Showing papers on "Finite difference published in 1983"
TL;DR: In this paper, the authors used mesh refnement and extrapolation to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls.
Abstract: Details are given of the computational method used to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls. Second-order, central difference approximations were used. Mesh refnement and extrapolation led to solutions for 103⩽Ra⩽10 6 which are believed to be accurate to better than 1 per cent at the highest Rayleigh number and down to one-tenth of that at the lowest value.
2,529 citations
TL;DR: In this paper, a new way of implementing radiation boundary conditions in finite difference schemes is reported, where instead of prescribing the incident field at the model boundary, waves are generated inside the boundary.
350 citations
TL;DR: In this article, the convergence of general approximation schemes to the Hamilton-Jacobi type is discussed and error estimates are obtained for general explicit and implicit finite difference schemes with error estimates.
306 citations
01 May 1983
TL;DR: In this paper, the class of one-dimensional stretching functions used in finite-difference calculations is studied for solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis.
Abstract: Abstract The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. It was first employed by Thomas et al. ( AIAA J. 10 (1972), 887). The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the one-dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The special case where the slopes were both equal and greater than one was first employed by Roberts. The general two-sided function has many applications in the construction of finite-difference grids. Examples of such applications are found in the listed references.
294 citations
TL;DR: The class of one-dimensional stretching functions used in finite-difference calculations is studied in this paper, for solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis.
282 citations
TL;DR: This article describes a time-domain finite-difference algorithm for solving Maxwell's equations in generalized nonorthogonal coordinates that is similar to conventional leapfrog-differencing schemes, but now an additional leapfrogging between covariant and contravariant field representations becomes necessary.
Abstract: This article describes a time-domain finite-difference algorithm for solving Maxwell's equations in generalized nonorthogonal coordinates. We believe this approach would be most useful for applications where a uniform, uncurved, but oblique, meshing scheme could be applied in lieu of staircasing. This algorithm is similar to conventional leapfrog-differencing schemes, but now an additional leapfrogging between covariant and contravariant field representations becomes necessary.
282 citations
TL;DR: A fully three-dimensional code has been written to compute the motion of a towed cable based on a robust and stable finite difference approximation to the differential equations derived from basic dynamics.
231 citations
TL;DR: In this article, a finite difference procedure that reflects the dominance of convection in incompressible flow in porous media is developed. But this method is not suitable for the case of two-phase, incompressibly flow.
Abstract: Two-phase, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Convection physically dominates diffusion, and the object of this paper is to develop a finite difference procedure that reflects this dominance. The pressure equation, which is elliptic in appearance, is discretized by a standard five-point difference method. The concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms. A convergence analysis is given for the method.
163 citations
TL;DR: In this paper, an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations is presented.
Abstract: The Osher algorithm for solving the Euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations. A first-order accurate version of the numerical method was derived by Osher circa 1980 for the one-dimensional non-isentropic Euler equations in Cartesian coordinates. In this paper, the extension of the scheme to arbitrary two-dimensional geometries is explained. Results are then presented for several example problems in one and two dimensions. Future work will include extension of the method to second-order accuracy and the development of implicit time differencing for the Osher algorithm.
160 citations
TL;DR: In this article, new explicit methods for the finite difference solution of a parabolic PDE are derived using stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages.
Abstract: In this paper, new explicit methods for the finite difference solution of a parabolic partial differential equation are derived. The new methods use stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages. By judicious use of alternating this strategy on the grid points of the domain results in an algorithm which possesses unconditional stability. The merit of this approach results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method is discussed and the results of numerical experiments presented.
150 citations
TL;DR: In this article, it was shown that standard monotone difference schemes, satisfying a fairly unrestrictive CFL condition, converge to the "correct" physical solution even in the case when a non-uniform spatial mesh is employed.
Abstract: Monotone finite difference schemes used to approximate solutions of scalar conservation laws have the advantage that these approximations can be proved to converge to the proper solution as the mesh size tends to zero. The greatest disadvantage in using such approximating schemes is the computational expense encountered since monotone schemes can have at best first order accuracy. Computation savings and effective accuracy could be gained if the spatial mesh were refined in regions of expected rapid solution variation. In this paper we prove that standard monotone difference schemes, (satisfying a fairly unrestrictive CFL condition), converge to the "correct" physical solution even in the case when a nonuniform spatial mesh is employed.
TL;DR: It is shown that the problem of finding the best of the Curtis, Powell and Reid type algorithms is NP-complete, and then two procedures for estimating the Jacobian that may use fewer function evaluations are proposed.
Abstract: When finding a numerical solution to a system of nonlinear equations, one often estimates the Jacobian by finite differences. Curtis, Powell and Reid [J. Inst. Math. Applics.,13 (1974), pp. 117–119] presented an algorithm that reduces the number of function evaluations required to estimate the Jacobian by taking advantage of sparsity. We show that the problem of finding the best of the Curtis, Powell and Reid type algorithms is NP-complete, and then propose two procedures for estimating the Jacobian that may use fewer function evaluations.
TL;DR: In this paper, a mathematical description of groundwater flow in fractured aquifers is presented, where the Galerkin finite element method is used to approximate the equation of flow in the fracture domain and a convolution integral is employed to describe the leakage flux between the fractures and porous matrix blocks.
Abstract: A mathematical description of groundwater flow in fractured aquifers is presented. Four alternative conceptual models are considered. The first three are based on the dual-porosity approach with different representations of fluid interactions between the fractures and porous matrix blocks, and the fourth is based on the discrete fracture approach. Two numerical solution techniques are presented for solving the governing equations associated with the dual-porosity flow models. In the first technique the Galerkin finite element method is used to approximate the equation of flow in the fracture domain and a convolution integral is used to describe the leakage flux between the fractures and porous matrix blocks. In the second the Galerkin finite element approximation is used in conjunction with a one-dimensional finite difference approximation to handle flow in the fractures and matrix blocks, respectively. Both numerical techniques are shown to be readily amendable to the governing equations of the discrete fracture flow model. To verify the proposed numerical techniques and compare various conceptual models, four simulations of a problem involving flow to a well fully penetrating a fractured confined aquifer were performed. Each simulation corresponded to one of the four conceptual models. For the three simulated cases, where analytical solutions are available, the numerical and the analytical solutions were compared. It was found that both solution techniques yielded good results with relative coarse spatial and temporal discretizations. Greater accuracy was achieved by the combined finite element-convolution integral technique for early time values at which steep hydraulic gradients occurring near the fracture-matrix interface could not be accommodated by the linear finite difference approximation. Finally, the results obtained from the four simulations are compared and a discussion is presented on practical implications of these results and the utility of various flow models.
TL;DR: In this paper, a numerical method for determining the flame speed and the structure of freely propagating, adiabatic flames is discussed. But the method is computationally faster than other methods, and it is potentially more accurate because it employs an adaptive gridding strategy.
Abstract: Abstract–We discuss a numerical method for determining the flame speed and the structure of freely propagating, adiabatic flames. The method uses a finite difference procedure in which the nonlinear difference equations are solved by a damped, modified, Newton method. This approach is in contrast to the traditional approach of solving a related transient problem until a steady-state solution i5 achieved. Our method is computationally faster than other methods, and it is potentially more accurate because it employs an adaptive gridding strategy. We demonstrate its use for the determination of hydrogen-air flame speeds.
TL;DR: This work presents an algorithm for computing a set of intervals to be used in a forward-difference approximation of the gradient, and shows how certain “standard” choices for the finite-Difference interval may lead to poor derivative approximations for badly scaled problems.
Abstract: When minimizing a smooth nonlinear function whose derivatives are not available, a popular approach is to use a gradient method with a finite-difference approximation substituted for the exact gradient In order for such a method to be effective, it must be possible to compute “good” derivative approximations without requiring a large number of function evaluations Certain “standard” choices for the finite-difference interval may lead to poor derivative approximations for badly scaled problems We present an algorithm for computing a set of intervals to be used in a forward-difference approximation of the gradient
TL;DR: In this paper, it was shown that the GKS stability criterion has a physical interpretation in terms of group velocity, and that if the finite difference model together with its boundary conditions can support a set of waves at the boundary with group velocities pointing into the field then it is unstable.
TL;DR: In this article, the minimum truncation error of one and two-dimensional Burgers' equations with moderate to severe internal and boundary gradients was compared with three, five-, and seven-point finite difference schemes with linear, quadratic, and cubic rectangular finite element schemes.
TL;DR: In this paper, implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady Euler equations, based on the mathematical theory of characteristics for hyperbolic systems of equations.
Abstract: Implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady Euler equations. This new boundary point treatment is based on the mathematical theory of characteristics for hyperbolic systems of equations. Along with the theoretical background, the practical application of the method to several types of boundaries is also explained using several examples. The specific boundary conditions covered include subsonic inflow and outflow, surface tangency, and shock waves. The example problems include one-dimensional Laval nozzle flow, dual-throat rocket engine nozzle flow, and supersonic flow past a sphere. The implicit boundary treatment permits the use of large time steps allowing the finite difference algorithm to converge to the asymptotic steady state much faster than schemes that use explicitly applied boundary conditions. At least an order of magnitude increase in computational speed is demonstrated in the examples shown.' Background T HE growing popularity of solutions to the Euler equations in transonics and their continued application in supersonics have increased the need for quicker solutions. The potential of implicit schemes in this direction has not been fully exploited for want of correct, implicit application of boundary conditions. The predominant use of implicit algorithms for the Navier-Stokes equations has partly been responsible for the neglect of implicit boundary point treatment for the Euler equations. Thus, there is a need for correct and stable procedures for the easy implicit application of boundary conditions. Such methods will serve the two purposes of 1) reaching time-asymptotic steady state faster and 2) permitting a time step for truly unsteady flow that is not necessarily restricted by the CFL stability criterion but is based upon the magnitude of the transients. For clues and information on how to construct such boundary condition procedures, one must turn to the mathematical theory of characteristi cs for hyperbolic systems of equations. The unsteady Euler equations belong to this category. The theory for hyperbolic systems is rich with information on signal propagation directions. The characteristics theory clearly points to the number of boundary conditions that may and need be prescribed without overdetermining the solution. Boundary condition procedures based on this theory have been known and applied for several years by Kentzer, 1 Porter and Coakley,2 de Neef, 3 and others. In earlier work by this author,4'5 easily understood and implementable methods for boundary point treatment were presented. However, all of the above techniques were developed for explicit finite difference schemes. It seems that it must be easy to extend such methodologies based on mathematical theory for hyperbolic systems to implicit finite difference schemes, and indeed, it is simple enough. The rest of this paper describes such implicit boundary condition procedures. The given examples illustrate in detail the application of the proposed methodology to specific types of boundaries and demonstrate the merits of the new scheme.
TL;DR: In this article, the origin and nature of spurious oscillation modes that appear in mixed finite element methods are examined and a modal analysis for the one-dimensional problem is developed, which provides an explanation of the phenomenon in question and permits the cause of the very complex behavior of spurious modes observed in numerical experiments with the shallow water equations and Navier-Stokes equations.
TL;DR: In this paper, a rapidly growing instability in energy-enstrophy conserving finite difference forms of the primitive equations is described, and modifications necessary to control the instability are discussed.
Abstract: A rapidly growing instability in energy-enstrophy conserving finite difference forms of the primitive equations is described. The instability is unusual in that it is purely internal. It arises because the linearized forms of the equations do not conserve momentum. The modifications necessary to control the instability are discussed.
TL;DR: In this paper, a finite difference or finite element discretization scheme is used to obtain a set of simultaneous linear equations, which are then solved using the eigenvalues and eigenvectors of a matrix.
Abstract: A procedure for solving the differential groundwater flow equation is presented herein. Using a finite difference or finite element discretization scheme, a set of simultaneous linear equations is obtained. The eigenvalues and eigenvectors of a matrix, which is a function of the coefficients of the set, are the key to the solution. A vector L is obtained straightforwardly by combining the eigenvector matrix A, the eigenvalue vector α, the pumping vector P, and the initial head vector H. Vector L, which depends on time, can be expressed simply and explicitly as a function of the eigenvalues. Piezometric heads can be obtained by combining A and L. L is the only vector that needs to be computed as P changes with time. In this way, influence functions of a piezometric head, flow velocity and flow depletion of a stream connected with the aquifer under a unit stress, can be obtained explicitly and continuously in time. The method can be applied to confined as well as to leaky aquifers and to one-, two-, or three- dimensional linear models. Its main advantage lies in the fact that it is unnecessary to repeatedly solve a matrix for every time increment. The method is particularly useful for groundwater management problems in which a large number of alternatives have to be evaluated.
TL;DR: In this paper, finite difference and reflectivity synthetic seismograms are compared for laterally homogeneous seafloor models with step and ramp discontinuities in elastic properties, given suitable numerical parameters in each case excellent agreement can be obtained.
Abstract: Summary
Finite difference and reflectivity synthetic seismograms are compared for laterally homogeneous seafloor models with step and ramp discontinuities in elastic properties. Given suitable numerical parameters in each case excellent agreement can be obtained.
For step discontinuities between a liquid and a solid the explicit finite difference formulation of the elastic wave equation for heterogeneous media is unstable. It is necessary to introduce the boundary conditions specifically at the interface and suitable formulations correct to first and second order in the space increment are given. For ramp discontinuities between a liquid and a solid the finite difference formulation for heterogeneous media is stable and agrees with the reflectivity method when, in the latter case, the gradient is approximated by homogeneous layers thinner than one-fifth of the minimum compressional wavelength at the upper half-power frequency.
For a step discontinuity between two solids the explicit finite difference formulation of the elastic wave equation for heterogeneous media, although stable, is inaccurate and boundary conditions must again be specifically introduced.
TL;DR: In this paper, two numerical methods for solving the problem of solute transport in unsteady flow in unconfined aquifers are studied: the method of characteristics (MOC) based on the finite difference method (FDM), and the finite element method(FEM).
Abstract: Two numerical methods for solving the problem of solute transport in unsteady flow in unconfined aquifers are studied. They are the method of characteristics (MOC) based on the finite difference method (FDM), and the finite element method (FEM). The FEM is further subdivided into four schemes: moving mesh, pseudo-Lagrangian (FEM1); stationary mesh, pseudo-Lagrangian (FEM2); pseudo saturated-unsaturated, Eulerian (FEM3); and non-stationary element, Eulerian (FEM4).
Experiments on a one-dimensional flow case are performed to illustrate the schemes and to determine the effect of discretization on accuracy. In two-dimensional flow the above methods are compared with experimental results from a sand box model. Results indicate that for a similar degree of accuracy, the FEM requires less computational effort than the MOC. Among the four FEM schemes, FEM4 appears to be most attractive as it is the most efficient and most convenient to apply.
TL;DR: In this article, the authors present a collection of stability results for finite difference approximations to the advection-diffusion equation, which are derived from a uniform framework based on the Schur-Cohn theory of Simple von Neumann Polynomials and are necessary and sufficient for the stability of Cauchy problem.
Abstract: We present a collection of stability results for finite difference approximations to the advection-diffusion equation $u_t\ = a u_x\ + b u_{xx}$. The results are for centered difference schemes in space and include explicit and implicit schemes in time up to fourth order and schemes that use different space and time discretizations for the advective and diffusive terms. The results are derived from a uniform framework based on the Schur-Cohn theory of Simple von Neumann Polynomials and are necessary and sufficient for the stability of the Cauchy problem. Some of the results are believed to be new.
TL;DR: In this article, the Von Karman equations are used to represent the behavior of thin rectangular glass plates subjected to lateral pressures, and the finite difference technique with iteration is used to solve the nonlinear equations.
Abstract: Von Karman equations are used to represent the behavior of thin rectangular glass plates subjected to lateral pressures. The glass plate is assumed to be simply supported without any restraint for inplane displacements at the supports. The finite difference technique with iteration is used to solve the nonlinear equations. The number of iterations are reduced substantially by using an under relaxation parameter. Results are compared with other available solutions for accuracy as well as efficiency.
TL;DR: In this article, the use of numerical methods in simulating and solving problems that arise in the electroheat industry is reviewed. But the focus is on the coupled electrothermal and induction stirring problems that are typical of this industry.
Abstract: This paper reviews the use of numerical methods in simulating and solving problems that arise in the electroheat industry. Particular attention is given to the coupled electrothermal and induction stirring problems that are typical of this industry. Following a brief review of the nature of electrothermal problems, the Finite Difference, Volume Integral Equation and Finite Element simulation techniques are critically examined. It is shown that each technique has a definite role and each is illustrated with practical examples. A brief discussion of unsolved problems is presented.
TL;DR: Several finite difference-schemes for approximating solutions of initial value problems associated with systems of linear hyperbolic differential equations and extensions to several space-like dimensions by splitting methods are considered.
TL;DR: This paper describes the procedure for analyzing nonlinear transient electromagnetic phenomena in electrical machines and devices with the finite element method and includes illustrative examples of application of the methods to one-and two-dimensional time-dependent eddy current problems in a conducting slab, a rotating machine under asynchronous operation, and a three- phase bus-bar enclosure.
Abstract: This paper describes the procedure for analyzing nonlinear transient electromagnetic phenomena in electrical machines and devices with the finite element method. Two time integration methods are used which are based on (1) an implicit forward difference scheme and (2) the Crank-Nicholson method. The quasilinearization of the nonlinear matrix equation is handled by a simple chord iteration method in the former and the Newton-Raphson scheme in the latter. The associated variational expressions for the time-dependent diffusion equation are obtained in terms of energy-related functionals. The paper includes illustrative examples of application of the methods to one-and two-dimensional time-dependent eddy current problems in a conducting slab, a rotating machine under asynchronous operation, and a three- phase bus-bar enclosure.
01 Jan 1983
TL;DR: In this paper, a facility has been constructed to study shear-driven, recirculating flow, namely, a three-dimensional lid-driven cavity flow, and the experimental results are compared to simulations by two different numerical codes, one employing finite differences and one employing infinite elements.
Abstract: A facility has been constructed to study shear-driven, recirculating flow, namely, a three-dimensional lid-driven cavity flow. In the extant case, the cavity depth-to-width aspect ratio is 1:1, while the span-to-width aspect ratio is 3:1. A description of the circulation cell structure obtained by flow visualization is given for two Reynolds numbers with and without temperature-induced density stratification. The experimental results are compared to simulations by two different numerical codes, one employing finite differences and one employing finite elements. The numerical codes reproduce the overall behavior observed in the experiments, but fail to simulate observed longitudinal vortices. There are some significant differences also between the numerical results, highlighting the effects of using the HYBRID (upwind/central) differencing scheme in one code.
TL;DR: In this paper, a new mathematical model for analyzing mat foundations under static loading is developed based on Reissner's concept of a simplified elastic continuum, where horizontal normal and shear stresses are assumed to be zero throughout an elastic layer of finite thickness.
Abstract: A new mathematical model for analyzing mat foundations under static loading is developed based on Reissner's concept of a simplified elastic continuum. Horizontal normal and shear stresses are assumed to be zero throughout an elastic layer of finite thickness. The relationship between applied surface pressure and settlement is defined by a single differential equation that can be solved readily using finite differences. Coefficients in the differential equation are functions of the elastic parameters and thickness of the layer only. Solutions for a Young's modulus constant with depth, as well as varying lin early and with the square root of depth, are discussed. The Reissner Simplified Continuum is shown to offer substantially better correlation with exact theory of elasticity solutions than does the commonly used modulus of subgrade re action (Winkler) model.