Showing papers on "Finite difference published in 1987"

The pseudospectral method: Comparisons with finite differences for the elastic wave equation

Abstract: The pseudospectral (or Fourier) method has been used recently by several investigators for forward seismic modeling. The method is introduced here in two different ways: as a limit of finite differences of increasing orders, and by trigonometric interpolation. An argument based on spectral analysis of a model equation shows that the pseudospectral method (for the accuracies and integration times typical of forward elastic seismic modeling) may require, in each space dimension, as little as a quarter the number of grid points compared to a fourth‐order finite‐difference scheme and one‐sixteenth the number of points as a second‐order finite‐difference scheme. For the total number of points in two dimensions, these factors become 1/16 and 1/256, respectively; in three dimensions, they become 1/64 and 1/4 096, respectively. In a series of test calculations on the two‐dimensional elastic wave equation, only minor degradations are found in cases with variable coefficients and discontinuous interfaces.

Forced Convection Cooling Across Rectangular Blocks

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.

Numerical investigation of the three-dimensional development in boundary-layer transition

Abstract: A numerical method for solving the complete Navier-Stokes equations for incompressible flows is introduced that is applicable for investigating three-dimensional transition phenomena in a spatially growing boundary layer. Results are discussed for a test case with small three-dimensional disturbances for which detailed comparison to linear stability theory is possible. The validity of our numerical model for investigating nonlinear transition phenomena is demonstrated by realistic spatial simulations of the experiments by Kachanov and Levchenko1 for a subharmonic resonance breakdown and of the experiments of Klebanoff et al.2 for a fundamental resonance breakdown.

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

Abstract: An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

A local convergence theory for combined inexact-Newton/finite-difference projection methods

Abstract: Finite-difference versions of some recently developed Krylov subspace projection methods are presented and analysed in the context of solving systems of nonlinear equations using Inexact-Newton Met...

A modified equation approach to constructing fourth order methods for acoustic wave propagation

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...

Computation of turbulent flows using an extended k-epsilon turbulence closure model

Abstract: An extended kappa-epsilon turbulence model is proposed and tested with successful results. An improved transport equation for the rate of dissipation of the turbulent kinetic energy, epsilon, is proposed. The proposed model gives more effective response to the energy production rate than does the standard kappa-epsilon turbulence model. An extra time scale of the production range is included in the dissipation rate equation. This enables the present model to perform equally well for several turbulent flows with different characteristics, e.g., plane and axisymmetric jets, turbulent boundary layer flow, turbulent flow over a backward-facing step, and a confined turbulent swirling flow. A second-order accurate finite difference boundary layer code and a nearly second-order accurate finite difference elliptic flow solver are used for the present numerical computations.

Difference Schemes: An Introduction to the Underlying Theory

Abstract: Ordinary Difference Equations. Difference Equations of First and Second Order. Examples of Difference Schemes. Boundary-Value Problems for Equations of Second Order. Basis of the FEBS Method. Difference Schemes for Ordinary Differential Equations. Elementary Examples of Difference Schemes. Convergence of the Solutions of Difference Equations as a Consequence of Approximation and Stability. Widely-Used Difference Schemes. Difference Schemes for Partial Differential Equations. Basic Concepts. Simplest Examples of the Construction and Study of Difference Schemes. Some Basic Methods for the Study of Stability. Difference Scheme Concepts in the Computation of Generalized Solutions. Problems with Two Space Variable. The Concept of Difference Schemes with Splitting. Elliptic Problems. Concept of Variational-Difference and Projection-Difference Schemes. Stability of Evolutional Boundary-Value Problems Viewed as the Boundedness of Norms of Powers of a Certain Operator. Construction of the Transition Operator. Spectral Criterion for the Stability of Nonselfadjoint Evolutional Boundary-Value Problems. Appendix: Method of Internal Boundary Conditions. Bibliographical Commentaries. Bibliography. Index.

The finite difference thermodynamic integration, tested on calculating the hydration free energy difference between acetone and dimethylamine in water

Abstract: A new technique is proposed to compute by Monte Carlo (or molecular dynamics) computer simulation the hydration free energy differences. The method, called finite difference thermodynamic integration, is a combination of the thermodynamic integration and the perturbation method. It was compared with thermodynamic integration over two different paths and the perturbation method on computing the solvation free‐energy difference between the dilute aqueous solution of acetone and dimethyl amine. Finite difference thermodynamic integration was found to have the best convergence characteristics among the methods tested.

Simulation of one-dimensional dam-break flows

Abstract: Two new finite-difference schemes - Gabutti, and Beam and Warming - are introduced and compared for the analysis of unsteady free-surface flows resulting from the breaking of a dam. These schemes split the fluxvector into positive and negative parts, each of which corresponds to the direction of a characteristic, thereby allowing use of proper finite differences for the space derivatives. Central finite differences are used for subcritical flow and upwind differences are used for supercritical flow. The details of these schemes are presented and the computed results are compared with the analytical solution to demonstrate their validity. Because of their simplicity, these schemes are attractive for solving the dam-break problem, especially when supercritical flow is present.

Electromagnetic scattering by stratified inhomogeneous anisotropic media

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.

Detailed Simulation of Unstable Processes in Miscible Flooding

Abstract: This paper describes applications of linear and nonlinear simulations to unstable miscible flooding. The first section describes a method of calculating linear growth of unstable modes by use of finite differences in the direction of flow and Fourier decomposition perpendicular to flow. This work extends the previous long- and short-wavelength analytic results to cover the whole wave-number range. Results obtained are used to help validate a two-dimensional (2D) code to study nonlinear evolution of viscous fingers and to identify likely fingering regimes in the computed solution by identifying the range of wave numbers that dominates the linear growth behavior. The second section describes the numerical scheme for calculations on a fine grid of the nonlinear development of an instability. Results are presented on calculations of nonlinear growth at several mobility ratios and levels of diffusion. Comparisons with Blackwell's experimental data are presented. Good agreement is obtained, suggesting that the physical processes governing fingering are being correctly modeled.

A simplified TVD finite difference sheme via artificial viscousity

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.

Calculating Rotordynamic Coefficients of Seals by Finite-Difference Techniques

Abstract: For modelling the turbulent flow in a seal the Navier-Stokes equations in connection with a turbulence (kappa-epsilon) model are solved by a finite-difference method. A motion of the shaft round the centered position is assumed. After calculating the corresponding flow field and the pressure distribution, the rotor-dynamic coefficients of the seal can be determined. These coefficients are compared with results obtained by using the bulk flow theory of Childs and with experimental results.

Numerical Calculations of Electrostatic Potentials of Protein-Solvent Systems by the Self Consistent Boundary Method

Abstract: A macroscopic dielectric model for a protein-solvent system has been solved by a numerical calculation using a supercomputer. The whole space including a target protein molecule was first divided into lots of cubes, some of which were inside the protein and the others were embedded in the solvent. Poisson equation and Poisson-Boltzmann equation were then numerically solved inside and outside of the protein, respectively. At the artificial boundary far from the protein for the finite difference procedure, neither the potential nor the field was set to be zero, but they were calculated self-consistently from the field and the potential inside the boundary calculated beforehand. The results by this method agreed well with those by the analytical calculations for some simple model cases. The results of actual protein-solvent systems were quantitatively discussed comparing with the experiments.

On the theory of the synthesis of single and dual offset shaped reflector antennas

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.

Simulation of contaminant transport in three dimensions: 1. The alternating direction Galerkin Technique

Abstract: The alternating direction Galerkin technique for the simulation of advective-dispersive transport in three dimensions is developed. The technique is designed for high efficiency in handling the large and detailed grids that are often required in simulations of natural groundwater systems. It is comparable to an alternating direction finite difference scheme, except for two advantages: first, the capability to handle certain curvilinear grids that may conform to either a flow net geometry or a natural stratigraphy and, second, the option to enhance solution accuracy through a choice between a finite difference or a finite element representation of the time derivative. Criteria for controlling numerical dispersion and accuracy can be applied easily. Three options for the time-stepping algorithm are developed and analyzed for stability, and their accuracy is investigated for simple flow systems. The technique is shown to be far more efficient than a conventional three-dimensional finite element model. In a companion paper, an application to a field-scale contaminant transport problem is described.

Theis' Solution by Nonlinear Least-Squares and Finite-Difference Newton's Method

Abstract: A method using the nonlinear least-squares and finite-difference Newton's method to determine the aquifer parameters via a pumping test in a homogeneous and isotropic confined aquifer system is proposed. The nonlinear least-squares is used to find the values of transmissivity and storage coefficient such that the sum of the squares of differences between the predicted drawdowns and observed drawdowns is minimized. The finite-difference Newton's method is used to solve the system of nonlinear least-squares equations for transmissivity and storage coefficient. Comparisons of the results between the proposed method and graphical methods including the Theis, Cooper-Jacob, and Chow methods are discussed in detail, showing data of a 6-hour pumping test. The proposed method has the advantages of high accuracy and quick convergence for most initial guesses.

3DFEMWATER: A three-dimensional finite element model of water flow through saturated-unsaturated media

Abstract: This report presents the development and verification of a one- dimensional finite element model of water flow through saturated- unsaturated media. 1DFEMWATER is very flexible and capable of modeling a wide range of real-world problems. The model is designed to (1) treat heterogeneous media consisting of many geologic formations; (2) consider distributed and point sources/sinks that are spatially and temporally variable; (3) accept prescribed initial conditions or obtain them from steady state simulations; (4) deal with transient heads distributed over the Dirichlet boundary; (5) handle time-dependent fluxes caused by pressure gradient on the Neumann boundary; (6) treat time-dependent total fluxes (i.e., the sum of gravitational fluxes and pressure-gradient fluxes) on the Cauchy boundary; (7) automatically determine variable boundary conditions of evaporation, infiltration, or seepage on the soil-air interface; (8) provide two options for treating the mass matrix (consistent and lumping); (9) provide three alternatives for approximating the time derivative term (Crank-Nicolson central difference, backward difference, and mid-difference); (10) give three options (exact relaxation, underrelaxation, and overrelaxation) for estimating the nonlinear matrix; (11) automatically reset the time step size when boundary conditions or source/sinks change abruptly; and (12) check mass balance over the entire region for every time step. The modelmore » is verified with analytical solutions and other numerical models for three examples.« less

Implementation of Design Sensitivity Analysis with Existing Finite Element Codes

Abstract: A numerical method is presented to implement structural design sensitivity analysis theory, using the versatility and convenience of existing finite element structural analysis programs. Design variables such as thickness and cross-sectional areas of components of individual members and built-up structures are considered. Structural performance functionals considered include displacement and stress. The method is also applicable for eigenvalue problem design sensitivity analysis. It is shown that calculations can be carried out outside existing finite element codes, using postprocessing data only. Thus, design sensitivity analysis software does not have to be embedded in an existing finite element code. Feasability of the method is shown through analysis of several problems, including a built-up structure. Accurate design sensitivity results are obtained without the uncertainty of numerical accuracy associated with selection of finite difference perturbations.

Averaging of difference schemes

Abstract: The author considers a natural class of difference equations whose coefficients have micro-inhomogeneities. A general compactness theorem is established, asserting that the solutions of these difference equations can converge only to solutions of differential equations as the lattice is refined. Difference equations with random micro-inhomogeneous coefficients are studied separately; their averaging properties are determined.Bibliography: 12 titles.

Calculation Of Light Scatter From Structures On Silicon Surfaces

Abstract: This paper demonstrates numerical solutions of Maxwell's equations for laser diffraction and scattering from submicron objects on silicon wafers. Discrete numerical solvers using time domain, explicit finite element (FE) and finite difference (FD) methods have been implemented and vectorized for supercomputers. Both 2-D and 3-D FE and FD simulations have been performed for fibers and spheres (.5 to 1.6 microns) in free-space and on bare silicon, illuminated at normal incidence by a He-Ne laser beam. The wave solvers were verified against exact solutions for cylinders and spheres in free-space. Laboratory scattering experiments have been conducted with latex spheres on bare silicon, and the results compared to the corresponding numerical simulations. Simulations of glass and silicon fibers on a layer of photoresist over silicon examined differences in exposure patterns as well as the implementation of bleaching in nonlinear calculations. The results indicate that there is no inherent limitation to rigorously simulating many of the optical problems associated with IC fabrication, inspection, and design. The idea here is not to obtain just "exact" solutions for unsolved problems, but rather to provide the analyst with a broader capability for numerical experiments and simulations. In this way computers and mathematical algorithms can augment ad hoc approximations and laboratory experiments. The ultimate goal of this research is to simulate imaging systems and resist exposure patterns, and incorporate some of these results in process simulation codes like SAMPLE.

CONDIF: A modified central‐difference scheme for convective flows

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.

Numerical Methods for Semilinear Parabolic Equations

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.