Solving the Finite-Difference Equation

A number of numerical techniques are available for solving the large set of algebraic equations of the form of (75) or (76). Selection depends to a large extent on the convenience of the programmer, the type of computer, and the required speed. 

An initial potential is guessed and assigned to each grid point. Typically, the same potential, e.g., = Oor(p = (Va+Vc)/2, 

This method uses a minimum of memory space. For N points only N real numbers are needed for storing the potentials. 

The method is iterative and requires many repeat steps, exceeding by a large factor the number of points. 

The same procedure as in the relaxation method is followed except that the potentials at each point are overcorrected or undercorrected according to certain rules. This often provides a faster convergence. 

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Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Equations (30), (31), and (32) are all highly nonlinear differential equations, so we will solve them by replacing derivatives with finite differences and use a high-speed digital computer to solve the resulting difference equations. 

Chemical reactions couple the matrix equations for each species so they cannot be solved independently. The easy way around this is to approximate the kinetic terms explicitly (using concentrations at the old time), for example in an ECE mechanism species C is made from species B. The finite difference equation for species C could therefore use the concentration of species B from the previous time step as in (115). 

M. E. Davis and J. A. McCammon, J. Comput. Chem., 10, 386 (1989). Solving the Finite Difference Linearized Poisson-Boltzmann Equation A Comparison of Relaxation and Conjugate Gradient Algorithms. 

Note that thermodynamic temperatures must be used in radiation heat transfer calculations, and ail temperatures should be expressed in K or R when a boundary condition involves radiation to avoid mistakes. We usually try to avoid the radiation boundary condition even in numerical solutions since it causes the finite difference equations to be nonlinear, wlu ch are more difficult to solve. 

Since h changes as a function of time (t), the finite difference form of equation (5.18) (5.61) becomes nonlinear. Equation (5.61) is solved in Maple below using the program developed for example 5.2.1 by solving the finite difference form of the moving boundary equation (equation (5.62) simultaneously with the governing equations for the concentration profiles ... 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

In solving the finite-difference approximation, we let FDA = 0 and, in fact, do not solve the differential equation, but rather the difference between the ODE and TE. For example, in Table 4.3, the deviation of the numerical results from those of the exact solution is caused by the truncation error, since Ax is not small enough to eliminate the effect of the truncated terms. 

To summarize the finite difference method, all we have to do is to replace all derivatives in the equation to be solved by their appropriate approximations to yield a finite difference equation. Next, we deal with boundary conditions. If the boundary condition involves the specification of the variable y, we simply use its value in the finite difference equation. However, if the boundary condition involves a derivative, we need to use the fictitious point which is outside the domain to effect the approximation of the derivative as we did in the above example at x =. The final equations obtained will form a set of algebraic equations which are amenable to analysis by methods such as those in Appendix A. If the starting equation is linear, the finite difference equation will be in the form of tridiagonal matrix and can be solved by the Thomas algorithm presented in the next section. 

For the backward difference in time, follow the same procedure as in Problem 12.12 to show that the finite difference equation for the error in solving the same problem is... 

The methods and codes applied to solve the 2D and 3D multigroup diffusion equation are well established. Most of the codes used for fast reactor analysis are based on the finite difference equation, although very efficient diffusion codes also exist using other kinds of solutions, such as finite element [4.43], coarse mesh [4.44] and nodal methods [4.45]. One of the main advantages of... 

The influence of a bend on the distribution of particles in a pipe cross-section of pneumatic conveying systems has been investigated numerically. The numerical model solved the finite-volume equations for the conservation of mass and momentum for two phases. It was evident that the cross-sectional concentration of the particles a few meters after a bend is not uniform and that the particles tend to concentrate around the pipe s wall. Various cross-sectional concentrations of particles were found for different pipe to bend radius ratios particles size and direction of gravity (i.e. horizontal to vertical flow, and horizontal to horizontal flow). Based on the (Efferent cross-sectional concentrations for different particle sizes, it was concluded that the paths taken by the particles after the bend were strongly dependent upon their sizes. As a consequence, segregation of particles downstream of a bend is expected. 

X. Zhexin, S. Yunyu, and X. Yinhu,/. Comput. Chem., 16, 200 (1995). Solving the Finite-Difference, Nonlinear, Poisson-Boltzmann Equation Under a Linear Approach. 

Note that the unknowns are at level j + 1, so all terms on the right-hand side are known. When the finite difference equation is applied for all /, a tridiagonal system of equations results. Such systems can be solved very efficiently (as compared to full-matrix linear systems) using a method called the Thomas algorithm (see below). The Matrix.xla function SYSLINT can also be used. 

To evaluate a given row of the A,B oxC matrix it can readily be seen what has to be done from the equation set. The partial derivatives of one of the functions must be evaluated with respect to each solution variable, each variable first derivative and each variable second derivative. These partial derivatives allow the elements in Eq. (11.60) to be evaluated row by row, or column by column if more convenient. In fact it is more efficient to evaluate the terms on a column by column basis as one then only has to select an increment value once for each variable and derivative and then apply this incremented value to all the functions. In terms of a single variable BV problem, there is roughly as much computational effort in setting up the finite difference equations, where N is the number of coupled equations. Then there is additional time required to solve the set of coupled matrix equations as represented by Eq. (11.58). 

The finite-element technique is based on dividing the cell domain into polygonal sections. The potential within each of the elements is assumed to be a linear combination of the value at the vertices. However, unlike the finite-dilference method, which solves the finite-difference approximation of the Laplace equation, the finite-elements method seeks a solution for the potential distribution within the cell, which best fits the Laplace equation and the boundary conditions. The degree of accuracy is similar to that of the finite-difference method however, curved boundaries and narrow corners can be described with more precision and ease. On the other hand, the presence of electrochemical nonlinear boundary conditions leads to ill-conditioned matrix equations which are more difficult to solve than the finite-difference system. 

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