Finite-difference method methods

Finite difference methods Methods used for the calculation of ntunerical solutions of systems of partial differential equations. The differential elements in the differential equations are replaced by corresponding finite differences, giving difference equations. Stability and accuracy conditions must be satisfied (Chapter 10, Section 10.3). 

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. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

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],... 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

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. 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

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 ... 

Because the coefficient depends on c, the equations are more compBcated. A finite difference method can he written in terms of the fluxes at the midpoints, -t- 1/2. 

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. 

Finite Difference Methods Solved with Spreadsheets A... 

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. 

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

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces 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 apphcable 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. 

W. Herrmann and L.D. Bertholf, Explicit Lagrangian Finite-Difference Methods,... 

A good starting point for understanding finite-difference methods is the Taylor expansion about time t of the position at time t + At,... 

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 ... 

Smith, G.D., 1985. Numerical Solution of Partial Differential Equations Finite Difference Methods, 3rd edition. Clarendon Press. 

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... 

Many numerical methods have been proposed for this problem, most of them finite-difference methods. Using a finite-difference technique, Brode (1955) analyzed the expansion of hot and cold air spheres with pressures of 2000 bar and 1210 bar. The detailed results allowed Brode to describe precisely the shock formation process and to explain the occurrence of a second shock. 

Guirao and Bach (1979) used the flux-corrected transport method (a finite-difference method) to calculate blast from fuel-air explosions (see also Chapter 4). Three of their calculations were of a volumetric explosion, that is, an explosion in which the unbumed fuel-air mixture is instantaneously transformed into combustion gases. By this route, they obtained spheres whose pressure ratios (identical with temperature ratios) were 8.3 to 17.2, and whose ratios of specific heats were 1.136 to 1.26. Their calculations of shock overpressure compare well with those of Baker et al. (1975). In addition, they calculated the work done by the expanding contact surface between combustion products and their surroundings. They found that only 27% to 37% of the combustion energy was translated into work. 

The three primary advantages of the finite element approach over finite difference methods are [9] ... 

Other methods for solving PDE s include Monte Carlo, spectral, and variational. Spectral methods in particular converge more rapidly than finite difference methods, but do not handle problems involving irregular geometries or discontinuities well. 

Forsythe, George E., and Wasow, Wolfgang R., Finite-Difference Methods for... 

Huang, R, Wang, H. Z., Xu, L. G., Meng, Y. G., and Wen, S. Z., Numerical Analysis of the Lubrication Performances for Ultrathin Gas Film Lubrication of Magnetic Head/Disk with a New Finite Difference Method, Proceedings of IMECE05, Paper No. IMECE2005-80707,2005. 

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. 

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