes the FDTD method and evaluates it by comparing its results to analytic solutions in two and three dimensions. The utility of the FDTD method is demonstrated by a 3D scan of the human torso. The results obtained demonstrate that the FDTD method is capable of calculating internal SAR distribution with acceptable accuracy. With the availability of supercomputers, such as the CRAY II, the calculation of SAR distribution in a man model of 50 000 cells (1.27 cm per cell) appears to be feasible.

Abstract: A linear model for neutral surface-layer flow over complex terrain is presented. The spectral approach in the two horizontal coordinates and the finite-difference method in the vertical combines the simplicity and computational efficiency of linear methods with flexibility for closure schemes of finite-difference methods. This model makes it possible to make high-resolution computations for an arbitrary distribution of surface roughness and topography. Mixing-length closure as well as E − e closure are applied to two-dimensional flow above sinusoidal variations in surface roughness, the step-in-roughness problem, and to two-dimensional flow over simple sinusoidal topography. The main difference between the two closure schemes is found in the shear-stress results. E − e has a more realistic description of the memory effects in length and velocity scales when the surface conditions change. Comparison between three-dimensional model calculations and field data from Askervein hill shows that in the outer layer, the advection effects in the shear stress itself are also important. In this layer, an extra equation for the shear stress is needed.

Abstract: The embedding of microwave devices is treated by applying the finite-difference method to three-dimensional shielded structures. A program package was developed to evaluate electromagnetic fields inside arbitrary transmission-line connecting structures and to compute the scattering matrix. The air bridge, the transition through a wall, and the bond wire are examined as interconnecting structures. Detailed results are given and discussed regarding the fundamental behavior of embedding.

Abstract: In this paper we use a modified equation analysis to develop formally fourth order accurate finite difference and pseudospectral methods for the one-dimensional wave equation. The difference scheme is constructed by performing a modified equation analysis of a centered, second-order conservative scheme to determine its dominant error term. Subtracting a centered discretization of this term from the scheme cancels the second order truncation errors. This technique yields a formally fourth order accurate explicit difference scheme that employs only three time levels. Similarly, the modified equation technique can be used to achieve fourth order time accuracy for the pseudospectral method with no increase in storage. The difference and pseudospectral schemes are fourth order convergent for constant coefficients even when a spatially singular forcing term is used for a source. Numerical results are given comparing the accuracy and efficiency of these methods for some model problems. Finally, we present a gene...

Abstract: In this paper we show that the total variation diminishing (TVD) finite difference scheme which was analysed by Sweby [8] can be interpreted as a Lax—Wendroff scheme plus an upwind weighted artificial dissipation term. We then show that if we choose a particular flux limiter and remove the requirement for upwind weighting, we obtain an artificial dissipation term which is based on the theory of TVD schemes, which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Finally, we conduct numerical experiments to examine the performance of this new method.

Abstract: An analytical formulation is presented for the computation of scattering and transmission by general anisotropic stratified material. This method employs a first-order state-vector differential equation representation of Maxwell's equations whose solution is given in terms of a 4 \times 4 transition matrix relating the tangential field components at the input and output planes of the anisotropic region. The complete diffraction problem is solved by combining impedance boundary conditions at these interfaces with the transition matrix relationship. A numerical algorithm is described which solves the state-vector equation using finite differences. The validation of the resultant computer program is discussed along with example calculations.

Abstract: Mesh Refinement and Redistribution Efficient Techniques for the Analysis of Pollutant Migration Direct General Finite Difference Techniques for Elliptic Problems Defined in Bounded and Unbounded Two-Dimensional Domains Solution Strategies for Elastic and Inelastic Contact Problems of Solids Recent Developments in Finite Difference Methods for the Computation of Transient Flows Numerical Analysis of Rain Effects on an Airfoil Solution Techniques for Boundary Integral Matrices Some Transient and Coupled Problems - A State-of-the- Art Review The Prediction of Instabilities using Bifurcation Theory Long Time Calculations and Non-linear Maps Modelling of Coupled Thermo-elastoplastic-hydraulic Response of Clays Subjected to Nuclear Waste Heat Numerical Modelling of Free- Surface Flows Transient Algorithms and Fluid-Structure Interaction - An Overview Nonlinear Transient Dynamic Analysis of Reinforced Concrete Structures using a Three-Dimensional Approach.

Abstract: A method is presented for obtaining the consolidation behaviour of a layered soil subjected to strip, circular, or rectangular surface loadings, or subjected to fluid withdrawal due to pumping. The solution method involves applying a Fourier or Hankel transform to the field quantities along with a Laplace transformation. The effect of the Fourier or Hankel transform is to reduce a two- or three-dimensional problem or one involving axial symmetry, to one involving only a single spatial dimension. In cases where the soil is horizontally layered, this has great advantages over conventional methods, such as finite element or finite difference methods, since very little computer storage and data preparation time is required. Solution of the time dependent problem is achieved by applying a Laplace transformation to the field variables, obtaining solutions in Laplace transform space, and then numerically inverting the transformed solutions to obtain the real time behaviour. This eliminates the need for ‘marching type’ schemes where a solution is found from one at a previous time. By direct inversion of the Laplace transform, a solution may be obtained directly at any given time.

Abstract: A brief description of a numerical technique suitable for solving axisymmetric, unsteady free‐boundary problems in fluid mechanics is presented. The technique is based on a finite‐difference solution of the equations of motion on a moving orthogonal curvilinear coordinate system, which is constructed numerically and adjusted to fit the boundary shape at any time. The initial value problem is solved using a fully implicit first‐order backward time differencing scheme in order to insure numerical stability. As an example of application, the unsteady deformation of a bubble in a uniaxial extensional flow for Reynolds numbers is considered in the range of 0.1≤R≤100. The computation shows that the bubble extends indefinitely if the Weber number is larger than a critical value (W>Wc). Furthermore, it is shown that a bubble may not achieve a stable steady state even at subcritical values of Weber number if the initial shape is sufficiently different from the steady shape. Finally, potential‐flow solutions as an ...

Abstract: Focused finite amplitude sound fields are investigated with numerical solutions of the Khokhlov‐Zabolotskaya‐Kuznetsov (KZK) equation. The numerical solution is based on the algorithm developed by Aanonsen et al. [J. Acoust. Soc. Am. 75, 749–768 (1984)], who used a Fourier series expansion of the sound pressure to reduce the KZK equation to a set of coupled parabolic equations. The basic algorithm has been modified by introducing a coordinate system that follows the convergent geometry of focused sound fields. In this way, more efficient numerical evaluation of the detailed field structure within the focal region is achieved. Arbitrary axisymmetric sources can be modeled. Here, circular sources having linear focusing gains of order 50 will be considered. The calculated time waveforms, propagation curves, and beam patterns illustrate clearly the combined effects of nonlinearity, diffraction, and absorption on finite amplitude sound that passes through a focal region. Among the new results are power curves ...

Abstract: Amathematical model was developed for the first time for predicting detailed flow patterns and temperature profiles during natural convection heating of canned liquids. Finite difference methods were used to solve the governing Navier-Stokes equations in axisymmetric cylindrical coordinates. A vorticity-stream function formulation of the equations was used. Details of the numerical techniques used are discussed. Plots of transient isotherms, streamlines and velocities are provided. From the standpoint of food processing, the slowest heating points migrated within the bottom 15% of the can with no particular pattern of migration.

Abstract: It is shown that if the expanded node three-dimensional TLM method is operated in a certain way, then it can be numerically equivalent to a finite-difference method. Some comments are made on comparisons between the two approaches.

Abstract: Since Kinber (Radio Technika and Engineering-1963) and Galindo (IEEE Trans. Antennas Propagat.-1963/1964) developed the solution to the circular symmetric dual shaped synthesis problem, the question of existence (and of uniqueness) for offset dual (or single) shaped synthesis has been a point of controversy. Many researchers thought that the exact offset solutions may not exist. Later, Galindo-Israel and Mittra (IEEE Trans. Antennas Propagat.-1979) and others formulated the problem exactly and obtained excellent and numerically efficient but approximate solutions. Using a technique similar to that first developed by Schruben for the single reflector problem (Journal of the Optical Society-1973), Brickell and Westcott (Proc. Institute of Electrical Engineering-1981) developed a Monge-Ampere (MA) second-order nonlinear partial differential equation for the dual reflector problem. They solved an elliptic form of this equation by a technique introduced by Rall (1979) which iterates, by a Newton method, a finite difference linearized MA equation. The elliptic character requires a set of finite difference equations to be developed and solved iteratively. Existence still remained in question. Although the second-order MA equation developed by Schruben is elliptic, the first-order equations from which the MA equation is derived can be integrated progressively (e.g., as for an initial condition problem such as for hyperbolic equations) a noniterative and usually more rapid type solution. In this paper, we have solved, numerically, the first-order equations. Exact solutions are thus obtained by progressive integration. Furthermore, we have concluded that not only does an exact solution exist, but an infinite set of such solutions exists. These conclusions are inferred, in part, from numerical results.

Abstract: The unipolar injection stability problem is characterized by a hysteresis loop between nonlinear and linear stability criteria. The problem of determining the corresponding finite amplitude convective solution is investigated for rolls in the weak injection case, taking into account the peculiarities of the charge conservation equation and the possible nonstationary character of the flow. The results of three different approaches (finite difference method, approximate analytical solution, and particle-type method) are presented and discussed.

Abstract: There are spurious phenomena in the numerical approximation of the hyperbolic equations of fluid dynamics that may be investigated by invoking concepts which originate from wave propagation theory. Many of the significant results which have been obtained by pursuing this kind of analysis are reviewed in this paper by using as an illustration a family of implicit approximations of the simple linear advection equation. Included in this family of algorithms are the common six-point implicit finite difference scheme, the linear finite element/Galerkin scheme and the ‘box’ method. The phase and group velocities of sinusoidal solutions are brought into the analysis of the accuracy and of the spurious reflection or scattering phenomena which are created at computational boundaries and in non-uniform grids. General properties become apparent in this Fourier/wave propagation approach to the analysis. One of these is in the form of an analogy with quantum mechanics. Another shows that certain energy norms of the errors are independent of time discretization, i.e. depend on space discretization alone.

Abstract: A new finite-difference scheme has been developed to solve efficiently the unsteady Euler equations for three-dimensional inviscid supersonic flows with subsonic pockets The technique utilizes planar Gauss-Seidel relaxation in the marching direction and approximate factorization n the crossflow plane An 'infinitely large' time step is used in parts of the flowfield where the component of velocity in the marching direction is supersonic - here the Gauss-Seidel sweeps are restricted to the forward direction only, and the procedure reduces to simple space-marching; a finite time step is used in parts of the flowfield where the marching component of velocity is subsonic - here, backward and forward Gauss-Seidel sweeps are employed to allow for upstream and downstream propagation of signals, and a time-asymptotic steady state is obtained The discretization formulas are based on finite-volume implementation of high accuracy (up to third-order) total variation diminishing formulations Numerical solutions are obtained for an analytically defined forebody, a realistic fighter configuration, and the Space Shuttle The results are in very good agreement with available experimental data and numerical solutions of the full-potential equation

Abstract: On construit une famille de schemas aux differences finies uniformement precis pour le probleme du type −eu''+a(x)u'+b(x)u=f(x). On donne une analyse complete de l'erreur de discretisation basee sur les resultats de stabilite. On presente des essais numeriques

Abstract: The paper presents a method, called CONDIF, which modifies the CDS (central-difference scheme) by introducing a controlled amount of numerical diffusion based on the local gradients. The numerical diffusion can be adjusted to be negligibly low for most problems. CONDIF results are significantly more accurate than those obtained from the hybrid scheme when the Peclet number is very high and the flow is at large angles to the grid.

Abstract: The method of upper-lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions. The idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem. The aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable. These two schemes are modified Jacobi method and Gauss–Seidel method for nonlinear algebraic equations. An advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration. On the other hand, the standard Picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations.