Finite difference formula

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

If the grid spacing is not uniform, the formulas must be revised. The notation is shown in Fig. 3-50. The finite-difference form of the equations is then... 

The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives such techniques are called indirect methods and include the following classes ... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

We can understand how —d2fix)/dx2 shows us where fix) is locally concentrated or depleted in another way by approximating it by the finite difference formula ... 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

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... 

If f(x) is not given by a formula, or the formula is so complicated that analytical derivatives cannot be formulated, you can replace Equation (5.5) with a finite difference approximation... 

Finite difference Newton method. Application of Equation (5.8) to/(jc) = x2 - x is illustrated here. However, we use a forward difference formula for f x) and a three-point central difference formula for/"(jc)... 

Two different formulas for quadratic interpolation can be compared Equation (5.8), the finite difference method, and Equation (5.12). 

From numerous tests involving optimization of nonlinear functions, methods that use derivatives have been demonstrated to be more efficient than those that do not. By replacing analytical derivatives with their finite difference substitutes, you can avoid having to code formulas for derivatives. Procedures that use second-order information are more accurate and require fewer iterations than those that use only first-order information(gradients), but keep in mind that usually the second-order information may be only approximate as it is based not on second derivatives themselves but their finite difference approximations. 

The Excel Solver. Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on the values and formulas of a spreadsheet model. Versions 4.0 and later include an LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formula cells whose values are to be constrained (the constraints), and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions, and approximates derivatives, using finite differences. The NLP solution engine for the Excel Solver is GRG2 (see Section 8.7). 

As a consequence, the gradient of the objective function and the Jacobian matrix of the constraints in the nonlinear programming problem cannot be determined analytically. Finite difference substitutes as discussed in Section 8.10 had to be used. To be conservative, substitutes for derivatives were computed as suggested by Curtis and Reid (1974). They estimated the ratio /x of the truncation error to the roundoff error in the central difference formula... 

On the other hand, Equation (91) may be easily used in conextion with an orbital theory with the electron density and the electrostatic potential obtained from a standard SCRF wavefunction. The third term may be also evaluated from finite difference approximation formula. The charm of Eq (91) comes from the fact that it introduces for the first time, the natural reactivity indices of DFT in the expression of the solvation energy. This feature should be of great importance for the study of solvation effects in... 

This provides us with an avenue for the direct evaluation of Fukui function without considering the cationic and anionic systems. However, this approach is not generally accepted due to many inherent limitations, and Fukui functions are evaluated from finite difference formula (Equation 12.10) using atomic charges. Once the Fukui function is evaluated following a particular scheme, condensed-to-atom softness can easily be evaluated from the relation (following Equation 12.7)... 

Y.P. Chiou, Y.C. Chiang, H.C. Chang, Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices , J. Lightwave Technol. 18, 243-251 (2000). 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

The classical Lagrange formula is not efficient numerically. One can derive more efficient, but otherwise naturally equivalent interpolation formulas by introducing finite differences. The first order divided differences are defined by... 

D8 Oftentimes the formula defining a cell can be complex and tedious to type. Once such a formula is typed, it is a very good idea to save it somewhere. Cell D8 is used to represent the finite-difference formula that was typed into cell D17. Recall that for this problem the difference formula was typed only once and dragged relatively to all other cells where it is needed. 

The columns of cells below row 16 contain the values of the dependent variables at the node points. They will all be iterated until a final solution is achieved. The formula in each cell represents an appropriate form of the difference equations. Each column represents an equation. Column B represents the continuity equation, column C represents the radial momentum equation, column D represents the circumferential momentum equation, and column E represents the thermal energy equation. Column F represents the perfect-gas equation of state, from which the nondimensional density is evaluated. The difference equations involve interactions within a column and between columns. Within a column the finite-difference formulas involve the relationships with nearest-neighbor cells. For example, the temperature in some cell j depends on the temperatures in cells j — 1 and j + 1, that is, the cells one row above and one row below the target cell. Also, because the system is coupled, there is interaction with other columns. For example, the density, column F, appears in all other equations. The axial velocity, column B, also appears in all other equations. 

These equations are now in convenient form for a finite-difference scheme along lines similar to that used above. Ail alternative approach developed in some detail employs the Lagrangian interpolation formula to follow the motion of the boundary in the (x/a, r) plane. This is a means of developing finite-difference approximations to derivatives based on functional values and not necessarily equally spaced in the argument. Crank points out that the application of Lagrangian interpolation formulas involves a relatively large number of steps in time, whereas the fixedboundary procedures require iterative solutions at each time interval, which are, however, far fewer in number. 

Based on the finite-difference formula Eq. (3.31) all Hessian updates Eire required to fulfill the quasi-Newton condition... 

Table 3-2 Summary of Nodal Formulas for Finite-Difference Calculations (Dashed Lines Indicate Element Volume.)...

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

The Lagrange interpolation polynomial was used to develop the spatial finite difference formulas used for the distance method of lines calculation. For example, the two point polynomial for the solids flux variable F(t,z) can be expressed by... 

Different combinations of spatial finite difference formulas were tried to determine the best set for our system of equations. The two point upwind formula was found to be best for the solids component molar fluxes. The low order formula was used because most of the gasifier reactions turn off abruptly when a component disappears and this creates sharp discontinuities. Higher order formulas tend to flatten out discontinuities, and in some cases, this causes material balances to be lost which then leads to numerical instability problems. Maintaining component material balance is an important aid to preserving numerical stability in the calculations. The low order formulas minimized these difficulties. 

The original system of partial differential equations is transformed into a system of ordinary differential equations by replacing the time differential terms with time finite difference formulas. The number of equations in the new system is the same as the original number of equations. However, it is necessary to store intermediate results at spatial nodes for both current and previous time increments. 

The Lagrange interpolation polynomial was again used to develop the finite difference formulas. To avoid additional iterations, only upwind differences were used. The two point upwind formula for the solids stream concentration variable at any location z within the reactor for time t is given by... 

In equation (42), the quantities j denote the vailues of the function / or those of its derivative / jj and / z. The details of the basis functions, the quantities j and the nodes of an element have been presented elsewhere [26], The first and second derivatives of the mapping function / can be obtained from derivatives of the basis fimctions. To compute the pressiure p and tensor components, the four nodal values at points A, B, C and D of each element are chosen as unknowns. Their derivatives are evaluated by using finite-difference formulae. In Fig 14 we present computed streamlines for 4/1 and 8/1 axisymmetric contractions, and the computed shear stress along the computed streamlines, for a K-BKZ fluid of the form given by Papanastasiou et al [19]. 

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