Showing papers on "Finite difference coefficient published in 1980"
TL;DR: In this article, the authors give convergence criteria for general difference schemes for boundary value problems in Lipschitzian regions, and prove convergence for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
93 citations
76 citations
TL;DR: A unified framework is presented for analyzing the accuracy of finite difference, finite element, and spectral methods in approximating evolutionary problems and demonstrates the importance of the interpretation given to the discrete data generated in any computation.
44 citations
TL;DR: In this paper, the buckling loads for tapered and stepped columns have been determined by a finite difference method using a matrix iteration solution technique (a BASIC program for which is appended).
41 citations
TL;DR: In this paper, a detailed study of the error growth associated with explicit difference schemes for a conduction-convection problem is made, and it is shown that the error can become arbitrarily large after a finite number of time steps even though it ultimately decays to zero.
Abstract: A detailed study is made of the error growth associated with explicit difference schemes for a conduction-convection problem. It is shown that the error can become arbitrarily large after a finite number of time steps even though it ultimately decays to zero. Certain ambiguities reported in the literature can thereby be resolved.
40 citations
01 Oct 1980
TL;DR: In this paper, an implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form, which is maintained by use of approximate factorization techniques, and the numerical algorithm is first order in time and second order in space.
Abstract: An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form Computational efficiency is maintained by use of approximate factorization techniques The numerical algorithm is first order in time and second order in space A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm
38 citations
TL;DR: In this article, the authors investigated several factors affecting the accuracy and efficiency of numerical determination of the bound state energy eigenvalues of the one dimensional Schrodinger equation, and concluded that the Numerov-Cooley method is the most efficient method for most complex potentials.
32 citations
TL;DR: In this paper, a time-split finite element recursion relation based on linear basis function is used to solve the two-dimensional equations of motion, which has the advantage of finite element accuracy and finite difference speed and simplicity.
19 citations
19 citations
TL;DR: In this paper, it was shown that for a reasonably small mesh size the numerical solutions obtained by using either the central or the upwind difference scheme do not differ appreciably at low Reynolds numbers.
Abstract: Both the central and upwind difference schemes are commonaly employed to solve the Navier-Stokes equations governing the two-dimensional laminar flow of an incompressible laminar flow of an incompressible viscous fluid. By a judicious choice of a damping parameter and using direct methods for solving the systems of linear equations appearing in an iterative procedure, the central difference scheme could be made to give convergent results even for large Reynolds numbers. Using a model problem of flow through a driven square cavity it is shown that the converged results so obtained become progressively oscillatory and inaccurate as the Reynolds number increases. It is known that the central difference scheme gives more accurate results than the upwind difference scheme, at least for small values of Reynolds number. However, for a reasonably small mesh size the numerical solutions obtained by using either the central or the upwind difference scheme do not differ appreciably at low Reynolds numbers. There is a small range of Reynolds numbers for which the central difference scheme may yield more accurate results than the upwind difference scheme; however, this range is not known a priori.
18 citations
TL;DR: In this article, the authors study finite difference approximations to weak solutions of the Cauchy problem for hyperbolic systems of conservation laws in one space dimension and establish stability in the total variation norm and convergence for a class of hybridized schemes which employ the random choice scheme together with perturbations of classical conservative schemes.
Abstract: : We study finite difference approximations to weak solutions of the Cauchy problem for hyperbolic systems of conservation laws in one space dimension. We establish stability in the total variation norm and convergence for a class of hybridized schemes which employ the random choice scheme together with perturbations of classical conservative schemes. We also establish partial stability results for classical conservative schemes. Our approach is based on an analysis of finite difference operators on local and global wave configurations. (Author)
TL;DR: In this article, the plate strain components are expressed in terms of discrete nodal displacements with the aid of the two dimensional Taylor expansion and a variational derivation of the equations of motion or equilibrium is presented.
TL;DR: In this paper, the authors considered certain structure conserving properties of finite difference methods for the solution of parabolic initial-boundary value problems and showed that these properties are stronger than the classical ones which imply the maximum principle for the finite difference equations.
Abstract: In this paper we consider certain structure conserving properties of finite difference methods for the solution of parabolic initial-boundary value problems. We are interested in conditions on the step size ratio μ=Δt/Δx 2 in one-step methods which guarantee that the number of sign changes of the discrete approximation does not increase while proceeding from one time level to the following one. This means that difference schemes of this type possess a so-called variation-diminishing property which is known to hold for continuous diffusion equations also. It turns out that our conditions on μ are stronger than the classical ones which imply the maximum principle for the finite difference equations. By means of an example we show that our sign stability condition is necessary too.
TL;DR: In this paper, a class of compact finite difference collocation methods is shown to be stable on general nonuniform meshes without the assumption of a bounded mesh ratio, and error estimates are included.
Abstract: A class of compact finite difference collocation methods is shown to be stable on general nonuniform meshes without the assumption of a bounded mesh ratio. Some error estimates are included.
TL;DR: In this paper, a topological solution for partial, linear inhomogeneous finite difference equations with variable coefficients and arbitrarily specified boundary conditions has been obtained, where the solution is homomorphic to a set of discrete paths constructed from vectors determined by the level differences of the equation.
Abstract: Using the discrete path formalism, we obtain a topological solution for ordinary, as well as partial, linear inhomogeneous finite difference equations with variable coefficients and arbitrarily specified boundary conditions. The solution is homomorphic to a set of discrete paths constructed from a set of vectors determined by the level differences of the equation.
TL;DR: In this article, the authors compared the behavior of closed-form solutions of the steady-state one-dimensional energy equation produced by quadratic finite element, linear finite elements, central differencing and upwind differences.
TL;DR: In this article, an invariant measure which is continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree, which gives the density of the population whose growth is governed by the difference equation.
Abstract: An invariant measure which is absolutely continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree. In a biological context this invariant measure gives the density of the population whose growth is governed by the difference equation. Further asymptotically universal results are obtained for a class of difference equations.
TL;DR: In this article, the construction of three-point finite difference approximations for the class of two-point boundary value problems is discussed, and a family of fourth-order discretizations for the differential equations are obtained.
TL;DR: The study tends to show that terms of the general form ∂(uvw)/∂x should be discretized by considering u,v and w as separate variables instead of taking the product uvw as a single variable as is often done with equations written in conservative (divergence law) form.
Abstract: A brief description of the finite element method to be used is given. It is then shown how various finite difference schemes for the wave and Burgers' equations can be achieved, in particular the predictor-corrector method of MacCormack. By the finite element method a more efficient predictor-corrector scheme is also obtained. Furthermore, the study tends to show that terms of the general form ∂(uvw)/∂x should be discretized by considering u,v and w as separate variables instead of taking the product uvw as a single variable as is often done with equations written in conservative (divergence law) form.
TL;DR: A priori error estimates for the simplest finite difference and finite element approximations to an inverse problem in which it is desired to identify an unknown constant coefficient in a differential equation whose general form is known are derived in this article.
Abstract: A priori error estimates are derived for the simplest finite difference and finite element approximations to an inverse problem in which it is desired to identify an unknown constant coefficient in a differential equation whose general form is known.
01 Jan 1980
TL;DR: In this article, the authors compared three finite element formulations of the linearized shallow water equations which are applied to the geostrophic adjustment process and three corresponding finite difference schemes are also included in the study.
Abstract: : This report compares three finite element formulations of the linearized shallow-water equations which are applied to the geostrophic adjustment process. The three corresponding finite difference schemes are also included in the study. The development follows Schoenstadt (1980) wherein the spatially discretized equations are Fourier transformed in x, and then solved with arbitrary initial conditions. The six schemes are also compared by integrating them numerically from an initial state at rest with a height perturbation at a single point. The finite difference and finite element primitive equation schemes with unstaggered grid points give very poor results for the small scale features. The staggered scheme B gives much better results with both finite differences and finite elements. The vorticity-divergence system with unstaggered points also is very good with finite differences and finite elements. It is especially important to take into account these results when formulating efficient finite element prediction models. (Author)
TL;DR: The notion of thermodynamic consistency of a finite difference scheme is introduced in this paper in connection with steady compressible flow simulation, which requires that the finite-difference scheme must lead to solutions which are consistent with the Second Law of Thermodynamics, whatever the grid sizes employed.
Abstract: The notion of thermodynamic consistency of a finite difference scheme is introduced in connection with steady compressible flow simulation. This concept requires that the finite difference scheme must lead to solutions which are consistent with the Second Law of Thermodynamics, whatever the grid sizes employed. Thus, in addition to the usual consistency with the conservation equations of mass, momentum and energy, the solutions of the finite difference equations must satisfy thermodynamic restrictions. The scheme proposed by Singhal and Spalding for the computation of plane, isentropic flow is generalized in a thermodynamically consistent way to handle inviscid, adiabatic but non-isentropic plane flow. Numerical solutions are obtained for a model problem using both the original and the extended methods. The results are compared with analytical predictions of shock-expansion theory. The effectiveness and the thermodynamic consistency of the new formulation is demonstrated.
TL;DR: In this paper, the first two elements of a sequence satisfying a second order difference equation are prescribed, and the remaining elements are evaluated from a continued fraction and a simple product, respectively.
Abstract: Abstract When the first two elements of a sequence satisfying a second order difference equation are prescribed, the remaining elements are evaluated from a continued fraction and a simple product.
TL;DR: In this article, the effects of suppression pool containment boundaries on the flow fields produced by air/steam blowdown have been obtained by exact solution for a point source in an annulus, i.e., the Green's function for this geometry, by the method of images (MOI), which solves Laplace's equation in a finite rectangular pool ‘trough, by use of a series of point sources and sinks, and by the finite difference technique with singularities developed by Chu and Raymont.
TL;DR: In this paper, a finite element algorithm for numerical calculation of fluid flows governed by the unsteady Navier-Stokes equations is presented. But the algorithm requires core storage and computer time.