Spatial finite difference formulation

The independent variables on which fJK depends are k and t. The principal advantage of using this formulation is that spatial derivatives become summations over wavenumber space. The resulting numerical solutions have higher accuracy compared with finite-difference methods using the same number of grid points. 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

Finite difference methods (FDM) are directly derived from the space time grid. Focusing on the space domain (horizontal lines in Fig. 6.6), the spatial differentials are replaced by discrete difference quotients based on interpolation polynomials. Using the dimensionless formulation of the balance equations (Eq. 6.107), the convection term at a grid point j (Fig. 6.6) can be approximated by assuming, for example, the linear polynomial. 

In this section, an explicit time advance scheme for unsteady flow problems is outlined [30]. The momentum equation is discretized by an explicit scheme, and a Poisson equation is solved for the pressure to enforce continuity. The continuity is discretized in an implicit manner. In the original formulation, the spatial derivatives were approximated by finite difference schemes. 

The Fast Implicit Finite Difference method, implemented by Rudolph [68, 69] has marginally higher computational requirements but has higher potential accuracy when tackling demanding problems. The FIFO formulation expresses the difiu-sional transport equation on an exponentially expanding spatial grid via... 

Methods that approximate the spatial derivatives with fourth order accuracy using only three points are commonly called compact methods. They are known under different names in numerical mathematics, like Mehrstel-lenformeln and Hermitian Formulas [6]. Compact methods have been used with success in fluid mechanics [7, 8]. Two-dimensional compact discretizations are considered e.g. in [9], non-aequidistant formulations are derived in [10]. To our knowledge, compact finite differences have so far not been used in the solution of Chemical Engineering problems. 

To study the dynamic behavior of the BZ gels, we numerically integrate Eqs (8.1 -8.3) in two [1, 2] or three [3] dimensions using our recently developed gLSM. This method combines a finite-element approach for the spatial discretization of the elastodynamic equations and a finite-difference approximation for the reaction and diffusion terms. We used the gLSM approach to examine 2D confined films and 3D bulk samples here, we briefly discuss the more general 3D formulation [3]. 

A finite difference Reynolds equation with the central difference formulation of the. first order spatial derivatives and a Crank-Nicolson scheme for incorporating the time dependent term [3] was used for the analyses presented in the current paper, and the results obtained with an alternative finite element formulation were indistinguishable. 

In these equations, a non-uniform spatial grid has been assumed and the subscript j refers to the row of the matrix element and the subscript k refers to the column of the matrix. The subscript i is used to refer to the spatial point around which the finite difference equation is being written. If this formulation of the coefficients is used for three equations with no coupling between the equations, only the diagonal elements would have non-zero terms. In many practical problems, several of the partial derivatives may be zero for example the second derivative may occur uncoupled in the form of Eq. (11.55) in which case the A matrix would be of diagonal form. However, to retain as much flexibility as possible with the equation formulation, all terms will be kept in the implementation here. 

With the above formulism a method is now defined for forming a finite difference set of equations for a partial differential equation of the initial value type in time and of the boundary value type in a spatial variable. The method can be applied to both linear and nonlinear partial differential equations. The result is an implicit equation which must be solved for the spatial variation of the solution... 

Solvents with vanishing molecular dipole moments but finite higher order multipoles, such as benzene, toluene, or dioxane, can exhibit much higher polarity, as reflected by its influence on the ET energetics, than predicted by the local dielectric theory [228], Full spatially dispersive solvent response formulation is required in this case [27-29, 104, 229-233], There are different approaches to the problem of spatial dispersion. The original formulation by Kornyshev and co-workers [27c, 28] introduces the frequency-dependent screening effect on the basis of heuristic arguments. More recent approaches are based on the density-function theory [104,197],... 

The finite volume method, which returns to the balance equation form of the equations, where one level of spatial derivatives are removed is the method of choice always for the pressure equation and nearly always for the saturation equation. Commercial reservoir simulators are, with the exception of streamline simulators, entirely based on the finite volume method. See [11] for some background on the finite volume method, and [26] for an introduction to the streamline method. The robustness of the finite volume method, as used in oil reservoir simulation, is partly due to the diffusive nature of the numerical error, known as numerical diffusion, that arises from upwind difference methods. An interesting research problem would be to analyse the essential role that numerical diffusion might play in the actual physical modelling process particularly in situations with unstable flow. In the natural formulation, where the character of the problem is not clear, and special methods applicable to hyperbolic, or near hyperbolic problems are not applicable, the finite volume method, in the opinion of the author, is the most trustworthy approach. 

The FE solvers developed in this chapter have made use of some of the approximate matrix solution techniques developed in the previous chapter. However, most of the code is new because of the different basic formulation of the finite element approach. In this work the method of weighted residuals has been used to formulate sets of FE node equations. This is one of the two basic methods typically used for this task. In addition, the development has been based upon flie use of basic triangular spatial elements used to cover a two dimensional space. Other more general spatial elements have been sometimes used in the FE method. Finally the development has been restricted to two spatial dimensions and with possible an additional time dimension. The code has been developed in modular form so it can be easily applied to a variety of physical problems. In keeping with the nonlinear theme of this work, the FE analysis can be applied to either linear or nonlinear PDEs. 

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