Abstract: A domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented. In this procedure, interface values between subdomains are found by an explicit finite difference formula. Once these values are calculated, interior values are determined by backward differencing in time. A natural extension of this method allows for the use of different time steps in different subdomains. Maximum norm error estimates for these procedures are derived, which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters.

Abstract: The analysis of a group of three simple antennas is used to illustrate the accuracy of the FDTD (finite-difference time-domain) method and to show that various geometrical features are handled correctly by the method. Each antenna is a well-posed electromagnetic boundary value problem that corresponds to a realizable experimental model. The three antennas considered are of increasing complexity: an open-ended parallel plate waveguide, a cylindrical monopole, and a conical monopole. The FDTD calculations for these antennas were compared with analytical results (open-ended parallel plate waveguide) and accurate measurements in the time and frequency domains (cylindrical and conical monopoles). In all cases the agreement was excellent. >

Abstract: SUMMARY The finite-difference method is applied to compute the seismic response of 2-D inhomogeneous structures for SH-waves. A technique is proposed which uses an irregular grid (a rectangular grid with varying grid spacing). A geological structure may be composed of blocks of media inside of which velocity and density vary linearly in horizontal and vertical directions. The technique allows better adjusted modelling of a medium and, in numerical examples presented, yields more efficient computations as compared to those with regular grids. The technique is tested through comparison with a discrete-wavenumber method. As an example, the seismic response of the sediment-filled Chusal Valley, Carm region, USSR, is computed. The numerical results are compared with observations.

Abstract: We have developed a new implicit finite difference method to simulate the interaction of intense nanosecond laser beams with semiconductors and metal-coated ceramic structures. This method is based upon a higher order implicit finite difference scheme with a smaller truncation error and is not restricted by any stability criterion, thereby allowing faster convergence to the exact solution. The temperature-dependent optical and thermal properties of the irradiated material as well as the temporal variation in the laser intensity have been taken into account. Finite difference equations have been set up for accurate determination of the temperature gradients at the liquid-solid interface, which control the melt-in and resolidification velocities. A new formulation is introduced to accomodate the effect of pulsed laser irradiation on layered composite structures (e.g. metal-coated ceramics) by incorporating the boundary conditions at the composite interface. Using this method, the thermal histories of laser-irradiated materials were predicted. The effects of variation in the pulse energy density, pulse duration and substrate temperature on the maximum melt depths, solidification velocities and surface temperatures were computed. The calculations on the depth of melting were found to be in good agreement with experimental results where complete annealing of the ion implantation damage was used as a measure of the melt depth. The surface temperatures and melt lifetimes in metal-coated ceramics were determined in order to understand the laser mixing process. Simple energy balance considerations were applied to calculate some of the effects of laser irradiation on materials. From these energy considerations, the maximum melt depths as a function of energy density, pulse duration and substrate temperature were obtained and compared with the exact solutions. The maximum surface temperatures, solidification velocities and melt lifetimes were also determined by this analytical method and compared with the detailed calculations. A good agreement between the analytical relations and the detailed numerical calculations provides an excellent guide to researchers in this field.

Abstract: A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Abstract: A mathematical model for low‐pressure chemical vapor deposition (LPCVD) in a single wafer reactor of the impinging jet type has been developed. The model includes the partial differential equations describing the balance of mass, momentum, heat, and species concentration, Stefan‐Maxwell equations for multicomponent diffusion, multicomponent thermodiffusion, multiple surface reactions, and variable gas properties. Gas‐phase chemistry is neglected. The equations are solved numerically in two‐dimensional, axisymmetric form, using a control‐volume‐based finite difference method. The model is applied to silicon LPCVD from silane. It is shown that in single wafer LPCVD modeling, the coupling of the flow equations to the species concentrations is very important, as are thermodiffusion effects. Multicomponent diffusion phenomena can be modeled accurately using Wilke's approximation in many cases. In some situations, however, the Stefan‐Maxwell equations should be used. The model is used to optimize both reactor geometry and process conditions, in order to obtain uniform deposition on large wafers at high growth rates. A series of parameter variations is presented, illustrating the power of the model as an aid in such an optimization study.

Abstract: The semi-analytical method of sensitivity analysis combines ease of implementation with computational efficiency. A major drawback to this method, however, is that severe accuracy problems have recently been reported. A complete error analysis for a beam problem with changing length is carried out in this paper. It is shown that the sensitivity error is proportional to the relative length difference, but in agreement with Eq. 3.8. The approximate pseudo loads thus violate moment equilibrium for rigid body motion while the correct pseudo loads do not. It might be a good idea to modify the approximate pseudo loads in order to obtain general load equilibrium with rigid body motions. Such a method would be readily applicable for any element type, whether analytical expressions for the element stiffnesses are available or not. This topic is postponed for a future study.

Abstract: A fourth-order high-accuracy finite difference method is presented for the bouyancy-driven flow in a square cavity with differentially heated vertical walls. The two bench mark solutions against which other solutions can be compared were obtained. The present solution is seemed to be accurate up to fifth decimal. The proposed scheme is stable and convergent for high Rayleigh number, and will be applicable to general problems involving flow and heat transfer, especially in three dimensions.

Abstract: A finite-difference method (FDM) with nonuniform discretization for the analysis of channel waveguides is presented. Application of the boundary conditions for either the quasi-TE or quasi-TM mode is illustrated. Flexible discretization of the grid structures minimizes memory size, resulting in much smaller computing time without sacrificing the accuracy of the solution. This nonuniform discretization FDM technique is used to model the well-guided small-mode-size Ti:LiNbO/sub 3/ waveguides. The model treats both finite and infinite source diffusion cases. Quasi-TM mode profiles and the corresponding eigenvalues are rigorously evaluated and the theoretical results agree very well with the experimental results. >

Abstract: This paper describes the application of three-dimensional finite-difference calculation procedures to the problem of a jet impinging on a flat plate through the influence of a confined crossflow One procedure uses the hybrid central/upwind difference scheme, and the other uses a quadratic upstream weighted difference scheme (QUICK) to calculate the convection terms The standard two-equation "& — c" turbulence model is used to calculate the distribution of the Reynolds stresses The difficulty of assessing turbulence model performance in these complex flows due to the intrusion of numerical diffusion errors is demonstrated by comparing the calculations on both coarse and fine meshes and by improving the accuracy of the convection terms discretization using the higher-order QUICK method The ability of the model calculations to simulate both the mean and the turbulence fields is examined, particularly in the vicinity of the stagnation point The results show the advantages of QUICK differencing scheme over the hybrid treatment, since the same level of numerical accuracy requires far less CPU time and computer memory when the QUICK scheme is used The calculations reveal the existence of large regions of low pressure, associated with an upstream recirculating flow region due to the interaction between the upstream wall jet and the crossflow, which may produce a substantial lift loss for a VSTOL aircraft

Abstract: We present a new rapid approach to the analysis of strongly guiding and longitudinally varying semiconductor rib waveguide structures. Our technique, which is a synthesis of the beam propagation and finite difference methods is Hermitian, second-order accurate and intrinsically stable.

Abstract: Some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained. In particular an estimate for the solutions of the continuous time versus the discrete time Hamilton–Jacobi–Bellman equations is given. The technique used is more analytic than probabilistic.

Abstract: The inverse heat conduction problem involves the calculation of surface heat flux and/or temperature histories from transient, measured temperatures inside solids. We consider the one dimensional semi-infinite linear case and present a new solution algorithm based on a data filtering interpretation of the mollification method that automatically determines the radius of mollification depending on the amount of noise in the data and finite differences. A fully explicit and stable space marching scheme is developed. We describe several numerical experiments of interest showing that the new procedure is accurate and stable with respect to perturbations in the data even for small dimensionless time steps.

Abstract: Some numerical aspects of finite-differ ence algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogeneous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a more efficient solution procedure. The primary motivation for constructing these schemes was to address large systems of thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be simplified, thus providing a more efficient solution procedure than one might have anticipated. An implicit algorithm with explicit coupling between fluid and species equations is also proposed.

Abstract: A numerical algorithm for the two-dimensional solidification problem in the twin-roll continuous casting system is presented in this paper Attention is focused on the elucidation of heat transfer and flow characteristics in both the liquid and the solid phases The present mathematical model can be applied to general full Navier-Stokes and energy equations, thereby covering the wide range of twin-roll casting conditions The boundary fixing method (BFM) is adopted to handle the moving boundary, and the resultant transformed governing equations for the solid and liquid regions are solved separately by using a usual explicit-type finite difference method In this paper, a general numerical methodology is presented, and the quantitative relationships between the important control parameters in continuous casting of twin-roll type (such as the roll speed, the roll gap, the initial temperature of molten materials, the material properties, the solidification profile, and the endpoint of solidification) are clarified in detail The present numerical results have been compared with experimental results obtained separately to check the validity of the proposed method

Abstract: An explicit finite difference algorithm is developed to approximate the solution of a nonlinear and nonlocal system of integro-differential equations that models the dynamics of a two-sex population. The algorithm is unconditionally stable. The optimal rate of convergence of the algorithm is demonstrated for the maximum norm. Results from a numerical simulation of U.S. population growth from 1970 to 1980 are presented; these compare favorably with the actual data. (EXCERPT)

Abstract: This paper examines mainly oscillatory behavior of a fluid-conveying collapsible tube using a two-dimensional flexible channel made of a pair of membranes. The equation of equilibrium of the membrane in a large deflection theory is coupled with the equations of continuity and momentum of an incompressible flow in a one-dimensional flow theory accounting for flow separation. An explicit finite difference method was used to solve the governing equations numerically. According to numerical results, the fluids in the inlet and outlet rigid channels have strong effects on the oscillation of the system. Depending on initial values for the numerical integration, there may exist both a stable static equilibrium and an oscillatory solution for the same parameter values, but only if the external pressure is sufficiently large.

Abstract: A hybrid finite-element method to discretize the continuity equation in semiconductor device simulation is given. Within each element of a finite element discretization, the current is uniquely determined by nodal values of the density and the potential. The authors use the integrability condition for a system of partial differential equations to obtain the equations that determine the current within the element. They then satisfy the continuity in the current flow across interelement boundaries in a weak sense. They have found that the method works in any dimension and for (d-dimensional) simplexes as well as for quadrilaterals, bricks, prisms, and so on, although they have no proof that it will not break down in particular cases. >

Abstract: Backward difference methods for the discretization of parabolic boundary value problems are considered in this paper. In particular, we analyze the case when the backward difference equations are only solved 'approximately' by a preconditioned iteration. We provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations (often independent of the spatial discretization and time step size) are used. Results are provided for the smooth as well as nonsmooth initial data cases. Finally, the results of numerical experiments illustrating the algorithms' performance on model problems are given.

Abstract: Truncation error estimates are considered as criteria for fine-grid placement and movement in nested and adaptive finite-difference atmospheric models. A simple method for calculating the truncation error at any time during an integration is described. Two cases using the shallow-water equations and the hydrostatic primitive equations are examined to demonstrate the accuracy of the method and illuminate the relationships among the truncation error, a particular discretization, the equations being solved and the flow physics. The relationship between the truncation error and the solution error is also discussed and it is argued that minimization of the truncation error is the necessary consideration for producing more accurate numerical solutions. Examples of use of the truncation error estimates in adaptive models are also presented.