Showing papers on "Finite difference published in 1974"

Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations

Abstract: Publisher Summary This chapter defines finite element and finite difference methods for hyperbolic partial differential equations. The advantage of the finite element method is that the resulting procedures are automatically stable and there is extreme flexibility in choosing the basic functions. Therefore, in very complicated domains or for problems with complicated interfaces, the method is the only feasible one. For hyperbolic partial differential equations it is essential to control the dispersion, dissipation, and the propagation of discontinuities. This is easily done by using suitable difference approximations. The main disadvantage of finite difference methods is that it may be difficult to handle boundaries properly.

Differences of fractional order

Abstract: Derivatives of fractional order, D af, have been considered extensively in the literature. However, little attention seems to have been given to finite differences of frac- tional order, A af. In this paper, a definition of differences of arbitrary order is presented, and A af is computed for several specific functions f (Table 2.1). We find that the operator A a is closely related to the contour integral which defines Meijer's G-function. A Leibniz rule for the fractional difference of the product of two functions is discovered and used to gen- erate series expansions involving the special functions.

An Initial Value Problem from Semiconductor Device Theory

Abstract: A system of three quasi-linear partial differential equations is considered, as a simplified model of the transport of mobile carriers in a semiconductor device. Assuming a convenient form of the boundary conditions, it is shown that the initial value problem is well-posed, and that the steady state solution is unique and stable. A finite difference approximation preserving reasonable bounds on the numerical solutions is also described.

Mesh selection for discrete solution of boundary problems in ordinary differential equations

Abstract: In order to use finite difference approximations with non-uniform meshes in boundary value problems, it is necessary to develop procedures for mesh selection. In this paper we extend techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes. By this term we mean meshes on which the local truncation error of the method is approximately constant in some norm. Improved error estimates for methods which use equidistributing meshes are obtained.

Three-level galerkin methods for parabolic equations*

Abstract: A priori error estimates are derived for algebraically linear discrete-time Galerkin methods for approximating solutions of nonlinear parabolic initial-boundary value problems. The methods discussed result from applying a Galerkin procedure in the space variables and using three- level finite difference approximations in the time variable.

Finite-amplitude thermal convection in a spherical shell

Abstract: The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and nonaxisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms.

Solution Schemes for Problems of Nonlinear Structural Dynamics

Abstract: A number of alternative solution schemes for problems of dynamic structural analysis involving large displacements and plastic deformations are compared. The equations of motion, derived from the finite element approximation, are integrated directly using a number of well-known time integration operators; namely, the Houbolt, Newmark, Wilson, and central difference operators. This latter operator serves as a basis for a comparison of accuracy and economy with all methods. Numerical results for nonlinear systems are used to indicate the true worth of integration operators whose properties have been determined theoretically by studying their behavior with linear systems only.

Theory of a Finite Difference Method on Irregular Networks

Abstract: A new theory is presented for the approximation of second order elliptic Dirichlet boundary value problems by a well-known finite difference scheme on irregular networks which cannot be handled by the finite element theory. Adequate regularity conditions are imposed on the network, a finite difference scheme is set up, and the convergence of its solution towards the solution of the continuous boundary value problems is established, as well as a bound for the discretization error. A numerical example illustrates the theory.

Semisimilar Solutions to Unsteady Boundary-Layer Flows Including Separation

Abstract: A study is made of a certain class of flows for which semisimilar solutions to the unsteady two-dimensional laminar boundary-layer equations may be obtained. It is shown that such solutions are possible when the velocity at the edge of the boundary layer, ud(x, f), is an arbitrary function of the variable £ where £ is either x/(l — Bt) or (x + Kt)/(l-Bt). If the constant K is choosen properly the latter case represents the same external flow as the former case but observed in a reference system moving with the unsteady separation point. It is possible, therefore, to study the unsteady separation phenomenon in this reference system where separation is most easily identified and analyzed. Numerical solutions have been obtained for an unsteady linearly retarded flow where ud(x, t) = U^ — A^. These solutions verify the Moore-Rott-Sears model for unsteady boundary-layer separation in which separation appears as a point of vanishing shear and velocity, within the boundary layer, in a reference system moving with separation. Solutions such as those presented here may also be used as test cases for the new numerical techniques where unsteady boundary layers are analyzed using three-dimensional finite difference techniques.

Finite Element Formulation and Solution of Contact-Impact Problems in Continuum Mechanics

Abstract: This report presents a general theory of contract-impact problems cast in a variational theorem suitable for implementation with the finite element method. The numerical scheme is described as is the structural analysis computer code in which it is contained.

Optimal Predictor-Corrector Method for Systems of Second-Order Differential Equations

Abstract: A very general class of predictor-corrector schemes for numerically integrating a system of equations of the type x + f(x, t) = 0 is examined. This class includes the schemes obtained by combining one explicit predictor with p implicit correctors (allowing the use of a different corrector for each evaluation). Both the predictor and the correctors are assumed to be two- or three-step operators. It is shown that all of the schemes within this class are conditionally stable, that is, are stable only if the time increment A/ does not exceed a critical value. Furthermore, it is shown that the central difference scheme is the optimal one in the sense that for all the schemes within the above class, the value of Ar divided by the number of evaluations of f(x, t) is never greater than for the central difference one. The same conclusions are valid for any scheme with a characteristic equation of the type p3+ d^p2 + d 2p + d3 = 0, where dk are polynomials in (coAr)2, where co is the largest natural frequency of the system.

Three‐dimensional computation of buoyant plumes

Abstract: The flow field resulting from the interaction of a stream with the much larger body of flowing water into which it debouches was analyzed. Local density of the water was expressed as a function of salinity and/or temperature. Three-dimensional unsteady conservation equations used to describe the interaction included the effects of buoyancy, inertia, and the difference in the hydrostatic heads of the two currents. Turbulence effects were modeled by using an effective eddy viscosity. For cases of interest, boundary conditions were essentially constant for the time required for the plume to converge to steady state. The conservation equations were solved with a time-dependent finite difference technique that continuously adjusted the flow field variables until a steady flow resulted. A sample case from a simulation of the South Pass of the Mississippi River revealed three-dimensional flow in the interaction region. Buoyancy near the river mouth induced a pair of standing vortices. Farther away from the mouth a strong crosscurrent dominated the flow. Numerical results agreed qualitatively with available field data.

The Construction and Comparison of Finite Difference Analogs of Some Finite Element Schemes

Abstract: 0. Summary. Three subjects are considered in this paper. First, the notion of the resolving power of approximation methods, i.e. the number of intervals (or function values) per wavelength necessary to attain a preassigned error when approximating a given frequency, permits the evaluation of the relative efficiency of three types of spatial differencing used in setting up differential-difference equations for the 1-periodic parabolic problem ut = uxx. The three types of high-order spatial schemes considered are (explicit) centered differencing, the super-convergent smooth spline-Galerkin schemes discovered by Thomee, and the very high-order implicit schemes (Mehrstellenverfahren) which generalize Numerov's method. Seven time discretizations are introduced, namely Euler, backward-Euler, duFort-Frankel, 2nd order explicit, trapezoidal, Calahan-Zlamal, and 4th order Pade. The computational work necessary to solve each full discretization is minimized, for given error requirements, by balancing the number of intervals per wavelength against the number of time intervals per eth-life. The resulting data is used to compare the relative efficiency of these finite element and finite difference methods. Secondly, some corresponding results for the hyperbolic problem ut = ux are briefly reviewed. Finally, as Numerov-trapezoidal differencing turns out to be almost optimal for the heat equation, a tridiagonal implicit difference formula which extends Numerov differencing to the general 2nd order linear differential operator in one space dimension is presented. The technique used in deriving this scheme inspires certain difference analogs of some finite element schemes. It also leads to a curious modification of the diamond difference scheme for ut = ux. This “alternating kite” scheme has O(Δ2 + Δx3) truncation error with no more computational work than the diamond scheme itself.

A Discussion of the Finite Difference Method in Computer Modelling of Electrical Conductivity Structures. A Reply to the Discussion by Williamson, Hewlett and Tammemagi

Abstract: It is apparent that the finite difference approximation to the Laplacian obtained by Williamson, Hewlett & Tammemagi (1974) differs from that given by Jones R: Pascoe (1971). The difference is related to the manner in which first derivatives are approximated in the two cases. Jones & Pascoe assunzed a central diflerence form for first derivatives from the outset, whereas the method of Williamson et al. implies a linear cornhination of backward and forward differences for first derivatives. It is instructive to examine the two approaches in greater detail. Following Jones & Pascoe, Taylor's theorem may bc used to obtain the field values around the point '0' as: for the real part of the field values and similar equations for the imaginary part (a misplaced minus sign which appeared in the earlier paper has been correctly located above). Combining the equations for f1 and f3 gives:

A Fully Implicit Finite Difference Approximation to the One-dimensional Wave Equation using a Cubic Spline Technique

Abstract: A cubic spline approximation to the wave equation is shown to produce a fully implicit finite difference representation, the more usual explicit and implicit approximations of this equation being shown as special cases of the formulation given here. The truncation error and stability condition are derived for the scheme produced, and the full computational procedure for the scheme's solution is developed.

Rapid finite-difference computation of subsonic and transonic aerodynamic flows

Abstract: Rapid iterative (or semidirect) computation methods are developed for the finite-difference solution of the nonlinear equations of subsonic and transonic aerodynamics. At each iteration, a fast, direct elliptic algorithm solves the entire computation field. In an application to subsonic flow over a lifting airfoil, the full nonlinear stream-function equation is solved. Finally, a direct Cauchy-Riemann solver is used for the nonlinear transonic small-disturbance equations for a biconvex airfoil. At M = 0.7, t/c = 0.1 (subcritical), three iterations on a 39 x 32 mesh (totaling 2.45 sec on an IBM 360/67 computer) obtain convergence within 0.1%. A slightly supercritical case requires seven iterations (6.75 sec) for convergence within 1%.

Application of finite element methods to thermohydrodynamic lubrication

Abstract: A finite element formulation is presented for the equations governing the steady thermohydrodynamic behaviour of liquid lubricated bearings. This formulation permits application of the iterative solution scheme to bearings of arbitrary geometry. A generalized Reynolds equation resulting from the combination of the mass and momentum conservation equations is cast into variational form and used to derive general finite element equations. The method of weighted residuals with Galerkin's criterion is used to generate finite element matrix equations for the thermal energy equation. In addition to the finite element formulation, a discussion of appropriate finite difference techniques is also given for problems without complex geometry. As an example, the formulations are applied to obtain numerical solutions for a three-dimensional sector thrust bearing operating in the thermohydrodynamic regime. Pressure, velocity and temperature distributions are give, and the thermohydrodynamic solutions are compared with the results of classical isothermal theory.

Numerical Solution of the Incompressible Navier-Stokes Equations in Doubly-Connected Regions

Abstract: A transient computational procedure for the numerical solution of the incompressible Navier-Stokes equations in doubly-connected domains is described. The tangential velocity boundary conditions are satisfied by introducing shear layers at the boundaries at each time-step. The requirement that the pressure be single-valued is explicitly included in the formulation. The computational procedure is applied to the viscous flow in the annulus between two eccentric cylinders when either of them is rotating. Transient and steady-state results are obtained for both Stokes flow (inertia neglected) and Navier-Stokes flow. Rotational speeds of the outer cylinder are explored for the entire extent of laminar flow and of the inner cylinder up to the point of incidence of Taylor vortices. The agreement between the steady-state numerical results and the exact analytical results for the Stokes flow is very good. It is believed that inclusion of inertia does not appreciably affect the solution accuracy.