The Finite-Difference Method

The grid may be rectangular such that hx need not be equal io hy. Also, the grid size may vary with position in the cell to accommodate necessary resolution in different regions. An extension is an adaptive grid, where the grid is adjusted to accommodate certain geometrical features (e.g., a corner) or variations in the current density. 

The Laplace equation is first written in terms of a finite-difference approximation. For simplicity, a two-dimensional Cartesian coordinate system is assumed  

The second derivatives are approximated in terms of the forward (70) and backward (71), difference equations  

The finite-difference form of the Laplace equation at any grid point (/, j) is accordingly 

Applying the finite-difference form of the Laplace equation (75) to each grid point in the cell yields a large set of equations to be solved. For example, on a 30 x 30 grid there are 900 equations to be solved. However, each equation has at most five unknowns. The unknown potentials are solved by a numerical iterative procedure. When calculating the secondary or tertiary distribution, exponential terms will appear at the boundaries. These terms are linearized about the potential of the previous iteration. The iterations are repeated until the potentials converge to within the specified error, typically 10 or smaller. 

The Unite difference method is the oldest method for numerical solution of PDF s, presumably introduced by Euler in the 18th century [71, 49, 174, 167]. It is a convenient method to use for simple geometries. 

Taylor series expansions can be used to obtain approximations to the first and second order derivatives of the variables with respect to the coordinates. When necessary, these methods are used to obtain variable values at locations other than grid nodes (interpolation). The Taylor series expansion can be defined by  

By forming linear combinations of the values of the function at various grid points z,z Az, z 2Az, and so on, we can obtain approximations to the derivatives. 

The truncation error is usually proportional to a power of the grid spacing Axi and/or the time step At. If the most important error term is proportional to (Z xj) or (Zlf) we call the method an nth-order approximation, often simply indicated by 0 Ax ) or 0 At ), respectively, n 0 is required to 

The simplest approximations for the first derivative are given below. 

Forward differences, first order explicit Euler method  

Backward differences, first order implicit Euler method dtp ip(z) - ip(z - Az) 

The simplest approximation for the second derivative can be obtained in a similar manner. 

Tht finite-difference method replaces the derivatives in the differential equations with finite difference approximations at each point in the interval of integration, thus converting the differential equations to a large set of simultaneous nonlinear algebraic equations. To demonstrate this method, we use, as before, the set of two differential equations  

we express the derivatives of y in terms of forward finite differences using Eq. (4.33)  

For higher accuracy, we could have used Eq, (4.41), which has error of order Jt instead of Eq. (4.33). In either case, the steps of obtaining the solution to the boundary-value problem are identical. 

We divide the interval of integration into n segments of equal length and write Eqs. (5.125) 

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For the determination of the approximated solution of this equation the finite difference method and the finite element method (FEM) can be used. FEM has advantages because of lower requirements to the diseretization. If the material properties within one element are estimated to be constant the last term of the equation becomes zero. Figure 2 shows the principle discretization for the field computation. 

Fig. 1. Explanation of the principles of the finite-difference method for solution of the Poisson-Boltzmann equation...

Equation (23) represents the essence of the finite-difference method [21, 22, 23, 24],... 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point x, is assigned the symbol Cj = c(Xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are ... 

Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems. 

Example A reaction diffusion problem is solved with the finite difference method. 

The Galerldn finite element method results when the Galerldn method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerldn finite element equations. The element integrals are defined as... 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

In the finite difference method an explicit technique would evaluate the right-hand side at the /ith time level. 

The varianee equation ean be solved direetly by using the Calculus of Partial Derivatives, or for more eomplex eases, using the Finite Difference Method. Another valuable method for solving the varianee equation is Monte Carlo Simulation. However, rather than solve the varianee equation direetly, it allows us to simulate the output of the varianee for a given funetion of many random variables. Appendix XI explains in detail eaeh of the methods to solve the varianee equation and provides worked examples. 

With reference to Appendix XI, we can solve each partial derivative term in equation 4.114 using the Finite Difference method to give ... 

The numerical solution, as mentioned earlier, was obtained by the finite difference method. The two regions (layers) indicated in Figure 4-52 are represented with a series of regularly spaced material points... 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

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 . ... 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

Mitchell, A. and Griffits, D. (1980) The Finite Difference Methods in Partial Differential Equations. Wiley New York. 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

Within the finite-difference method (FDM), the derivative terms appearing in Eq. (32) are approximated by finite-difference expressions at each grid node. As an... 

While most authors have used the finite-difference method, the finite element method has also been used—e.g., a two-dimensional finite element model incorporating shrinkable subdomains was used to de.scribe interroot competition to simulate the uptake of N from the rhizosphere (36). It included a nitrification submodel and found good agreement between ob.served and predicted uptake by onion on a range of soil types. However, while a different method of solution was used, the assumptions and the equations solved were still based on the Barber-Cushman model. 

The Finite Difference Method (FD)168 169. This is a general method applicable for systems with arbitrary chosen local dielectric properties. In this method, the electrostatic potential (RF) is obtained by solving the discretized Poisson equation ... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

The numerical methods for partial differential equations can be classified according to the type of equation (see Partial Differential Equations ) parabolic, elliptic, and hyperbolic. This section uses the finite difference method to illustrate the ideas, and these results can be programmed for simple problems. For more complicated problems, though, it is common to rely on computer packages. Thus, some discussion is given to the issues that arise when using computer packages. 

The finite-difference method can be combined with the perturbation technique that was previously used to derive the basic formulas in Sect. 2.2. This yields another single-state perturbation formula [49, 50]. Starting from (2.66), we get... 

Each nodal block represents a distinct system, as we have defined it (Fig. 2.1). Conceptually, the properties of the entire block are projected onto a nodal point at the block s center (Fig. 20.2). A single value for any variable is carried per node in a transport or reactive transport simulation. There is one Ca++ concentration, one pH, one porosity, and so on. In other words, there is no accounting in the finite difference method for the extent to which the properties of a groundwater or the... 

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