Showing papers on "Finite difference published in 1972"
TL;DR: In this article, it was shown that the upwind difference scheme of formulating differential expressions, in problems involving transport by simultaneous convection and diffusion, is superior to the central differences scheme, when the local Peclet number of the grid is large.
Abstract: It is shown that the upwind difference scheme of formulating differential expressions, in problems involving transport by simultaneous convection and diffusion, is superior to the central differences scheme, when the local Peclet number of the grid is large. Even better schemes are derived and discussed. It is pointed out that the best finite differences analogues are found by approximating differential expressions as a whole, and that simple (e.g. one-dimensional) exact solutions form a useful, legitimate and independent source of these optimum algebraic formulae.
1,140 citations
TL;DR: In this paper, a class of conservative finite-difference approximations of the primitive equations is given for quasi-uniform spherical grids derived from regular polyhedrons, where the earth is split into several contiguous regions, and a coordinate system derived from central projections is used, instead of the spherical coordinate system, to avoid the use of inconsistent boundary conditions at the poles.
Abstract: A class of conservative finite-difference approximations of the primitive equations is given for quasi-uniform spherical grids derived from regular polyhedrons. The earth is split into several contiguous regions. Within each region, a coordinate system derived from central projections is used, instead of the spherical coordinate system, to avoid the use of inconsistent boundary conditions at the poles. The presence of artificial internal boundaries has no effect on the conservation properties of the approximations. Examples of conservative schemes, up to the second order in the case of a cube, are given. A selective damping operator is needed to remove the two-grid interval waves generated by the existence of internal boundaries.
314 citations
01 Jan 1972
TL;DR: In this paper, a technique for computing multidimensional time-dependent flow fields is presented, which avoids much of the inefficiency typically found in finite difference calculations by dividing the flow field into regions, each containing a mesh of general quadrilateral cells chosen to provide spatial resolution of local features of the flow.
Abstract: A technique is presented for computing multidimensional time-dependent flow fields that avoids much of the inefficiency typically found in finite difference calculations. The technique initially divides the flow field into regions, each containing a mesh of general quadrilateral cells chosen to provide spatial resolution of the local features of the flow. A finite difference operator of second order accuracy, consisting of a sequence of one-dimensional operators (each operating at near maximum Courant-Friedrich-Lewy number) is then constructed for each region. Numerical results illustrating the technique for inviscid flows about simple bodies that generate shock waves, embedded shock waves, and expansion fans are presented and compared with exact theory.
167 citations
TL;DR: In this paper, the Galerkin procedure was used to generate the approximating equations and the integrals in the resulting equations may be efficiently evaluated by using isoparametric quadrilateral elements and numerical integration.
Abstract: The groundwater flow equations may be solved using the Galerkin procedure to generate the approximating equations. The integrals in the resulting equations may be efficiently evaluated by using isoparametric quadrilateral elements and numerical integration. A comparison of solutions for an idealized problem obtained by using Galerkin techniques and finite difference techniques indicates they achieve approximately the same degree of accuracy. A field application of the two methods shows that the Galerkin procedure provided satisfactory solutions with far fewer nodes than were required for the finite difference approach. In selecting one of the above methods for a particular hydrologic problem the flexibility of the irregular subspaces used in the Galerkin approach must be weighed against the very efficient equation solving schemes applicable to the finite difference equations.
167 citations
TL;DR: Preliminary studies indicate that for many shell problems this variable grid technique will yield improved efficiency as well as a simple method for handling curved boundaries and varying stress patterns.
165 citations
TL;DR: With these bounds it becomes possible to study the effect of element geometry, the order of interpolation functions and other intrinsic and discretization parameters on Cn(K) and to isolate the factors that may lead to ill-conditioning.
Abstract: N a previous Note1 it has been shown (see also Refs. 2 and 3) that the spectral condition number Cn(K) of the global (stiffness) matrix K arising from a uniform mesh of finite elements (or of finite differences) discretization can be expressed by Cn(K) = cNes2m where 2m is the order of the differential equation and c a coefficient independent of Nes, the number of elements per side, but dependent on the order of the interpolation polynomials inside the element. This condition is "natural", since it is inherently associated with the approximation of the continuous problem by the discrete (algebraic) one. Nonuniform meshes of finite elements introduce many additional factors which may adversely affect the condition of the system. It is the purpose of this Note to describe a technique for establishing bounds on the condition number for irregular meshes of finite elements. With these bounds it becomes possible to study the effect of element geometry, the order of interpolation functions and other intrinsic and discretization parameters on Cn(K) and to isolate the factors that may lead to ill-conditioning. The matrix K is termed ill-conditioned when \Q~sCn(K) = 1, where s denotes the number of decimals in the computer. The bounds on Cn(K) are expressed in terms of the extremal eigenvalues of the element matrices. Since the element matrices are of restricted size, derivation of the bounds on Cn(K) as a function of the discretization parameters become straightforward for any problem and any element. Particular attention is focused on the possibility of improving the condition of the matrix by scaling. Bounds on the Extremal Eigenvalues
153 citations
TL;DR: The finite difference boundary value method for obtaining eigenvalues and eigenfunctions of the one-dimensional Schroedinger equation is discussed in this article, where the method is noniterative and may be applied to one dimensional problems on (- ∞, ∞) or to the radial equation on (0, ∾).
139 citations
TL;DR: The influence coefficient algorithm as mentioned in this paper is a simple, easily implemented, and rapidly convergent computational procedure for the solution of the parameter identification problem in unsteady open channel flow from field observations on stage hydrograph and velocity distribution at one or more points along the channel.
Abstract: This paper introduces the influence coefficient algorithm, a simple, easily implemented, and rapidly convergent computational procedure for the solution of the parameter identification problem in unsteady open channel flow from field observations on stage hydrograph and velocity distribution at one or more points along the channel. (Identification is a mathematical process whereby the parameters embedded in a differential equation defining a system are determined from observations of system input and output.) The parameters specifically chosen for identification are the two ‘friction slope’ characteristics: the channel roughness coefficient and the exponent of the hydraulic radius in the empirical friction slope relation, a number usually assumed to be 4/3. These parameters are not physically measurable and have to be determined from the solutions of the mathematical model using concurrent input and output measurements. This new procedure is related to both quasilinearization and gradient methods. Additionally, an effective formulation of the algorithm is shown to depend on certain stability and convergence features related to the finite difference solutions of the governing flow equations but often ignored or glossed over.
126 citations
TL;DR: In this paper, it was shown that the usual analysis for the implicit artificial viscosity of finite difference analogs of the linear advection equation is ambiguous, with different results obtained for transient and steady-state problems.
117 citations
109 citations
TL;DR: In this paper, the design of a general circulation model using the primitive equations in spherical form is described, including a statement of the finite difference forms used to integrate the system and explanations of the motives for unusual aspects of finite difference scheme.
Abstract: The design of a general circulation model using the primitive equations in spherical form is described, including a statement of the finite difference forms used to integrate the system and explanations of the motives for unusual aspects of the finite difference scheme. The model incorporates the hydrological cycle, topography, a simple scheme for the radiative exchanges and arrangements for the simulation of deep free convection (sub grid-scale) and for the representation of exchanges of momentum, sensible and latent heat with the underlying surface. An experiment performed with the model forms the subject of a separate paper.
TL;DR: In this paper, the fundamental and higher order hybrid modes in both single and coupled microstrip are investigated by the use of a graded finite difference mesh, and a spurious nonphysical class of solutions is found to exist in this and similar formulations, the characteristics of which are described.
Abstract: By means of finite difference methods, dispersion curves are obtained for the fundamental and higher order hybrid modes in both single and coupled microstrip. Structures of realistic proportions are investigated by the use of a graded finite difference mesh. Variational methods are used in deriving the finite difference equations. The higher order modes are found to be similar to LSM slab line modes. A spurious nonphysical class of solutions is found to exist in this and similar formulations, the characteristics of which are described.
TL;DR: An iterative approach which combines the advantages of a finite element method and a standard finite difference technique is developed for the analysis of plates on elastic foundation subjected to general loadings and arbitrary edge support conditions as mentioned in this paper.
01 Jan 1972
TL;DR: This paper focuses on recently developed direct methods other than Gaussian elimination, which compare very favorably with standard band or block Gaussian Elimination methods even in that respect.
Abstract: A well-known source of sparse matrix problems is the systems of linear algebraic equations which arise when we solve elliptic boundary value problems by finite difference methods, A great deal of effort has been devoted to the design and study of iterative methods for the solution of such linear systems (Varga (1962A), Wachspress (1966A) and Young (1971B)). Among direct methods, i.e. methods which give an exact solution of the finite difference equations in absence of round-off errors, Gaussian elimination and its variants are undoubtedly the best known. In this paper we will concentrate on recently developed direct methods other than Gaussian elimination. The best known of these are due to Hockney and Buneman (Buzbee, Golub and Nielson (1970A), Dorr (1970A), Golub (1971A) and Hockney (1965A), (1970A)). Originally the methods of Buneman and Hockney were used only for Poisson’s equation on rectangular domains. The two methods are of comparable speed and very fast. According to Hockney (1970A) his method produces an accurate solution of the standard five-point difference approximation of Poisson’s equation on a 128 × 128 mesh in a time corresponding to that of 3 steps of a successive over-relaxation method for a problem of the same size. The amount of storage needed is about the same as that required for an iterative method and these direct methods therefore compare very favorably with standard band or block Gaussian elimination methods even in that respect.
TL;DR: In this article, the binomial coefficients in the expansion of (x - l)2r as the elements in each row and column were used to obtain explicit formulas for det A(2r + 1, n), A-1(2 r + 1, n), IIA-1, n)jj,, and an upper triangular matrix U such that A( 2 r+ 1,,t)U is lower triangular.
Abstract: Let A(2r + 1, n) denote the n X n band matrix, of bandwidth 2r + 1, with the binomial coefficients in the expansion of (x - l)2r as the elements in each row and column. Using the fact that the rows of A(2r + 1, ni) provide the coefficients for the 2rth central difference, a number of properties of A(2r + 1, ni) are obtained for all positive integers r and n. These include obtaining explicit formulas for det A(2r + 1, n), A-1(2r + 1, n), IIA-1(2r + 1, n)jj,,, and an upper triangular matrix U such that A(2r + 1, ,t)U is lower triangular. 1. Introduction. We consider the set of band-diagonal matrices of bandwidth 2r + 1, with the binomial coefficients in the expansion of (x - 1)2r displayed sym- metrically about the diagonal in each row and column. If A(2r + 1, n) denotes the nth order member of this set, then, for example
TL;DR: In this paper, an automatic switched Shuman filter was constructed to preserve second or higher order accuracy in smooth regions, and demonstrated the usefulness and accuracy of the method with third and fourth order accurate finite difference schemes.
Abstract: When using Shuman's filtering operator in the numerical computation of shock waves, nonlinear instabilities are prevented, but high order accuracy is lost even in smooth regions. In order to preserve second or higher order accuracy in these regions, an automatic switched Shuman filter is constructed. Nonsteady shock calculations in one and two spatial dimensions, demonstrate the usefulness and accuracy of the method, including examples with third and fourth order accurate finite difference schemes.
TL;DR: In this paper, a new method for solving two-point boundary value problems by finite difference methods has been developed, based on the observation that local truncation errors associated with central difference analogues of the defining differential equation become arbitrarily small as the interior node points are arranged in an optimal sequence.
TL;DR: The running mill time was about 1.2 sec, compared to 4.3 sec (to get the same accuracy) by using finite difference approach by using infinite difference approach.
Abstract: 1.73 0.47591 0.47597 0.54778 0.54776 0.28897 0.28893 3.46 1.77701 1.77706 0.92169 0.91267 0.10679 0.10678 5.19 3.48062 3.48064 0.99367 0.99565 0.01159 0.01157 The running mill time was about 1.2 sec, compared to 4.3 sec (to get the same accuracy) by using finite difference approach.
TL;DR: In this article, it is shown that the finite difference formulation of the governing fourth-order differential equation yields results of acceptable accuracy which converge to the exact values of the natural frequencies from below.
TL;DR: In this article, a digital computer program for the geometrically nonlinear static and dynamic response of arbitrarily loaded shells of revolution is described, which is based upon Sanders' nonlinear thin shell theory for the conditions of small strains and moderately small rotations.
01 Jan 1972
TL;DR: In this paper, the authors focus on spline approximation and difference schemes for the heat equation, and present an example of such a problem and show how a specific discrete time Galerkin scheme can be applied to initial data with and without smoothing.
Abstract: Publisher Summary This chapter focuses on spline approximation and difference schemes for the heat equation. The chapter discusses some preliminaries concerning certain trigonometric polynomials related to the B-splines, and reviews some properties of the spline interpolation operator. The finite difference operator Fk is analyzed by studying its symbol. It is proved that it is parabolic and accurate of order, 2μ - 2. Certain mixed initial-boundary value problems can be rephrased as pure initial-value problems by periodicity considerations. The chapter presents an example of such a problem and shows how a specific discrete time Galerkin scheme can be applied to initial data with and without smoothing. Some well-known facts about Toeplitz operators on sequences are reviewed. The continuous time Galerkin method, and the application of the finite difference theory is discussed. The chapter also presents a class of discretizations in time and analyzes the corresponding difference schemes. Various lemmas are also reviewed.
01 Aug 1972
TL;DR: In this paper, a theory for nonlinear collapse analysis of shells with general shape is presented, which combines energy principals and finite difference methods to obtain a system of nonlinear equations; these are solved by a modified Newton-Raphson technique.
Abstract: : F33615-69-C-1523, AF-1467146703, AFFDLTR-71-8-Vol-1(*shells(structural forms), failure(mechanics)), structural properties, numerical analysis, buckling, plastic properties, tensor analysis, computer programming newton- raphson method, collapse, plates(structural members), finite difference theory, stags computer program, structural analysis. The report presents a theory for nonlinear collapse analysis of shells with general shape. The theory combines energy principals and finite difference methods to obtain a system of nonlinear equations; these are solved by a modified Newton-Raphson technique. For greater economy and flexibility in the analysis a capability is provided for use of variable spacing finite difference grids. Inelastic material behavior, as predicted by the White-Besseling Theory, is incorporated into the analysis. A computer code, STAGS, based on the theory has been written and used to solve a number of sample problems. Results for these problems are presented.
TL;DR: In this article, the authors considered the mathematical description of gas transport across inert membranes permeable to all components, and showed that no completely accurate integration of local transport equations is possible without detailed information on the dependence of the local coefficients on state variables, and that attempt to bypass the difficulty by assuming an overall form that correctly represents the entropy production gives flux equations inferior to those obtained using simple arithmetic means.
Abstract: The mathematical description of gas transport across inert membranes permeable to all components is considered from two points of view: a phenomenological approach through non-equilibrium thermodynamics, and a kinetic-theory approach through the dusty-gas model. The results are shown to be equivalent in local (differential) forms. Extension of the equations to overall or finite-difference linear forms is made by use of arithmetic means for concentrations and pressure inside the membrane. Comparison with the accurately integrated kinetic-theory equations shows that the results are valid only over a relatively narrow regime that becomes rapidly constricted as either or both of the driving forces departs significantly from zero. Use of logarithmic mean concentrations that preserve the overall entropy production yields results of even more restricted validity. Data for the system He + Ar in a graphite membrane are compared with the theoretical results, and used to test the overall Onsager relation for the arithmetic-mean equations. The results suggest that “second-order” coefficients in flux-force equations may be artifacts of the integration of linear differential equation over finite differences. It is concluded that no completely accurate integration of local transport equations is possible without detailed information on the dependence of the local coefficients on state variables, and that the attempt to bypass the difficulty by assuming an overall form that correctly represents the entropy production gives flux equations inferior to those obtained using simple arithmetic means.
TL;DR: In this article, the nonlinear relaxation method in conjunction with finite difference approximation is utilized to solve the governing differential equations of flat membranes, and the results are presented in a form convenient for direct engineering use.
TL;DR: Setting the Reynolds number equal to zero, in a method for solving the Navier-Stokes equations numerically, results in a fast numerical method for biharmonic problems.
Abstract: Setting the Reynolds number equal to zero, in a method for solving the Navier-Stokes equations numerically, results in a fast numerical method for biharmonic problems. The equation is treated as a system of two second order equations and a simple smoothing process is essential for convergence. An application is made to a crack-type problem.
TL;DR: In this article, a finite element method is applied to isothermal incompressible two-dimensional slow flows of power-law fluids, including rectangular and axisymmetric converging channel flows, recirculating flows in rectangular channels and flow round cylinders.
Abstract: A finite element method is applied to isothermal incompressible two-dimensional slow flows of power-law fluids. Examples considered are rectangular and axisymmetric converging channel flows, recirculating flows in rectangular channels, and flow round cylinders. Results are successfully compared with both finite difference and analytical solutions. The flexibility of finite element methods makes them very suitable for problems involving complex boundary geometries. The method used is particularly suitable for non-Newtonian flows and can treat both rectangular and axisymmetric geometries.
TL;DR: In this article, the boundary value problem with finite differences is considered, and it is proved that the approximate solutions exist and converge tox for a large class of problems and convergence tox.
Abstract: The boundary value problem $$\ddot x(t) = g(t,x(t)) + (\mathfrak{F}x)(t)$$ , 0
TL;DR: In this article, the authors derived expressions for the pseudo forces and contributions to the tangent stiffness matrix for geometric and material nonlinearities using the basic principle of virtual work and a computional procedure is presented which evaluates the effects of non-linearities through the use of finite difference expressions.
01 Apr 1972
Abstract: A digital computer program known as SATANS (static and transient analysis, nonlinear, shells) for the geometrically nonlinear static and dynamic response of arbitrarily loaded shells of revolution is presented. Instructions for the preparation of the input data cards and other information necessary for the operation of the program are described in detail and two sample problems are included. The governing partial differential equations are based upon Sanders' nonlinear thin shell theory for the conditions of small strains and moderately small rotations. The governing equations are reduced to uncoupled sets of four linear, second order, partial differential equations in the meridional and time coordinates by expanding the dependent variables in a Fourier sine or cosine series in the circumferential coordinate and treating the nonlinear modal coupling terms as pseudo loads. The derivatives with respect to the meridional coordinate are approximated by central finite differences, and the displacement accelerations are approximated by the implicit Houbolt backward difference scheme with a constant time interval. The boundaries of the shell may be closed, free, fixed, or elastically restrained. The program is coded in the FORTRAN 4 language and is dimensioned to allow a maximum of 10 arbitrary Fourier harmonics and a maximum product of the total number of meridional stations and the total number of Fourier harmonics of 200. The program requires 155,000 bytes of core storage.
TL;DR: In this paper, the authors studied the initial-boundary value problems for a general class of nonlinear parabolic equations and showed that the solutions of certain associated finite difference equations converge to the solution of the problem with O(h^2 )$ rate of convergence.
Abstract: In this paper, initial-boundary value problems for a general class of nonlinear parabolic equations are studied. We show the solutions of certain associated finite difference equations converge to the solution of the initial-boundary value problem with $O(h^2 )$ rate of convergence..