Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Transient shell response by numerical time integration

Abstract: In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes: the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Using exact discrete solutions compared to the exact solutions of the differential equations, the results of these choices are displayed. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy.

The Direct Solution of the Discrete Poisson Equation on a Rectangle

Abstract: where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue. These discrete models for (1) consist of linear systems of equations of very large dimension, and it is widely recognized that the usual direct methods (e.g., Gaussian elimination) are unsatisfactory for such systems [18, ?? 21.2-21.3]. Theoretical investigation has, therefore, been primarily directed toward the development of effective iterative methods for the solution of these problems [64], [66]. In recent years, however, direct methods that take advantage of the special block structure of these linear equations have appeared. For the rectangular regions under consideration, these methods can be considerably faster than iterative methods. The purpose of this survey paper is to provide brief summaries and a list of references for methods which can be used to directly solve the finite difference equations. Some of these methods can be applied to problems in more general domains. However, the extensions generally include only simple rectilinear regions, such as L-shaped or T-shaped domains. This is basically due to the fact that the direct methods require a great degree of regularity in the block structure of the matrix equation. In our discussion, we will indicate whether the methods are easily adaptable to more general regions, and to more general elliptic partial differential equations.

The effect of numerical errors and turbulence models in large-eddy simulations of channel flow, with and without explicit filtering

Abstract: Turbulent channel flow simulations are performed using second- and fourth-order finite difference codes. A systematic comparison of the large-eddy simulation (LES) results for different grid resolutions, finite difference schemes, and several turbulence closure models is performed. The use of explicit filtering to reduce numerical errors is compared to results from the traditional LES approach. Filter functions that are smooth in spectral space are used, as the findings of this investigation are intended for application of LES to complex domains. Explicit filtering introduces resolved subfilter-scale (RSFS) as well as subgrid-scale (SGS) turbulence terms. The former can be theoretically reconstructed; the latter must be modelled. The dynamic Smagorinsky model, the dynamic mixed model, and the new dynamic reconstruction model are all studied. It is found that for explicit filtering, increasing the reconstruction levels for the RSFS stress improves the mean velocity as well as the turbulence intensities. When compared to LES without explicit filtering, the difference in the mean velocity profiles is not large; however the turbulence intensities are improved for the explicit filtering case.