Derivative, central-difference form approximation

An alternative approximation to the derivative is the central-difference form... 

When the user, whether working on stand-alone software or through a spreadsheet, supplies only the values of the problem functions at a proposed point, the NLP code computes the first partial derivatives by finite differences. Each function is evaluated at a base point and then at a perturbed point. The difference between the function values is then divided by the perturbation distance to obtain an approximation of the first derivative at the base point. If the perturbation is in the positive direction from the base point, we call the resulting approximation a forward difference approximation. For highly nonlinear functions, accuracy in the values of derivatives may be improved by using central differences here, the base point is perturbed both forward and backward, and the derivative approximation is formed from the difference of the function values at those points. The price for this increased accuracy is that central differences require twice as many function evaluations of forward differences. If the functions are inexpensive to evaluate, the additional effort may be modest, but for large problems with complex functions, the use of central differences may dramatically increase solution times. Most NLP codes possess options that enable the user to specify the use of central differences. Some codes attempt to assess derivative accuracy as the solution progresses and switch to central differences automatically if the switch seems warranted. 

The derivatives in eq 188 are approximated by central differences. This leads to a discrete representation of the problem in the form of linear equations. For j = 1 and j = N we have slightly modified equations, since here the boundary conditions have to be considered. Because the concentration of three isomers has to be calculated for each of the N volume elements as a whole, we have a system of N x 3 linear equations. This can be expressed in matrix notation ... 

For the discretization of spatial derivatives, the particular method uses a family of central difference operators, while a parametric expression with an extra degree of freedom for the temporal derivatives is employed [57], In two dimensions, these approximants have the general forms... 

We now express first, second, and mixed partial derivatives in terms of finite differences. We show the development of these approximations using central differences, and in addition we summarize in tabular form the formulas obtained from using forward and backward differences. 

In the limit as Ax 0, all three formulas agree if the derivative indeed exists. In the method of finite differences, we use finite, but small, values of Ax in one of (1.240) to approximate the derivative by an algebraic form. We study this method in further detail in Chapter 6 however, for now we merely note that the first approximation formula given above, the central-difference approximation, is the most accmate. 

A different situation obtains in compounds such as the basic beryllium monocarboxylates (140-142). In particular the acetate derivative, Be40(02CCH3)6, has been known in the solid state for nearly 100 years (143). A partial structure of this compound is in Fig. 13 (144). The central oxide ion is surrounded by a tetrahedron of beryllium atoms. The acetate ions form bridges across the six edges of the Be4 tetrahedron, with each acetate bonding with two Be atoms. Each berylium atom in surrounded by an approximate tetrahedron of... 

As shown in this chapter for the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaced by their finite-differenced equivalents. These finite-differ-enced forms of the model equations are shown to evolve as a natural consequence of the balance equations, according to Franks (1967), and as derived for the various examples in this book. The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end-sections can be better approximated by the forward and backward differences as derived in the previous examples. The various forms of approximation based on the use of central, forward and backward differences have been listed by Chu (1969). 

In this scheme, the temporal derivative is formed by the central (second-order ) difference between the upper and lower points, the second spatial derivative being approximated as usual. This makes the discretisation at the index i in space,... 

In the FDM, the differential form of the conservation equations (cf. (19.12) or (19.13)) are discretized by approximating the spatial and temporal derivatives by means of an appropriate difference quotient, such as a forward, central, or backward difference. The spatial derivatives utilize the cell nodes in one form or the other to achieve this discretization, while the temporal derivatives use a given time step. FDMs require a structured grid, that is, meshes that are topologically equivalent to a right hexahedron in integer space, called the logical space, where the nodes... 

By approximating the spatial derivatives according to the central three-point difference formula and using the backward implicit scheme, the form of the resulting equations for species A and B are analogous to those discussed in previous chapters ... 

Tortillas represent 30% of aU baked product sales in the United States and continue to be the most popular food in Mexico and Central America. Approximately, 120 million tortillas are consumed yearly in the United States, making these the second most popular baked product after white bread. An average Mexican consumes more than 80 kg of maize tortillas annually. Today, derived products such as flat tostadas and tortilla chips are extensively sold as snacks and in the preparation of fast foods (Chapter 12). Tostadas are the base for the preparation of a wide array of meals. Taco shells are the American version of tostadas, the only difference being that they are usually fried bent (U form) and rarely colored. Regular and low-fat tostadas and taco shells are available in grocery stores in Mexico and the United States. 

---