Special finite difference techniques

This chapter discusses finite-difference techniques for the solution of partial differential equations. Techniques are presented for pure convection problems, pure diffusion or dispersion problems, and mixed convection-diffusion problems. Each case is illustrated with common physical examples. Special techniques are introduced for one- and two-dimensional flow through porous media. The method of weighted residuals is also introduced with special emphasis given to orthogonal collocation. 

For those cases of explosive initiation in which the important mechanism of energy transfer is heat conduction, the usual method of numerical solution is by a finite difference technique as is described in Appendix A for the SIN code. The technique for two-dimensional geometries is described in Appendixes B and C. If one is interested only in heat conduction, one can use special purpose codes such as TEPLO described in reference 1. 

Abstract The aim of this chapter is to introduce special numerical techniques. The first part covers special finite element techniques which reduce the size of the computational models. In the case of the substructuring technique, internal nodes of parts of a finite element mesh can be condensed out so that they do not contribute to the size of the global stiffiiess matrix. A post computational step allows to determine the unknowns of the condensed nodes. In the case of the submodel technique, the results of a finite element computation based on a coarse mesh are used as input, i.e., boundary conditions, for a refined submodel. The second part of this chapters introduces alternative approximation methods to solve the partial differential equations which describe the problem. The boundary element method is characterized by the fact that the problem is shifted to the boundary of the domain and as a result, the dimensionality of the problem is reduced by one. In the case of the finite difference method, the differential equation and the boundary conditions are represented by finite difference equations. Both methods are introduced based on a simple one-dimensional problem in order to demonstrate the major idea of each method. Furthermore, advantages and disadvantages of each alternative approximation methods are given in the light of the classical finite element simulation. Whenever possible, examples of application of the techniques in the context of adhesive joints are given. 

Without a solution, formulated mathematical systems (models) are of little value. Four solution procedures are mainly followed the analytical, the numerical (e.g., finite different, finite element), the statistical, and the iterative. Numerical techniques have been standard practice in soil quality modeling. Analytical techniques are usually employed for simplified and idealized situations. Statistical techniques have academic respect, and iterative solutions are developed for specialized cases. Both the simulation and the analytic models can employ numerical solution procedures for their equations. Although the above terminology is not standard in the literature, it has been used here as a means of outlining some of the concepts of modeling. 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

The dependence of e on position allows to describe both local saturation effects, and, in salt solution, the dependence of e on concentration. In numerical grid methods (there are families of methods, called Finite Elements Methods, FEM, and Finite Differences Methods, FDM - see Tomasi and Persico (1994) for a short explanation of differences) there is nothing but a little increase in complexity in treating eq.(lll) instead of eqs.(109-110). The treatment of the latter model by means of a MPE method is more difficult, and a specialized PCM version (Cossi et al., 1994) is necessary to exploit it with a BEM-derived method (see Juffer et al., 1991, for another proposal based on the BEM technique). 

Boundary value problems are encountered so frequently in modelling of engineering problems that they deserve special treatment because of their importance. To handle such problems, we have devoted this chapter exclusively to the methods of weighted residual, with special emphasis on orthogonal collocation. The one-point collocation method is often used as the first step to quickly assess the behavior of the system. Other methods can also be used to treat boundary value problems, such as the finite difference method. This technique is considered in Chapter 12, where we use this method to solve boundary value problems and partial differential equations. 

Figure 5 shows a representative log-log plot of error versus problem size and emphasizes the fact that h-p techniques provide the best available way to get the most out of one s computational effort. Even more significant is the observation that these special adaptive techniques can produce numerical solutions to problems which are impossible to obtain by conventional finite difference or finite element techniques on the largest existing supercomputers Indeed, to reproduce the accuracy obtainable by h-p methods on some model elliptic problems, a finite difference mesh consisting of over ten million grid points would be required. 

In the case of alternant hydrocarbons it is possible to show that the finite changes in SCF charges, bond orders, and free valences due to changes in the parameters Wf and y (Greenwood and Hayward, 1960), have properties which are completely analogous to those of the corresponding quantities used in the Hiickel approach of Section IVB. The SCF results incorporate those of the Hiickel method as a special case, in which electron repulsion terms can be dropped from the non-linear equations without invalidating the derivations. The theoretical techniques used to obtain the analytical properties are essentially different from those described previously for Hiickel theory, but the result can be stated briefiy in similar terms, and this will sufiice for present purposes. 

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