Showing papers on "Finite difference published in 1985"

The boundary value problems of mathematical physics

Abstract: I Preliminary Considerations.- II Equations of Elliptic Type.- III Equations of Parabolic Type.- IV Equations of Hyperbolic Type.- V Some Generalizations.- VI The Method of Finite Differences.

Curvature and the evolution of fronts

Abstract: The evolution of a front propagating along its normal vector field with speedF dependent on curvatureK is considered. The change in total variation of the propagating front is shown to depend only ondF/dK only whereK changes sign. Analysis of the caseF(K)=1−eK, where e is a constant, shows that curvature plays a role similar to that of viscosity in Burgers equation. For e=0 and non-convex initial data, the curvature blows up, corners develop, and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity. We then numerically show that the solution as e goes to zero converges to the constructed weak solution. Numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds. As a remedy, we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution.

Artificial Dissipation Models for the Euler Equations

Abstract: Various artificial dissipation models that are used with central difference algorithms for the Euler equations are analyzed for their effect on accuracy, stability, and convergence rates. In particular, linear and nonlinear models are investigated using an implicit approximate factorization code (ARC2D) for transonic airfoils. Fully implicit application of the dissipation models is shown to improve robustness and convergence rates. The treatment of dissipation models at boundaries will be examined. It will be shown that accurate, error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model. I. Introduction T HE solution of the Euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme. This is because the Euler equations do not provide any natural dissipation mechanism (such as viscosity in the Navier-Stokes equations) that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks. A variety of numerical algorithms and computer codes for the Euler equations have been developed. Methods such as MacCormack's1 explicit

Transport of Immiscible Fluids Within and Below the Unsaturated Zone: A Numerical Model

Abstract: A numerical model is developed that describes the simultaneous flow of water and a second immiscible fluid under saturated and unsaturated conditions in porous media. The governing equations are a simplified subset of the three-phase flow equations commonly used in petroleum reservoir simulation. The simplification is analogous to that used to derive the Richard's equation for the flow of water in the unsaturated zone. The assumption that pressure gradients in the air phase are negligible leads to two partial differential equations. The proposed formulation is posed in terms of volumetric water saturation and fluid pressure in the immiscible fluid. The two-dimensional equations for flow in a vertical plane are approximated by finite differences. The fully implicit equations are solved by a direct matrix technique and Newton-Raphson iteration on nonlinear terms. The resulting numerical model is potentially applicable to many problems associated with immiscible contaminants in groundwater. Unfortunately, data such as relative permeabilities and capillary pressures for the types of fluids and porous materials present in hazardous waste sites are not readily available. As this type of data becomes available and field investigation techniques improve, applications of this type of model will become more practical. Examples are used to demonstrate themore » potential application of the model and sensitivity of results to fluid properties.« less

A modular system of algorithms for unconstrained minimization

Abstract: We describe a new package, UNCMIN, for finding a local minimizer of a real valued function of more than one variable. The novel feature of UNCMIN is that it is a modular system of algorithms, containing three different step selection strategies (line search, dogleg, and optimal step) that may be combined with either analytic or finite difference gradient evaluation and with either analytic, finite difference, or BFGS Hessian approximation. We present the results of a comparison of the three step selection strategies on the problems in More, Garbow, and Hillstrom in two separate cases: using finite difference gradients and Hessians, and using finite difference gradients with BFGS Hessian approximations. We also describe a second package, REVMIN, that uses optimization algorithms identical to UNCMIN but obtains values of user-supplied functions by reverse communication.

Numerical simulation of three‐dimensional flow in a cavity

Abstract: SUMMARY Previous three-dimensional simulations of the lid-driven cavity flow have reproduced only the most general features of the flow. Improvements to a finite difference code, REBUFFS, have made possible the first completely successful simulation of the three-dimensional lid-driven cavity flow. The principal improvement to the code was the incorporation of a modified QUICK scheme, a higher-order upwind finite difference formulation. Results for a cavity flow at a Reynolds number of 3200 have reproduced experimentally observed Taylor-Gortler-like vortices and other three-dimensional effects heretofore not simulated. Experimental results obtained from a unique experimental cavity facility validate the calculated results.

Computationally Efficient Algorithms for Parameter Estimation and Uncertainty Propagation in Numerical Models of Groundwater Flow

Abstract: Finite difference and finite element methods are frequently used to study aquifer flow; however, additional analysis is required when model parameters, and hence predicted heads are uncertain. Computational algorithms are presented for steady and transient models in which aquifer storage coefficients, transmissivities, distributed inputs, and boundary values may all be simultaneously uncertain. Innovative aspects of these algorithms include a new form of generalized boundary condition; a concise discrete derivation of the adjoint problem for transient models with variable time steps; an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation; and a new efficient first-order second-moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values. The techniques are presented in matrix form, and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations. Details of matrix structures are provided for a two-dimensional linear triangular finite element model.

Lagrangian Statistical Simulation of Concentration Mean and Fluctuation Fields.

Abstract: Lagrangian statistical (Monte Carlo) simulations of the mean and fluctuating concentration fields due to turbulent dispersion are critically reviewed. Attention has been restricted to work in which particle trajectories are modeled directly, which in effect means simulations based on the Langevin equation (or in finite difference form, Markov-chains) and its generalization. The material covered has been selected and presented so as to achieve an orderly progression from simple to more complex turbulent flows. At present this field involves many heuristic modeling assumptions and an attempt is made to justify some of these by appeal to special and limiting cases.

Review of Numerical Methods for the Analysis of Arbitrarily-Shaped Microwave and Optical Dielectric Waveguides

Abstract: This paper presents are view of the numerical methods for the analysis of the homogeneous and inhomogeneons,isotropic and anisotropic, microwave and optical dielectric waveguides with arbitrarily-shaped cross sections.The characteristics of various methods are compared,and a set of qualittative criteria to guide the selection of an appropriate method for a given problem is proposed. The main approaches discussed are those of point matching, integral equations, finite difference, and finite element.

A comparison of laboratory measurements and numerical simulations of baroclinic wave flows in a rotating cylindrical annulus

Abstract: Quantitative and qualitative comparisons are made between laboratory measurements of rotating annulus flows and corresponding numerical model simulations. Two laboratory annuli, of similar dimensions but differing in instrumentation, are used. One contains a thermocouple array for temperature measurement: the other contains no sensor array but the working fluid is seeded with minute neutrally buoyant beads (600 mm diameter) which enable the horizontal velocity field to be measured. Each annulus has a rigid insulating lid in contact with the working fluid. the numerical model is a finite difference formulation based on the Navier-Stokes equations for baroclinic flow of a Boussinesq liquid. Although the atmosphere and the laboratory annulus are both rotating baroclinic fluid systems, the forcing processes acting in the annulus are much simpler than those acting in the atmosphere, and may be accurately represented by established formulae: under a wide range of conditions no parametrizations of subgrid-scale dynamical and diabatic processes are required. Comparison of numerical model results with laboratory measurements therefore enables the explicit dynamical formulation of numerical models of rotating, baroclinic flow to be verified to an extent which would be very difficult, if not impossible, to achieve using atmospheric data. Detailed quantitative comparisons for a steady wave flow reveal good agreement for major features of the temperature and horizontal flow fields, although a significant discrepancy in total heat flux is found. Qualitative comparisons are made by investigating the ability of the numerical model to reproduce the main flow types and phenomena of the laboratory system. Numerical simulations of intransitivity, hysteresis, wavenumber transitions, amplitude vacillation and a weak structural vacillation are described. Several suggestions for further comparative studies are made in conclusion.

Grading functions and mesh redistribution

Abstract: We consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two-point boundary-value problems by finite difference or finite element methods. An analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary-value problems in one dimension. Error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria.

Refraction-diffraction model for linear water waves

Abstract: A numerical model is presented that predicts the transformation of monochromatic waves over complex bathymetry and includes both refractive and diffractive effects. Finite difference approximations are used to solve the governing equations, and the solution is obtained for a finite number of rectilinear grid cells that comprise the domain of interest. Model results are compared with data from two experimental tests, and the capability and utility of the model for real coastal applications are illustrated by application to an ocean inlet system.

Large-scale computations in fluid mechanics

Abstract: Part 1: Semi-Lagrangian advective schemes and their use in meteorological modeling by J. R. Bates Adaptive mesh refinement for hyperbolic equations by M. J. Berger Averaged multivalued solutions and time discretization for conservation laws by Y. Brenier Computing with high-resolution upwind schemes for hyperbolic equations by S. R. Chakravarthy and S. Osher Mathematical modeling of the steam-water condensation in a condenser by C. Conca Some theoretical and computational considerations of atmospheric fronts by M. J. P. Cullen and R. J. Purser Numerical solution of Euler's equation by perturbed functionals by S. K. Dey A mathematical model and numerical study of platelet aggregation during blood clotting by A. L. Fogelson Computational methods for discontinuities in fluids by O. A. McBryan Problems and numerical methods of the incorporation of mountains in atmospheric models by F. Mesinger and Z. I. Janjic Discrete shocks for systems of conservation laws by D. Michelson Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems by D. Michelson A mixed finite element method for $3d$ Navier-Stokes equations by J. C. Nedelec Stability of the flow around a circular cylinder to forced disturbances by V. A. Patel Some contributions to the modelling of discontinuous flows by P. Roe Techniques for numerical simulation of large-scale eddies in geophysical fluid dynamics by R. Sadourny Finite difference techniques for nonlinear hyperbolic conservation laws by R. Sanders Conforming finite element methods for incompressible and nearly incompressible continua by L. R. Scott and M. Vogelius Vortex methods and turbulent combustion by J. A. Sethian Numerical solutions of rotating internal flows by C. G. Speziale High resolution TVD schemes using flux limiters by P. Sweby Stability of hyperbolic finite-difference models with one or two boundaries by L. N. Trefethen Matching the Navier-Stokes equations with observations by T. Gal-Chen Dynamics of flame propagation in a turbulent field by A. F. Ghoniem New stability criteria for difference approximations of hyperbolic initial-boundary value problems by M. Goldberg and E. Tadmor A modified finite element method for solving the incompressible Navier-Stokes equations by P. M. Gresho Computational fusion magnetohydrodynamics by R. C. Grimm A finite amplitude eigenmode technique to solve and analyze nonlinear equations by R. Grotjahn Numerical boundary conditions by B. Gustafsson Numerical simulation in three space dimensions of time-dependent thermal convection in a rotating fluid by D. H. Hathaway and R. C. J. Somerville Static rezone methods for tensor-product grids by J. M. Hyman and M. J. Naughton A nonoscillatory shock capturing scheme using flux limited dissipation by A. Jameson Part 2: The use of spectral techniques in numerical weather prediction by M. Jarraud and A. P. M. Baede Improved flux calculations for viscous incompressible flow by the variable penalty method by H. Kheshgi and M. Luskin TVD schemes in one and two space dimensions by R. J. LeVeque and J. B. Goodman Upwind-difference methods for aerodynamic problems governed by the Euler equations by B. van Leer An MHD model of the earth's magnetosphere by C. C. Wu Application of TVD schemes for the Euler equations of gas dynamics by H. C. Yee, R. F. Warming, and A. Harten Recent applications of spectral methods in fluid dynamics by T. A. Zang and M. Y. Hussaini.

Finite difference model for vertical axis wind turbines

Abstract: A numerical technique based on Patankar's "SIMPLER" algorithm is developed to determine the flow characteristics and performance of a two-dimensional vertical axis wind turbine. The conservation of mass and momentum equations are solved using a finite difference procedure without the necessity of introducing an irrotationality constraint. The computational domain is subdivided into control volumes in cylindrical coordinates and the turbine blades are modeled as a porous cylindrical shell of one control volume thickness. The characteristics of the turbine are computed and compared with previous investigations. The results show a very good agreement.

Monotone iterative methods for finite difference system of reaction-diffusion equations

Abstract: This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.

An efficient algorithm for finite-difference analyses of heat transfer with melting and solidification

Abstract: A simple algorithm incorporating the equivalent heat capacity model is described for the finite-difference heat transfer analysis involving melting and solidification. The latent heat of fusion is ...

Numerical and experimental study of driven flow in a polar cavity

Abstract: SUMMARY A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function-vorticity form of the governing equations was approximated by upwind or central finite-differences. Both types of finite-difference approximations were solved by a multi-grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0 1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.

A new explicit method for the diffusion-convention equation

Abstract: We consider here the finite difference approximation to the diffusion-convection equation, from which new explicit formulation are obtained which are asymmetric. These explicit schemes can then be used to develop a new class of methods called Group Explicit as introduced in [2]. Theoretical aspects of the stability, consistency, convergence and truncation errors of this new class of methods is briefly discussed and numerical evidence presented to confirm our recomendations.

Selecting step sizes in sensitivity analysis by finite differences

Abstract: This paper deals with methods for obtaining near-optimum step sizes for finite difference approximations to first derivatives with particular application to sensitivity analysis. A technique denoted the finite difference (FD) algorithm, previously described in the literature and applicable to one derivative at a time, is extended to the calculation of several simultaneously. Both the original and extended FD algorithms are applied to sensitivity analysis for a data-fitting problem in which derivatives of the coefficients of an interpolation polynomial are calculated with respect to uncertainties in the data. The methods are also applied to sensitivity analysis of the structural response of a finite-element-modeled swept wing. In a previous study, this sensitivity analysis of the swept wing required a time-consuming trial-and-error effort to obtain a suitable step size, but it proved to be a routine application for the extended FD algorithm herein.

Numerical Prediction of Turbulent Flow over Surface-Mounted Ribs

Abstract: Turbulent flows over surface-mounted thin and thick ribs are investigated using a new computational method for incompressible recirculating flows. The method employs a differencing scheme which simultaneously satisfies the requirements of low numerical diffusion and positivity of coefficients. The effects of turbulence are modeled by a variant of the k-e turbulence model incorporating curvature corrections. The governing equations are solved by an efficient pressure-implicit split-operator algorithm. The predictions are compared with experimental data and previous calculations. The present results compare satisfactorily with most of the measurements and show a definite improvement over previous results. Nomenclature A = finite difference coefficient Ci,c2,cfji,ak,ffe,K = turbulence constants: 1.44, 1.92, 0.09, 1.0, *?/[(c2-ct)tf ],0.4187

Sensitivity calculations for iteratively solved problems

Abstract: The calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated. A modified finite difference procedure is presented which improves the accuracy of the calculated derivatives. The procedure is demonstrated for a simple algebraic example as well as an element-by-element preconditioned conjugate gradient iterative solution technique applied to truss examples.

Spectral methods for the Euler equations. II - Chebyshev methods and shock fitting

Abstract: The Chebyshev spectral collocation method for the Euler gas-dynamic equations is described. It is used with shock fitting to compute several two-dimensional, gas-dynamic flows. Examples include a shock-acoustic wave interaction, a shock/vortex interaction, and the classical blunt body problem. With shock fitting, the spectral method has a clear advantage over second order finite differences in that equivalent accuracy can be obtained with far fewer grid points.