Finite difference solution

Davis, M. E., McCammon, J. A. Dielectric boundary smoothing in finite difference solutions of the poisson equation An approach to improve accuracy and convergence. J. Comp. Chem. 12 (1991) 909-912. 

R. J. BShando, A Finite Difference Solution of Onsager s Modelfor Flow in a Gas Centrifuge Rept. UVA-ER-822-83U, University of Virginia, ChadottesviUe, 1983. 

D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of Some Adaptive Node-Movement Techniques in Finite-Element and Finite-Difference Solutions of Partial Differential Equations, J. Comput. Phys. 95 (1991). 

The first step in obtaining a finite difference solution is to divide the domain into nodal blocks, as shown in Figure 2.11. This process is known as discretization, or more simply as gridding. In a one-dimensional simulation, the domain is divided into Nx nodal blocks, so for a domain of length L, each nodal block is of length Ax = L/Nx. In two dimensions, there are Nx x Ny blocks, each of dimensions Ax x Ay, where for a domain of width W, Ay = W/Ny. 

The study of fire in a compartment primarily involves three elements (a) fluid dynamics, (b) heat transfer and (c) combustion. All can theoretically be resolved in finite difference solutions of the fundamental conservation equations, but issues of turbulence, reaction chemistry and sufficient grid elements preclude perfect solutions. However, flow features of compartment fires allow for approximate portrayals of these three elements through global approaches for prediction. The ability to visualize the dynamics of compartment fires in global terms of discrete, but coupled, phenomena follow from the flow features. 

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. ). 

At this point, specification of the finite-difference solution method is complete in that we have chosen to utilize the finite-difference scheme and have specified the mesh properties and the sampling of points required to provide the desired approximation to the derivatives of U. Such systems can be solved efficiently even for N and M large (>1000), although time scales of typical calculations range from several minutes to hours, depending on the type of computers used, eliminating some from consideration in those time-critical situations. We defer additional discussion of achieving solutions until the next step. 

This is actually of very low order in comparison to traditional finite difference solutions. 

Write a simulation program that implements a finite-difference solution of the system. Using this simulation, reproduce the results shown in Figs. 4.4 and 4.5. 

Discuss any pro s and con s that may be identified between the approach discussed in Section 4.2 and the finite-difference solution of the differential equations. 

This is a linear ordinary-differential-equation boundary-value problem that can be solved analytically (see Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, 1960). Here, however, proceed directly to numerical finite-difference solution, which can be implemented easily in a spreadsheet. Assuming a cone angle of a = 2° and a rotation rate of 2 = 30 rpm, determine f(0) — v /r. 

Figure 6.5 illustrates a stencil for the finite-difference discretization described by Eqs. 6.42 and 6.43. A spreadsheet that implements the finite-difference solution is described in detail in Appendix D (Section D.2). 

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

Implicit Euler finite difference solution for a cooling amorphous polymer plate. 

Lattice Boltzmann (LB) is a relatively new simulation technique and it represents an alternative numerical approach in the hydrodynamics of complex fluids. The LB method can be interpreted as an unusual finite-difference solution of the continuity... 

Figure 1. Comparison of the iterative finite difference solution of this work to Gears method...

Nodal points used in finite-difference solution of the continuigy equation. 

Nodal points used in obtaining finite difference solution. 

Instead of using the method of separation of variables, a numerical solution to Eq. (4.144) can be easily obtained. Here, a numerical finite-difference solution will be discussed. A series of grid lines in the Z and R directions are introduced as shown in Fig. 4.16, a uniform grid spacing, ARt being used in the radial coordinate direction. 

A simple finite-difference solution to the above set of equations will be discussed here. A series of nodal lines in the Z- and / -directions are introduced. 

Because the velocity field is fully developed, the variations of U and E with R are known. The solution to Eq. (7.93) can therefore be obtained using a similar procedure to that used in Chapter 4 to solve for thermally developing laminar pipe flow, i.e., using separation of variables. Here, however, a numerical finite-difference solution procedure will be used because it is more easily adapted to the situation where the wall temperature is varying with Z. 

Approximate solutions for the two limiting cases discussed above can be obtained (see below). However, most real flows are not well described by either of these two limiting solutions. For this reason, a numerical solution of the governing equations must usually be obtained. To illustrate how such solutions can be obtained, a simple forward-marching, explicit finite-difference solution will be discussed here. 

An iterative finite-difference solution procedure will be discussed here. Many other more efficient solution procedures are available but the simple procedure considered here should serve to indicate the main features of such methods. 

A forward-marching implicit finite-difference solution of the energy equation will again be considered. In order to obtain this solution, a series of nodal lines running parallel to the x and y-axes are again introduced as shown in Fig. 10.15. 

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