Showing papers on "Finite difference coefficient published in 1982"

Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures*

Abstract: Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal order error estimates in $L^2 $ and $W^{1,2} $ are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems $(b \gg a)$, these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods

Abstract: Introduction to Groundwater Modeling presents an overview of the fundamental concepts and applications of computerized groundwater modeling.

Finite difference methods and their convergence for a class of singular two point boundary value problems

Abstract: We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0

Comparison of First-Order Finite Difference and Finite Element Algorithms for the Analysis of Magnetic Fields. Part I: Theoretical Analysis

Abstract: Finite difference and finite element methods are investigated as applied to Laplace and Poisson linear equations in two dimensions. The comparative analysis concentrates on first-order square, rectangular, triangular and polar algorithms which are characteristic to each method. The results are found to be independent of the method used to establish the matrix equations but dependent upon the choice of grid systems employed to discretize the problem. It is found that for special cases the solutions are also independent of the type and structure of the grid systems used.

Wave propagation and stability for finite difference schemes

Abstract: : This dissertation investigates the behavior of finite difference models of linear hyperbolic partial differential equations. Whereas a hyperbolic equation is nondispersive and nondissipative, difference models are invariably dispersive, and often dissipative too. We set about analyzing them by means of existing techniques from the theory of dispersive wave propagation, making extensive use in particular of the concept of group velocity, the velocity at which energy propagates. The first three chapters present a general analysis of wave propagation in difference models. We describe systematically the effects of dispersion on numerical errors, for both smooth and parasitic waves. The reflection and transmission of waves at boundaries and interfaces are then studied at length. The key point for this is a distinction introduced here between leftgoing and rightgoing signals, which is based not on the characteristics of the original equation, but on the group velocities of the numerical model. The last three chapters examine stability for finite difference models of initial boundary value problems.

General Mapping Procedure for Variable Area Duct Acoustics

Abstract: A general mapping procedure is described and applied to the study of noise propagation in variable area ducts. The mapping provides a boundary fitted co-ordinate system which is ideal for the finite difference solution of acoustic fields with irregular boundaries, without the burden of large matrices required by finite element methods. The procedure is first described in general and then applied to a particular two-dimensional geometry under current experimental investigation. This method should be ideally suited to the study of high frequency noise propagation in variable area ducts and in cases where the far field is included in the calculation procedure. Moreover, the current approach can be directly extended to three-dimensions, resulting in numerical calculation over a rectangular parallelepiped in the transformed plane.

Compact Finite Difference Schemes for Mixed Initial-Boundary Value Problems

Abstract: This paper discusses a class of compact second order accurate finite difference equations for mixed initial-boundary value problems for hyperbolic and convective-diffusion equations Convergence is proved by means of energy arguments and both types of equations are solved by similar algorithms For hyperbolic equations an extension of the Lax–Wendroff method is described which incorporates dissipative boundary conditions Upwind-downwind differencing techniques arise as the formal hyperbolic limit of the convective-diffusion equation Finally, a finite difference “chain-rule” transforms the schemes from rectangular to quadrilateral subdomains

Numerical solution of a first‐order conservation equation by a least square method

Abstract: Least square methods have been frequently used to solve fluid mechanics problems. Their specific usefulness is emphasized for the solution of a first-order conservation equation. On the one hand, the least square formulation embeds the first-order problem into equivalent second-order problem, better adapted to discretization techniques due to symmetry and positive-definiteness of the associated matrix. On the other hand, the introduction of a least square functional is convenient for finite element applications. This approach is applied to the model problem of the conservation of mass (the unknown is the density ρ) in a nozzle with a specified velocity field (u, v), possibly including jumps along lines simulating shock waves. This represent a preliminary study towards the solution of the steady Euler equations. A finite difference and a finite element method are presented. The choice of the finite difference scheme and of a continuous finite element representation for the groups of variables (ρu, ρv) is discussed in terms of conservation of mass flux. Results obtained with both methods are compared in two numerical tests with the same mesh system.

Conjugate Gradient Methods Applied to Transonic Finite Difference and Finite Element Calculations

Abstract: Most transonic finite difference and finite element calculations are obtained by SLOR. Recently, approximate factorization methods (ADI, AF2, SIP) have been used with finite differences (application of such iterative methods to finite elements is not straightforward). In this paper incomplete LU decomposition and SSOR are used as preconditioning to second-degree iterative methods with adaptive acceleration parameters as in conjugate gradient algorithms for both finite difference and finite element calculations based on the artificial density formulation. Different cases are tested and the results are demonstrated. The present method is certainly more efficient than pure SLOR for obtaining results with reasonable accuracy.

Comparison of First-Order Finite Difference and Finite Element Algorithms for the Analysis of Magnetic Fields. Part II. Numerical Results

Abstract: Numerical results obtained by first- order finite difference and finite element algorithms are compared with closed-form solutions to the finite- depth-slot problem and to the section of an annulus of magnetic and nonmagnetic material, the magnetic field of which is caused by a current-carrying conductor of circular section. Magnitude and boundary condition errors are calculated with respect to these exact solutions, and both methods are compared numerically as applied to a finite slot pitch where a uniform current density region covers the slot area. The simultaneous difference equations resulting from the finite difference and finite element method are solved by Gaussian elimination. Both methods are compared with regard to computing time, storage requirements and errors for Laplace's and Poisson's equations in two dimensions on a numerical basis.

A comparison of the finite difference and finite element methods for heat transfer calculations

Abstract: The finite difference method and finite element method for heat transfer calculations are compared by describing their bases and their application to some common heat transfer problems. In general it is noted that neither method is clearly superior, and in many instances, the choice is quite arbitrary and depends more upon the codes available and upon the personal preference of the analyst than upon any well defined advantages of one method. Classes of problems for which one method or the other is better suited are defined.

Techniques for efficient implementation of pseudo-spectral methods and comparisons with finite difference solutions of the Navier-Stokes equations

Abstract: The results of this study have shown that: (i) direct matrix inversions are competitive with FFT's for calculating pseudo-spectral representations; (ii) all real space Poisson solvers are possible with the use of a predictor-corrector procedure; (iii) the iterative time integration scheme is very attractive for use in incompressible flows; (iv) energy conservation in stratified flow using finite difference techniques can be accomplished using a combination of conservative and Piacsek-Williams differencing; and (v) computation times for pseudo-spectral calculations are faster than finite difference calculations of equivalent accuracy.

Analysis of a finite difference grid

Abstract: Some means of assessing the suitability of a mesh network for a finite difference calculation are investigated in this study. This has been done by a study of the nonlinear truncation errors of the scheme. It turns out that the mesh can not be properly assessed a priori. The effect of the mesh on the numerical solution depends on several factors including the mesh itself, the numerical algorithm, and the solution. Several recommendations are made with regard to generating the mesh and to assessing its suitability for a particular numerical calculation.

Improving the accuracy of finite difference methods for solving boundary value ordinary differential equations

Abstract: It is shown that by introducing one or three interpolated values in each subinterval the local truncation error of finite difference (box, gap and deferred correction) methods can be decreased by two to four orders of magnitude when solving two point boundary value ordinary differential equations.