Showing papers on "Finite difference published in 1968"
TL;DR: In this paper, first, second, and fourth order finite difference approximations to the color equation in both advection and conservation form are considered in one and two space dimensions.
Abstract: First, second, and fourth order finite difference approximations to the color equation in both advection and conservation form are considered in one and two space dimensions. All schemes considered are based on forward time differences and most involve centered space differences. All are shown to be numerically stable for |uΔt/Δx| ≤ 1. Test calculations indicate that for the same order of accuracy, the conservation form produces more accurate solutions than the advection form. For either conservation or advection form, fourth order schemes are shown to be more accurate than second or first order schemes in terms of both amplitude and phase errors.
233 citations
TL;DR: In this article, a quasi-homogeneous net of points over a sphere for numerical integration is defined, which consists of almost equal-area, equilateral spherical triangles covering the sphere.
Abstract: A quasi-homogeneous net of points over a sphere for numerical integration is defined. The grid consists of almost equal-area, equilateral spherical triangles covering the sphere. Finite difference approximations for a nondivergent, barotropic model expressed in terms of a streamfunction are proposed for an arbitrary triangular grid. These differences are applied to the spherical geodesic grid. The model is integrated for 12-day periods using analytic initial conditions of wave number six and four. The numerical solution with these special initial conditions follows the analytic solution quite closely, the only difference being a small phase error. Small truncation errors are noticeable in the square of the streamfunction averaged over latitude bands. DOI: 10.1111/j.2153-3490.1968.tb00406.x
191 citations
01 Jun 1968
TL;DR: In this article, a finite difference equation analysis of in-depth response of materials exposed to high temperature environment is presented. But the analysis is limited to finite difference equations. But it is not restricted to a single material.
Abstract: Computer program for finite difference equation analysis of in-depth response of materials exposed to high temperature environment
162 citations
01 Oct 1968-Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences
105 citations
TL;DR: In this paper, a numerical formulation of high-order accuracy, based on variational methods, is proposed for the solution of multi-dimensional diffusion-convection type equations, without the difficulties that standard finite difference approximations present.
Abstract: A numerical formulation of high-order accuracy, based on variational methods, is proposed for the solution of multi-dimensional diffusion-convection type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that very accurate solutions of a one-domensional problem can be obtained in the neighborhood of a sharp front without doing a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in 2 dimensions. (14 refs.)
102 citations
TL;DR: In this article, a numerical experiment is carried out to investigate the circulation of an ocean, driven by a prescribed density gradient and wind stress at the surface, including in one model most of the physical effects that have been considered in previous theoretical studies.
Abstract: A numerical experiment is carried out to investigate the circulation of an ocean, driven by a prescribed density gradient and wind stress at the surface. The mathematical formulation includes in one model most of the physical effects that have been considered in previous theoretical studies. Starting out from conditions of uniform stratification and complete rest, an extensive numerical integration is carried out with respect to time. Care is taken in the final stages of the calculation to use a finite difference net which resolves the very narrow boundary layers which form along the side walls of the basin. A detailed description is made of the three-dimensional velocity and temperature patterns obtained from the final stage of the run. Since inertial effects play an important role in the western boundary current, it is possible to verify with a baroclinic model two results obtained previously with barotropic ocean models: 1) a concentrated outflow from the western boundary takes place along the...
94 citations
TL;DR: In this article, an integral equation technique is presented which may be used to efficiently compute the Maxwellian capacitance matrix, i.e., the coefficients of capacitance and inductance, for any system of zero-thickness strip conductors located parallel to and between two ground planes, the TEM characteristic admittances for various operating modes and the coupling coefficients can then be obtained from the elements of this matrix.
Abstract: An integral equation technique is presented which may be used to efficiently compute the Maxwellian capacitance matrix, i.e., the coefficients of capacitance and inductance, for any system of zero-thickness strip conductors located parallel to and between two ground planes, The TEM characteristic admittances for various operating modes and the coupling coefficients can then be obtained from the elements of this matrix. A single computer program based upon this technique can be used to compute the capacitance matrix for any particular strip line configuration desired and would thus be especially valuable to the design engineer who would like to quickly obtain accurate design curves for a previously unstudied configuration of ship conductors. This procedure gives much more accurate results in but a fraction of the computer time required when the more common finite difference equation approach is used, and it avoids the necessity for a separate mathematical analysis for each new strip line configuration, as would be required when using a conformal mapping technique. Illustrative results are given for several different strip-line configurations.
88 citations
TL;DR: In this paper, a revised procedure is described in which the pore pressure is treated as another unknown, and the form of the procedure used for finite elements, and that used for a lumped parameter model equivalent to a finite difference method.
Abstract: Stress distribution in incompressible or undrained soil is important to the engineer, but the usual finite difference and finite element methods break down for incompressible materials. A revised procedure is described in which the pore pressure is treated as another unknown. Specific application is made to linearly elastic materials; however, the technique can be used for other stress-strain relations. The paper shows the form of the procedure used for finite elements, and that used for a lumped parameter model equivalent to a finite difference method. Results for a sample problem illustrate the insensitivity of vertical stress to compressibility and show the larger effects on the other stress components. Comparison of undrained and drained analysis shows the motion of soil towards the load during consolidation.
70 citations
TL;DR: In this paper, a system of finite difference equations for storm surge prediction using forward time differences was constructed, and the scheme was tested for special simple geometrical configurations, and it was found to be stable without introducing smoothing operators.
Abstract: A system of finite difference equations for storm surge prediction has been constructed, using forward time differences. The scheme was tested for special simple geometrical configurations, and it was found to be stable without introducing smoothing operators. The variation with time of the total energy was, in each case, the test of stability. The small-scale oscillation of the energy with time (characteristic of forward difference schemes) was studied in detail. A method of reducing this effect is suggested. A completely implicit finite difference scheme is discussed from the point of view of stability and convergence. It is shown how the requirement of a convergent iterative process actually introduces a severe restriction on the ratio Δt/Δs, thus canceling the advantages of the otherwise unconditionally stable implicit schemes.
69 citations
TL;DR: In this paper, a general method of studying the deflexion of variable-thickness elastic plates using the Dynamic Relaxation method is described, where non-linear terms arising in the equations to the large deflexions of plates can be included directly by the iterative finite difference technique on which the dynamic relaxation method is based.
61 citations
TL;DR: In this article, an implicit method for the numerical solution of the equations of unsteady flow, which play an important role in the field of water resources, is established based on a centered difference scheme for representing the primary differential equations in finite difference form, and simultaneous solutions of the finite difference equations for each time step.
Abstract: There is great need for fast and accurate methods for the numerical solution of the equations of unsteady flow, which play an important role in the field of water resources. An implicit method which is both fast and accurate can be established based on (1) a centered difference scheme for representing the primary differential equations in finite difference form, and (2) simultaneous solutions of the finite difference equations for each time step. The difference equations constitute a system of nonlinear algebraic equations that can be solved on a digital computer by rapidly convergent procedures based on the generalized Newton iteration method. Application of the implicit method to the movement of floods in long channels confirms the efficiency and accuracy of the method.
30 Apr 1968
TL;DR: Partial differential equations involving one space dimension and time can be solved by hybrid computers using the serial (or continuous space-discrete time) method.
Abstract: Partial differential equations involving one space dimension and time can be solved by hybrid computers using the serial (or continuous space-discrete time) method. In so doing, the continuous integration capability of the analog computer is used along the space axis while integration along the time axis is performed in a discrete fashion by making use of finite differences.
TL;DR: In this article, a numerical study of the transient elastic stress distribution in cylindrical and spherical bodies subjected to a radially symmetric transient normal pressure (or velocity) at an internal or external surface is made.
Abstract: A numerical study is made of the transient elastic stress distribution in cylindrical and spherical bodies subjected to a radially symmetric transient normal pressure (or velocity) at an internal or external surface. The numerical procedure employs the characteristic relations on boundaries and on interfaces between media with different material properties while using an explicit finite difference scheme at all other points. In addition, as an option, the procedure can incorporate the exact jump conditions across discontinuous wave fronts. Problems with and without external boundaries are solved, and results are shown which clearly indicate the propagation of discontinuities in stress due to a step loading on the boundary. Comparison of these results with those available from other investigators yields excellent agreement. Results are also given for stress wave propagation in multilayered media consisting of two layers with different material properties.
TL;DR: In this article, a steady-state free surface seepage problem from trapezoidal, triangular and rectangular cross sections through homogeneous isotropic and anisotropic porous media to drained layers at finite depths is obtained by methods of finite differences.
Abstract: Solutions to steady-state free surface seepage problems from canals of trapezoidal, triangular and rectangular cross sections through homogeneous isotropic and anisotropic porous media to drained layers at finite depths are obtained by methods of finite differences. In the formulation of the problems, the coordinate directions x and y are considered the dependent variables and the velocity potential and the stream function the independent variables. The solutions are obtained by two independent methods. The first method combines solutions for y and x from the plane of the velocity potential and stream function with a solution of the stream function from the physical plane in a cyclic manner. The second method uses complex arithmetic to obtain the solution entirely in the plane of the velocity potential and stream function. A relationship between the seepage quantity and the dimensions of the canal and distance to the drained layer is developed by summarizing solutions obtained for canals of various sizes and shapes.
TL;DR: In this paper, a discrete flexural model of a plate is deduced from a discrete model of three-space solids, and all the flexural relations and equations, including boundary conditions, pertaining to a discrete set of field quantities can then be formulated directly through the model; these relations can also be shown consistently to be central difference analogs of the corresponding classical differential equations.
Abstract: By invoking the usual assumptions of the classical theory of plates and shells, a discrete flexural model of a plate is deduced from a discrete model of three-space solids. All the flexural relations and equations, including boundary conditions, pertaining to a discrete set of field quantities can then be formulated directly through the model; these relations can also be shown consistently to be central difference analogs of the corresponding classical differential equations. On this basis, the treatment of nonlinear-inelastic material properties is quite transparent; material properties are treated in their most basic form and general properties are handled in the same manner as that of linearly elastic material. All these lead to a simple set of recursive equations which constitute the basis for an algorithmic approach to the flexural analysis of nonlinear-inelastic problems of plate structures. The solution for several square plates of elastic-perfectly plastic material are illustrated with different boundary conditions. In all cases, the numerically predicted load-carrying capacities are shown to be consistently within the bounds of limit analysis.
TL;DR: Stability and error bounds in numerical integration of ordinary differential equations, determining highest possible degree of stable finite difference form are given in this article, where the maximum possible degree is determined by determining the highest possible variance.
Abstract: Stability and error bounds in numerical integration of ordinary differential equations, determining highest possible degree of stable finite difference form
TL;DR: In this paper, a comparison of the Point Successive Over- Relaxation (PSOR), 2-Line Cyclic Chebyshev Semi-Iterative (SOR) and Alternating Direction Implicit (ADI) methods was made with respect to speed of solution of a test problem.
Abstract: A study was made of numerical techniques for solving the large sets of simultaneous equations that arise in the mathematical modeling of oil reservoir behavior. It was found that noniterative techniques, such as the Alternating Direction Implicit (ADI) method, as well as some other finite difference approximations, produce oscillatory or unsmooth results for large time steps. Estimates of time step sizes sufficient to avoid such behavior are given. A comparison was made of the Point Successive Over- Relaxation (PSOR), 2-Line Cyclic Chebyshev Semi-Iterative (SOR) (2LCC), and iterative ADI methods, with respect to speed of solution of a test problem. It was found that, when applicable, iterative ADI is fastest for problems involving many points, while 2LCC is preferable for smaller problems. (28 refs.)
TL;DR: In this paper, a finite difference analysis for slip flow in the entrance region of the parallel plate channel and the plain tube is presented, and results are given for Knudsen numbers equal to 0, 0.01,0.03,0,05, and 0.1.
Abstract: A finite difference analysis is presented for slip flow in the entrance region of the parallel plate channel and the plain tube. Results are given for Knudsen numbers equal to 0, 0.01, 0.03, 0.05, and 0.1. The present results for the pressure distribution in the entrance region do not favour any particular one of the linear solutions to the problem which have been given previously. There is, however, very good agreement between the finite difference prediction of the developing velocity profiles and that of a certain linear solution.
24 Jun 1968
01 Jan 1968
TL;DR: Perturbation techniques for error analysis of finite difference approximations for linear differential equations have been proposed in this paper, where the error analysis is based on finite difference approximation.
Abstract: Perturbation techniques for error analysis of finite difference approximations for linear differential equations
TL;DR: In this paper, an integral equation method is developed for the analysis of thin elastic plates based on a Green's formula and adapted for the solution of the biharmonic problem of plate analysis.
Abstract: An integral equation method is developed for the analysis of thin elastic plates. The integral equations are based on a Green's formula and are adapted for the solution of the biharmonic problem of plate analysis. By means of this method, a numerical solution is obtained for a simply-supported, uniformly loaded corner plate. Deflections and bending moments along the diagonal and shear forces along the edges are found. Values of these quantities previously calculated by means of a finite difference technique for a right-angled plate are confirmed by this new method and the analysis is extended to corner plates with different angles at the corner. Furthermore, for a right-angled corner plate, detailed contour plots of bending and twisting moments inside the plate are presented. With a digital computer these results are obtained without undue difficulty, and it appears that the integral equation method is well suited to this type of problem.
TL;DR: In this paper, the solution of seepage through a homogeneous isotropic or anisotropic earth dam is obtained by finite differences, where the x and y coordinates are considered the dependent variables and the velocity potential and the stream function the independent variables.
Abstract: The solution of seepage through a homogeneous isotropic or anisotropic earth dam is obtained by finite differences. The x and y coordinates are considered the dependent variables and the velocity potential and the stream function the independent variables. The boundary value problems for y and x are formulated in the complex potential plane. Since the size of the flow field in the complex potential plane is unknown from a specification of the size and shape of the dam, the final solution is obtained from a cycle of solutions each subsequent one of which is obtained in a field of more nearly correct size. Comparisons are given between the coordinates on the phreatic line obtained by this finite difference method and those obtainable from a solution by the velocity hodograph. Summarizing the results from a number of solutions of different geometries, a graphical relation is given relating the geometric parameters of the dam and the seepage quantity.
TL;DR: In this paper, it is shown that with any two solutions of Lu(z) = 0 there is associated a new solution which is represented as a convolution product, which may be considered as a type of line integral and is based upon a discrete analogue of Green's formula.
Abstract: This paper is concerned with linear partial difference operators L having constant coefficients. The functions considered are defined only on the lattice points of the complex plane. It is shown that with any two solutions of Lu(z) = 0 there is associated a new solution which is represented as a convolution product. This product may be considered as a type of line integral and is based upon a discrete analogue of Green's formula. This development may be regarded as an anology to the pioneering work of H. Lewy concerning the composition of solutions of partial differential equations. It may also be considered a continuation of the investigation pursued by Duffin and Duris in the introduction of a convolution product for discrete analytic functions. HUNT LIBRARY CARNEGIE-MELLON UNIVERSITY A Convolution Product for the Solutions of Partial Difference Equations* by R. J. Duffin and Joan Rohrer CARNEGIE INSTITUTE OF TECHNOLOGY
TL;DR: In this article, a finite difference method has been applied to solve seismic pulse propagation problems mainly in cases when analytic solutions have not been found or are lengthy, and the following problems are discussed: the motion of an elastic quarter space due to an explosive line source, showing diffraction of surface and body waves at the corner; the response of a layered halfspace to an impulsive point-source, showing excellent agreement with previous analytic results, and giving additional information about refraction arrivals and interface waves.
Abstract: Finite difference methods have been applied to solve seismic pulse propagation problems mainly in cases when analytic solutions have not been found or are lengthy. The following problems are discussed: The motion of an elastic quarter space due to an explosive line-source, showing diffraction of surface and body waves at the corner; the response of a layered halfspace to an impulsive point-source-showing excellent agreement with previous analytic results, and giving additional information about refraction arrivals and interface waves.The same method has been applied to pulse propagation in a sphere. As an example, the motion of a fluid sphere is given showing effects of heterogeneity and of deviations from sphericity.