Finite difference methods application

Finite-Difference Methods. The numerical analysis literature abounds with finite difference methods for the numerical solution of partial differential equations. While these methods have been successfully applied in the solution of two-dimensional problems in fluid mechanics and diffusion (24, 25), there is little reported experience in the solution of three-dimensional, time-dependent, nonlinear problems. Application of these techniques, then, must proceed by extending methods successfully applied in two-dimensional formulations to the more complex problem of solving (7). The various types of finite-difference methods applicable in the solution of partial differential equations and their advantages and disadvantages are discussed by von Rosenberg (26), Forsythe and Wasow (27), and Ames (2S). 

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. 

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 ... 

Finally the BERTHA technology has been applied to relativistic density functional theory by Quiney and Belanzoni [36]. This showed that the method works well for closed shell atoms as compared with benchmark calculations using finite difference methods, and there have been promising parallelization studies [37] which should in future greatly extend the range of application of the code. 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

This treatment leads to a system of stiff, second-order partial differential equations that can be solved numerically to yield both transient and steady-state concentration profiles within the layer (Caras et al., 1985a). Because the concentration profile changes most rapidly near the x = L boundary an ordinary finite-difference method does not yield a stable solution and is not applicable. Instead, it is necessary to transform the distance variable x into a dummy variable y using the relationship... 

The finite difference method (FDM) is probably the easiest and oldest method to solve partial differential equations. For many simple applications it requires minimum theory, it is simple and it is fast. When a higher accuracy is desired, however, it requires more sophisticated methods, some of which will be presented in this chapter. The first step to be taken for a finite difference procedure is to replace the continuous domain by a finite difference mesh or grid. For example, if we want to solve partial differential equations (PDE) for two functions 4> x) and w(x, y) in a ID and 2D domain, respectively, we must generate a grid on the domain and replace the functions by functions evaluated at the discrete locations, iAx and jAy, (iAx) and u(iAx,jAy), or 4>i and u%3. Figure 8.1 illustrates typical ID and 2D FDM grids. 

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

When q is zero, Eq. (5-18) reduces to the familiar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplace s equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger (Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations applicable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Fixed Coordinate Approaches. In the fixed coordinate approach to airshed modeling, the airshed is divided into a three-dimensional grid for the numerical solution of some form of (7), the specific form depending upon the simplifying assumptions made. We classify the general methods for solution of the continuity equations by conventional finite difference methods, particle in cell methods, and variational methods. Finite difference methods and particle in cell methods are discussed here. Variational methods involve assuming the form of the concentration distribution, usually in terms of an expansion of known functions, and evaluating coeflBcients in the expansion. There is currently active interest in the application of these techniques (23) however, they are not yet suflBciently well developed that they may be applied to the solution of three-dimensional time-dependent partial differential equations, such as (7). For this reason we will not discuss these methods here. 

There have thus far been reported only two applications of finite-difference methods to the solution of (7) as they pertain to urban airsheds, both for the Los Angeles Basin. Eschenroeder and Martinez (21) applied the Crank-Nicolson implicit method to the simplified version of, (7),... 

SOLUTION Wo have solved this problem in Example 5-1 for the steady case, and here we repeat it for the transient case to demonstrate the application of the transient finite difference methods. Again we assume one-dimensional heat transfer in rectangular coordinates and constant thermal conductivity. The number of nodes is specified to be W = 3, and they arc chosen to be at the two surfaces of the plate and at the middle, as shown in the figure. Then the nodal spacing Ax becomes... 

Gray. S.K. and Goldfield. E.M. (2001) Dispersion fitted finite difference method with applications to molecular quantum mechanics J. Chem. Phys. 115, 8331-8344. 

Two methods are available for the numerical solution of initial-boundary-value problems, the finite difference method and the finite element method. Finite difference methods are easy to handle and require little mathematical effort. In contrast the finite element method, which is principally applied in solid and structure mechanics, has much higher mathematical demands, it is however very flexible. In particular, for complicated geometries it can be well suited to the problem, and for such cases should always be used in preference to the finite difference method. We will limit ourselves to an introductory illustration of the difference method, which can be recommended even to beginners as a good tool for solving heat conduction problems. The application of the finite element method to these problems has been described in detail by G.E. Myers [2.52]. Further information can be found in D. Marsal [2.53] and in the standard works [2.54] to [2.56]. 

An alternative way of solving the Poisson Boltzmann equation is the finite element method, which uses nonuniform and not rectangular grids. For example, the grid may be made finer around an active site to accurately evaluate ligand binding, and coarser elsewhere. This achieves comparable accuracy with the finite difference methods, but with a smaller number of grid points. Unfortunately, the finite element method has not been used extensively in applications only implementations of the method have been reported to date [45 47],... 

Initially (time zero) the values of c are known and therefore the values of cj (concentrations at t = At) can be calculated by direct application of (25.93). This approach can be repeated and the values of c"+1 can be calculated from the previously calculated values of c . This is an example of an explicit finite difference method, where, if approximate solution values are known at time t = nAt, then approximate values at time tn+1 = (n+ 1) At may be explicitly and immediately calculated using (25.93). Typically, explicit techniques require that constraints be placed on the size of Af that may be used to avoid significant numerical errors and for stable operation. In a stable method, unavoidable small errors in the solution are suppressed with time in an unstable method a small initial error may increase significandy, leading to erroneous results or a complete failure of the method. Equation (25.93) is stable only if At < (Ax)2/2, and therefore one is obliged to use small integration timesteps. 

Unsteady net flow calculation was developed by Weimann et.al. 151, originally as a simulation technique in process-computers for network control. Using a finite difference method the unsteady one dimensional balance equations for mass and momentum were solved, to predict pressure and flow development in high pressure networks. An application of this method at VEW is described in /3/. 

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