Finite difference methods fixed grid

Throughout the entire chapter, the functions u(x) of the continuous argument x G G are the elements of some functional space Hq- The space Hh comprises all of the grid functions yii(x), providing a possibility to replace within the framework of the finite difference method the space Hq by the space Hh of grid functions yh x). Recall that although the fixed notation is usually adopted, there is a wide variety of possible choices of the functional form of . ... 

Fixed Coordinate Approaches. In the fixed coordinate approach to airshed modeling, the airshed is divided into a three-dimensional grid for the numerical solution of some form of (7), the specific form depending upon the simplifying assumptions made. We classify the general methods for solution of the continuity equations by conventional finite difference methods, particle in cell methods, and variational methods. Finite difference methods and particle in cell methods are discussed here. Variational methods involve assuming the form of the concentration distribution, usually in terms of an expansion of known functions, and evaluating coeflBcients in the expansion. There is currently active interest in the application of these techniques (23) however, they are not yet suflBciently well developed that they may be applied to the solution of three-dimensional time-dependent partial differential equations, such as (7). For this reason we will not discuss these methods here. 

Inside each fluid, p and p are constants. Equations (95) and (96) were solved by using a finite difference method on a fixed two- or three-dimensional grid. The spatial terms were discretized by second-order finite differences on a staggered Eulerian grid. The discretization of time was achieved by an expKdt Euler method or a second order Adams Bashforth method. The boundary conditions used in their study were either periodic or full sKp in the horizontal directions and rigid, stress-free on the top and bottom. 

A number of standard numerical techniques are available to solve the resultant partial differential equation derived from Darcy s law, including the finite difference method (FDM), the boundary element method (BEM) and the finite element method (FEM) [16]. Since mould filling is a moving boundary problem, these numerical techniques can be broadly divided into two approaches into moving grid or fixed grid [17-18] ... 

The simulation of a continuous, evaporative, crystallizer is described. Four methods to solve the nonlinear partial differential equation which describes the population dynamics, are compared with respect to their applicability, accuracy, efficiency and robustness. The method of lines transforms the partial differential equation into a set of ordinary differential equations. The Lax-Wendroff technique uses a finite difference approximation, to estimate both the derivative with respect to time and size. The remaining two are based on the method of characteristics. It can be concluded that the method of characteristics with a fixed time grid, the Lax-Wendroff technique and the transformation method, give satisfactory results in most of the applications. However, each of the methods has its o%m particular draw-back. The relevance of the major problems encountered are dicussed and it is concluded that the best method to be used depends very much on the application. 

Method of lines. This method, which was introduced in the crystallization research by Tsuruoka and Randolph (JJ, transforms the population balance into a set of ordinary differential equations by discretization of the size axis in a fixed number of grid points. The differential 3 G L,t)n(L,t) /3L is then approximated by a finite difference scheme. As has been shown (X) a fourth-order accurate scheme using an equidistant grid (i- l.z) results in ... 

The closed equation (12.383) does not necessary conserve the moments of the distribution due to the macroscopic or finite grid resolution employed in the size domain, thus some sort of ad hoc numerical correction must be induced to enforce the conservative moment properties. It is noted that it is mainly at this point in the formulation of the numerical algorithms that the class method of Hounslow et al. [88], the discrete fixed pivot method of Kumar and Ramkrishna [112] and the multi-group approach used by Carrica et al. [30], among others, differs to some extent. The problem in question is related to the birth terms only. Following the discrete fixed pivot method of Kumar and Ramkrishna [112], the formation of a particle of size in size range... 

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