Showing papers on "Finite difference coefficient published in 1975"
TL;DR: In this article, the authors proposed an orthogonal collocation method for heat and mass transfer with chemical reaction in a catalyst pellet, which is especially suited to situations corresponding to high Thiele modulus when the solution is confined to a thin boundary region near the catalyst surface.
287 citations
TL;DR: In this paper, a two-dimensional finite-difference technique for irregular meshes is formulated for derivatives up to the second order, where the domain in the vicinity of a given central point is broken into eight 45 degree pie shaped segments and the closest finite difference point in each segment to the center point is noted.
257 citations
TL;DR: In this article, the authors investigated the achievable accuracy of various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level, and concluded that the Crank-Nicolson scheme with a simple averaging process is superior to the other methods investigated.
Abstract: This paper investigates the phenomenon of ‘noise’ which is common in most time-dependent problems. The emphasis is on the achievable accuracy that is obtained with various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level. A series of experiments are made where the space domain is discretized using the finite element method and the variation with time is approximated by several finite difference methods. The conclusion is reached that the Crank–Nicolson scheme with a simple averaging process is superior to the other methods investigated.
130 citations
TL;DR: In this paper, a finite difference approximation to the similinear heat equation with special emphasis on the case when the exact solution blows up with the blowing-up time was studied.
Abstract: This paper studies a finite difference approximation to the similinear heat equation (1) with special emphasis on the case when the exact solution blows up with the blowing-up timeT
∞. The key results will be given in Propositions 1 and 2. Proposition 1 states the local convergence, i.e., the convergence of the proposed finite difference solution to the exact solution in any fixed time interval 0 ⩽t ⩽ T, whereT < T
∞. Proposition 2 states the convergence of the numerical blowing-up time to the exact oneT
∞.
85 citations
TL;DR: In this article, a finite difference scheme is developed for seismological problems, containing a new treatment of surface and interface boundary conditions, associated with truncation errors of the second order, while previous representations were of the first order only.
Abstract: Summary A finite difference scheme is developed for seismological problems, containing a new treatment of surface and interface boundary conditions. The new representations of boundary conditions are associated with truncation errors of the second order, while previous representations were of the first order only. We show by comparison with analytical solutions that increasing the accuracy of the representation of boundary conditions increases the order of accuracy of the whole solution. The present scheme can be programmed easily and applied to seismological problems. We illustrate this by solving the problem of pulse propagation in two models of rift valley, and in the vicinity of an elastic wedge embedded in another elastic space.
66 citations
TL;DR: It is shown that this high accuracy difference method for the solution of evolution equations of parabolic type has better computational performance than classical schemes on non-uniform and coarse meshes.
Abstract: A high accuracy difference method (hermitian method) for the solution of evolution equations of parabolic type is presented. Its most original feature is to use several unknowns (the value of the solution and its spatial derivatives) at every nodal point of the computational grid. It is shown that this method has better computational performance than classical schemes on non-uniform and coarse meshes.
64 citations
TL;DR: In this article, the finite element method was applied to the radially symmetric case of the hydrogen atom, which has computational advantages over the finite difference and Rayleigh−Ritz methods.
Abstract: The finite element method, which in other fields has replaced finite difference and variational methods, is applied to the radially symmetric case of the hydrogen atom. The method is shown to have computational advantages over the finite difference and Rayleigh−Ritz methods. (AIP)
46 citations
TL;DR: In this paper, two explicit finite difference schemes of fourth order accuracy (in space and time) are presented for numerical solution of quasi-linear divergence free one-dimensional hyperbolic systems.
Abstract: Two explicit finite difference schemes of fourth order accuracy (in space and time) are presented for the numerical solution of quasi-linear divergence free one-dimensional hyperbolic systems. Both of these schemes are four-step methods, one being a two level scheme, the other using three levels. These algorithms are compared in numerical examples with both second order schemes and with the Kreiss–Oliger method which is fourth order in space and second order in time. The results show that it is most advantageous to use the true fourth order schemes.
25 citations
TL;DR: In this paper, a finite element method is presented for analyzing the response of periodic structures to convected random pressure fields, where the problem reduces to one of finding a single periodic section to a harmonic pressure wave.
22 citations
22 citations
TL;DR: In this paper, discrete equations are formed by applying Ampere's circuital law around each node, and a variational formulation for transient conditions in the presence of dissipation is formulated.
Abstract: The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and element methods. The finite difference method requires a large number of nodes and is slow to converge as permeability is variable. The finite element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient conditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.
TL;DR: In this article, a stochastic analysis of a single degree of freedom system subjected to random wind and seismic excitations to study the response characteristics was undertaken by the authors, where the exciting force was assumed to be nonstationary in character, and was represented by the product of a deterministic shape function and a stationary random process characterized by its power spectral density.
Abstract: Conclusion Compared to the rigorous procedures the solution to the previously stated problem, given by Eqs (4) and (5) is approximate, but avoids the cumbersome calculations involved in the former In this connection, the stochastic analysis of a single degree of freedom system subjected to random wind and seismic excitations to study the response characteristics was undertaken by the authors The exciting force was assumed to be nonstationary in character, and was represented by the product of a deterministic shape function and a stationary random process characterized by its power spectral density The choice of deterministic function and power spectral density was based on certain characteristics observed in a large number of past records of excitation process The application of Eqs (4) and (5) to study the peak response characteristics of the system revealed that the probability estimates for various appropriate values of X are about 05% below those obtained by an exact procedure
TL;DR: In this paper, the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations, where no assumption regarding the contact pressure distribution has been made.
Abstract: In the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non-dimensional form and also the results have been obtained in non-dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.
TL;DR: In this paper, the authors present a new proof that aq-step backward difference scheme for the approximate solution of a first order ordinary differential equation is stable in the sense of Dahlquist iff 1≦q≦6.
Abstract: We present a new proof that aq-step backward difference scheme for the approximate solution of a first order ordinary differential equation is stable in the sense of Dahlquist iff 1≦q≦6.
TL;DR: In this paper, the electron distribution function plays a key role in the quantitative study of various properties of electrons moving through a gas and is characterized as a solution of the Boltzmann transport equation.
TL;DR: In this article, a comparison between the finite difference method and the finite element method for solving the linear two-dimensional heat conduction equation is presented, and it is shown that the FEM method is superior to the FDM method in all areas except computer core storage.
TL;DR: In this article, a finite difference approximation to the ordinary differential equations governing the flow between a pair of rotating coaxial discs is examined, and it is shown that when the Reynolds number is small or large the finite difference equations can be solved to give the analytic expansions that would be obtained from the continuous equations.
Abstract: A finite difference approximation to the ordinary differential equations governing the flow between a pair of rotating coaxial discs is examined. It is shown that when the Reynolds number is small or large the finite difference equations can be solved to give the analytic expansions that would be obtained from the continuous equations. As the Reynolds number tends to infinity there is more information retained in the finite difference approximations than in the corresponding limiting continuous equations and this fact is used to eliminate some of the possible flows outside the boundary layers on the discs. The method is quite general and can be applied to other singular perturbation problems.
01 Jan 1975
TL;DR: In this article, Lagrangian finite element methods for solving a Poisson problem produces systems of linear equations, the global stiffness equations, whose components of the vectors which are the solutions of these systems are approximations to the exact solution of the problem at nodal points in the region of definition.
Abstract: The use of Lagrangian finite element methods for solving a Poisson problem produces systems of linear equations, the global stiffness equations. The components of the vectors which are the solutions of these systems are approximations to the exact solution of the problem at nodal points in the region of definition. There is thus associated with each nodal point an equation which can be thought of as a difference equation. Difference equations resulting from the use of polynomial trial functions of various orders on regular meshes of square and isosceles right triangular elements are derived. The rival merits of this technique of setting up a standard difference equation, as distinct from the more usual practice with finite elements of the repeated use of local stiffness matrices, are considered.
TL;DR: In this paper, a finite difference interpretation of the differential operator in the Pocklington form of the integro-differential equation representing the junction problem is presented, which is capable of producing accurate results even with relatively simple basis and testing functions, e.g.
Abstract: An approach to treating the thin-wire junction geometry, which arises in the computer modeling of a great many electromagnetic radiation and scattering problems, is presented. The method is based upon a finite-difference type interpretation of the differential operator in the Pocklington form of the integro-differential equation representing the junction problem. An important advantage of the method is that it is capable of producing accurate results even with relatively simple basis and testing functions, e.g., pulse and \delta -functions. Furthermore, the method does not require the imposition of additional constraints, such as the Kirchhoff current law or the conservation of charge, at the junction points. The method is versatile in that it applies to L-shaped structures as well as to junctions of thin wires of dissimilar radii. Numerical results based on the present finite difference approach have been computed and good agreement with results derived by other independent methods has been observed. An important conclusion of this work is that the conventional interpretation of the differential operator leads to erroneous results since the sampling interval in the conventional finite difference scheme is different from the correct value of the sampling interval found in this paper.
TL;DR: Finite difference techniques used to calculate the capacitance of a ring capacitor involves the solution of a Dirichlet boundary value problem and the calculation of the gradient of the solution obtained.
Abstract: This paper describes finite difference techniques used to calculate the capacitance of a ring capacitor. The determination of capacitance involves the solution of a Dirichlet boundary value problem and the calculation of the gradient of the solution obtained. Circular cylindrical coordinates are used. Nine point difference approximations are used for the Laplacian and the first derivatives of a function. If this function satisfies Laplace's equation and is sufficiently differentiable, the discretization error of each approximation isO(h
4) whereh is the maximum mesh size.
TL;DR: A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations with discrete form of a complicated operator by performing simple numerical operations on elementary matrix factors.
Abstract: A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations. The process is based on a novel approach to finite difference analysis in which differential operators are approximated by rectangular matrices. In this approach, the discrete form of a complicated operator is obtained by performing simple numerical operations on elementary matrix factors. The analysis is augmented by a listing of a computer program based on the method for the automatic generation of finite difference formulas.
TL;DR: In this paper, a highly accurate finite difference method is proposed for the numerical solution of partial differential equations that describe initial value problems, which includes a spatial smoothing operation and avoids the computational mode.
01 Jan 1975
TL;DR: In this paper, a finite difference scheme for the three-dimensional boundary layer equations based on the physics of the convective and diffusive momentum transport in the boundary layer is proposed.
Abstract: A new finite difference scheme has been formulated for the three-dimensional boundary layer equations based on the physics of the convective and diffusive momentum transport in the boundary layer. It is shown that the scheme is a physically optimum one and that it is consistent with the usual specification of initial conditions. A stability analysis of the linearized equations shows that a relative restriction is necessary on the step sizes along the convective coordinates even though the difference scheme is “implicit”. The method is then applied to a problem that taxes the method and the laminar boundary layer equations to their limit; that problem being the supersonic flow over a spinning sharp cone at angle of attack. The results of the spinning cone calculation also yields some very useful insight into the “Magnus” problem and to the contributions to the “Magnus” force by the boundary layer flow.
TL;DR: In this paper, a finite difference scheme for the two-dimensional incompressible advection-diffusion equation is derived for the model, which is stable at all grid mesh sizes or cell Reynolds numbers.
01 Jan 1975
TL;DR: The finite difference analog of the linear vorticity equation for the mass transport in the ocean does not hold good unless the grid size is smaller than a certain number as mentioned in this paper, which is not the case in this paper.
Abstract: The finite difference analog of the linear vorticity equation for the mass transport in the ocean does not hold good unless the grid size is smaller than a certain number.
TL;DR: In this article, the authors present the application of the calculus of finite differences to obtain an explicit expression for the natural frequencies of the finite strip model of a simply supported orthotropic rectangular plate.