Abstract: Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...

Abstract: Simulations of optical guided waves in three-dimensional waveguide structures by a full vector beam propagation method are described. Two sets of coupled equations governing the propagation of the transverse electric and magnetic fields are derived systematically. Polarization dependence and coupling due to the vectorial nature of the electromagnetic fields are considered in the formulations. The governing equations are solved subsequently by finite-difference schemes. The vector BPM is first assessed for a step-index circular fiber by comparing the numerical results with the exact analytical solutions. The guided modes of a rib waveguide are then investigated in detail. Comparisons among the scalar, semi-vector and full-vector simulations of the rib waveguide are made. Finally polarization rotation of a periodically loaded rib waveguide operated fully based on the vector nature of the electromagnetic waves is modeled and simulated. >

Abstract: We have developed a finite-difference solution for three-dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell's equations in time using a staggered-grid technique. The time-stepping uses a modified version of the Du Fort-Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first-order Maxwell's equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence-free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times.An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth-order difference method at early times and a second-order method at other times. Numerical checks against analytical, integral-equation, and spectral differential-difference solutions show that the solution provides accurate results.Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains 100 X 100 X 50 grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Omega . m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much-needed insight about TEM responses in complicated 3-D situations.

Abstract: Part 1 Basic concepts of finite difference methods: introduction to finite difference methods parabolic equations elliptic equations hyperbolic equations. Part 2 Pressure-based algorithms and their applications: pressure-based algorithms practical applications. Part 3 Interfacial transport: basic concepts of thermodynamics thermofluid phenomena involving capillarity and gravity physical and computational issues in phase-change dynamics.

Abstract: The stiffness and damping coefficients of an elastically supported gas foil bearing are calculated. A perfect gas is used as the lubricant, and its behavior is described by the Reynolds equation. The structural model consists only of an elastic foundation. The fluid equations and the structural equations are coupled. A perturbation method is used to obtain the linearized dynamic coefficient equations. A finite difference formulation has been developed to solve for the four stiffness and the four damping coefficients. The effect of the bearing compliance on the dynamic coefficients is discussed in this paper.

Abstract: The popularity of the finite-difference time-domain (FDTD) method stems from the fact that it is not limited to a specific geometry and it does not restrict the constitutive parameters of a scatterer. Furthermore, it provides a direct solution to problems with transient illumination, but can also be used for harmonic analysis. However, researchers have limited their investigation to materials that are either isotropic or that have diagonal permittivity, conductivity, and permeability tensors. The authors derive the necessary extension to the FDTD equations to accommodate nondiagonal tensors. Excellent agreement between FDTD and exact analytic results is obtained for a one-dimensional anisotropic scatterer. >

Abstract: The traditional, simple numerical differentiation of finite-element stiffness matrices by a forward difference scheme is the source of severe error problems that have been reported recently for certain problems of finite-element-based, semi-analytical shape design sensitivity analysis. In order to develop a method for elimination of such errors, without a sacrifice of the simple numerical differentiation and other main advantages of the semi-analytical method, the common mathematical structure of a broad range of finite-element stiffness matrices is studied in this paper. This study leads to the result that element stiffness matrices can generally be expressed in terms of a class of special scalar functions and a class of matrix functions of shape design variables that are defined such that the members of the classes admit “exact” numerical differentiation (exact up to round-off error) by means of very simple correction factors to upgrade standard computationally inexpensive first-order finite di...

Abstract: A comparison is made between several different methods that have recently been proposed for efficiently modeling electrically thin material sheets in the finite-difference-time-domain (FDTD) method. The test problems used in the comparison are parallel-plate waveguides loaded with electrically thin dielectric (lossless) and conducting sheets for which exact solutions are available. The accuracy of the methods is illustrated by comparison with analytical results for model problems that have exact solutions. >

Abstract: This paper introduces a generalization of the three-dimensional finite-difference-time-domain (FDTD) method, the three-dimensional contour FDTD (CFDTD) method, is introduced. The FDTD method represents curved media boundaries as stepped edges. Through the use of subcell modeling, the CFDTD method conformably models bodies with curved surfaces, yet retains the ability to model corners and edges. Electromagnetic scattering from single and multiple bodies is presented. >

Abstract: An efficient method to include frequency-dependent materials in finite difference time domain calculations based on the recursive evaluation of the convolution of the electric field and the susceptibility function has previously been presented. The method has been applied to various materials, including those with the Debye, Drude, and Lorentz forms of complex permittivity, and to anisotropic magnetized plasmas. Previous demonstrations of this approach have been confined to total field calculations in one dimension. In this paper the recursive convolution method is extended to three-dimensional scattered field calculations. The accuracy of the method is demonstrated by calculating scattering from spheres of various sizes composed of three different types of frequency-dependent materials. >

Abstract: The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.

Abstract: An adiabatic global multilevel primitive equation model using a two time-level, semi-Lagrangian semi-implicit finite-difference integration scheme is presented A Lorenz grid is used for vertical discretization and a C grid for the horizontal discretization The momentum equation is discretized in vector form, thus avoiding problems near the poles The 3D model equations are reduced by a linear transformation to a set of 2D elliptic equations, whose solution is found by means of an efficient direct solver The model (with minimal physics) is integrated for 10 days starting from an initialized state derived from real data A resolution of 16 levels in the vertical is used, with various horizontal resolutions The model is found to be stable and efficient, and to give realistic output fields Integrations with time steps of 10 min, 30 min, and 1 h are compared, and the differences are found to be acceptable

Abstract: A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow-water equations in open channels, together with an extension to two-dimensional flows. A linearized problem, analogous to that of Riemann for gas dynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearized problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. The scheme is applied to a one-dimensional dam-break problem, and to a problem of flow in a river whose geometry induces a region of supercritical flow. The scheme is also applied to a two-dimensional dam-break problem. The numerical results are compared with the exact solution, or other numerical results, where available.

Abstract: A simple and efficient approximate numerical technique is presented to obtain solutions to a wide class of two-point boundary value similarity problems in fluid mechanics. This technique is based on the common finite difference method with central differencing, a tridiagonal matrix manipulation and an iterative procedure. The technique described in this paper has been successfully applied to three different representative similarity problems of fluid mechanics. Each one of these problems is described by a coupled, non-linear system of three ordinary differential equations and has already been solved elsewhere using a different numerical method. So, the obtained numerical results, by our efficient numerical technique, permit a comparative study and show the accuracy and the effectiveness of this technique.

Abstract: Finite-difference acoustic-wave modeling and reverse-time depth migration based on the full wave equation are general approaches that can take into account arbitary variations in velocity and density, and can handle turning waves well. However, conventional finite-difference methods for solving the acousticwave equation suffer from numerical dispersion when too few samples per wavelength are used. Here, we present two flux-corrected transport (FCT) algorithms, one based the second-order equation and the other based on first-order wave equations derived from the second-order one. Combining the FCT technique with conventional finite-difference modeling or reverse-time wave extrapolation can ensure finite-difference solutions without numerical dispersion even for shock waves and impulsive sources. Computed two-dimensional migration images show accurate positioning of reflectors with greater than 90-degree dip. Moreover, application to real data shows no indication of numerical dispersion. The FCT correction, which can be applied to finite-difference approximations of any order in space and time, is an efficient alternative to use of approximations of increasing order.

Abstract: The boundary element method is used to calculate the electric field profiles at needle tips commonly used for electrical treeing tests. Field distributions are also obtained for polyethylene containing a space charge, at the needle tip, and are compared with the values previously obtained by the finite difference method. >

Abstract: The finite-difference time-domain (FDTD) method is extended to include nonlinear active regions embedded in distributed circuits. The procedures necessary to produce a stable algorithm are described, and a single device cavity oscillator is simulated with this method. >

Abstract: A general algorithm for modeling arbitrary shape planar metal strips by the finite-difference-time-domain (FDTD) method is presented. With this method, fields in the entire computation domain are computed by the regular FDTD algorithm except near metal strips, where special techniques proposed herein are applied. Unlike the case for globally conformed finite-difference algorithms, the computation efficiency of the regular FDTD method is maintained while high space-resolution is obtained by this locally conformed finite-difference method. Numerical tests have verified that a higher computation accuracy is achieved by this scheme than by the conventional staircase approximation. The modeling of electrical characteristics of two crossed strip lines is provided as an example. >

Abstract: The numerical properties of approximation schemes for a model that simulates water transport in root-soil systems are considered. The model is derived in detail. It is based on a previously proposed model which is reformulated completely in terms of the water potential. The system of equations consists of a parabolic partial differential equation that contains a nonlinear capacity term coupled to two linear ordinary differential equations. A closed form solution is obtained for one of the latter equations. Finite element and finite difference schemes are defined to approximate the solution of the coupled system. Some new techniques which have wide applicability for analyzing the nonlinear capacity term are used, and optimal order error estimates are derived. A postprocessed water mass flux computation is also presented and shown to be superconvergent to the true flux. Computational results which verify the theoretical convergence rates are given.

Abstract: Details of an efficient semivectorial beam propagation method are presented. The propagation is determined by solving the finite-difference equation in alternating directions. Unlike the traditional split-operator beam propagation method, in the semivectorial method the index term is not separated from the spatial differentiations. Since boundary conditions at dielectric interfaces can be satisfied by two orthogonal polarizations individually, the method is semivectorial. The performance can be enhanced by using a nonuniform grid. By modifying the routine, it can also be applied to periodic structures. Numerical results for rib waveguides are presented. >

Abstract: The heat dissipation and temperature distribution in multilayered soil surrounding a buried cable are calculated using the finite difference method and the energy conservation principle. The numerical technique proposed is very suitable for modeling any real cable installation configuration in multilayered soil. The development of the model and the effect of the parameters that influence the conversion and the stability of the numerical solution of the heat dissipation from the underground cable system are studied. >

Abstract: This work introduces a new numerical algorithm that can be used to analyze complex problems of penetrant transport. Penetrant transport in polymers often deviates from the predictions of Fick's law because of the coupling between penetrant diffusion and the polymer mechanical behavior. This phenomenon is particularly important in glassy polymers. This leads to a model consisting of two coupled differential equations for penetrant diffusion and polymer stress relaxation, respectively. If the polymer relaxation is the rate-limiting step, both the concentration and stress profiles are very steep. A new algorithm based on a finite difference method is proposed to solve the model equations. It features the development of a tridiagonal iterative method to solve the nonlinear finite difference equations obtained from the finite difference approximation of the differential equations. This method was found to be efficient and accurate. Numerical simulation of penetrant diffusion in glassy polymers was performed, showing that the integral sorption Deborah number is a major parameter affecting the transition from Fickian to anomalous diffusion behavior. © 1993 John Wiley & Sons, Inc.