Central-difference formulae

Third card FORMAT(8F10.2), size of increments to be used in central difference formula for calculating derivatives with respect to the independent variables. 

The leap-frog algorithm uses the simplest central difference formula for a derivative... 

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)... 

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... 

For stationary flows, the time-averaged values should be used in place of X, y in the central-difference formula in order to improve the smoothness of the estimated fields. For non-stationary flows, it may be necessary to filter out excess statistical noise in u Uj X, iy before applying (7.71). In either case, the estimated divergence fields are given by... 

Physically, the diffusive terms use a conservative central-difference formula. In the energy equation, for example,... 

In this case, then, the truncation error in the expression for yn+i is dominated by a term proportional to k2. The central-difference formula gives... 

The classical example of the instability that results is the difference formulation given by Richardson (R2) for the heat-conduction equation. He proposed using a central-difference formula [as in Eq. (5-15)] for the derivative with respect to time, together with the usual central-difference formula for the space derivative. The resulting equation has a truncation error of the third order in both time and space steps, but the solution is unstable for any length of step. Thus, this natural and accurate formulation is not available for use if more than a few steps are to be taken. 

There are two practical approaches in formulating the working difference equations for the packed tubular reactor. The simple one is to use a forward-difference equation for the axial derivative and central-difference formulas for the radial derivatives. The leading terms in the truncation error are then proportional to k2 and to kh2, where h and k are written for the radial and axial steps. This means that, in order to take advantage of the accuracy of the approximations for the radial derivatives, k must have the same order of magnitude as h2, so that k2 and kh2 will be comparable. This is a serious limitation on the length of the axial step that can be used. 

PROG 17 uses the interpolation methods of determine the values of F, Fj, and F3 at t = 575°F. The values from the Stirling s central difference formula are ... 

For the distributed-parameter skin compartment model, the concentration Csc is calculated by discretizing the stratum corneum compartment into a set of iV -i-2 equidistant nodes and using the central difference formula. This results in the representation of the one-dimensional Fickian diffusion equation to calculate mass flux at any depth within the stratum corneum (46) ... 

The central difference formula of Stirling thus furnishes the same result as the ordinary difference formula of Newton. We get different results when the higher orders of differences are neglected. For instance, if we neglect differences of the second order in formulae (7) and (20), Stirling s formula would furnish more accurate results, because, in virtue of the substitution A1 = A1 - JA2 v we have really retained a portion of the second order of differences. If, therefore, we take the difference formula as far as the first, third, or some odd order of differences, we get the same results with the central and the ordinary difference formulae. One more term is required to get an odd order of differences when central differences are employed. Thus, five terms are required to get... 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

By simple addition of forward to backward series expression a central difference formula for the second derivative is obtained ... 

In some cases, where iXi + iXj — 1 < 0 so that Go oo, it is not possible to correct the Numerov scheme for the Poisson equation [491,492]. In these cases one can use a five-point central difference formula [498] (also with truncation error of order h ) which leaves the coefficient function on the diagonal. However, this five-point formula leads to difficulties at the boundaries, where values at grid points s i (at the lower boundary) and s +2 (at the upper boundary) would be needed, which lie outside the range of definition. 

M-point Bickley central-difference formula with sufficiently high n may guarantee sufficient accuracy even in cases where relativistic effects on the gradient are negligible. 

Comparing these chords with the actual tangent, it will be clear that the central difference formula, Eq. 3.12, is the most satisfactory on average (it is possible to draw three points where it it is not the best). Any of the above three formulae can be used in digital simulation. 

Mathematically, the system consists of parabolic PDEs, which were solved numerically by discretization of the spatial derivatives with finite differences and by solving the ODEs thus created with respect to time (Appendix 2). Typically, 3-5-point difference formulae were used in the spatial discretization. The first derivatives of the concentrations originating from a plug flow (Equations 9.1 through 9.3) were approximated with BD formulae, whereas the first and second derivatives originating from axial dispersion in the bulk phases and diffusion inside the catalyst particles were approximated by central difference formulae. Some simple backward (Equation 9.14) and central difference (Equation 9.15) formulae are shown here as examples ... 

In the mathematical description of the model, the parabolic PDFs were converted into ODEs by the method of lines and, consequently, a large number of ODEs were solved. The conversion of PDFs to ODEs is carried out using central difference formulae for the derivatives d c,/dx. The kinetic model for the components can be described as follows ... 

Despite their lesser accuracy, backward and forward difference formulas are often used for practical reasons. For example, they are applied at the boundaries of computational domains. At such boundaries, central difference formulas (e.g.. Equations 20-4 and 20-5) require values of i that are outside the domain, and hence, undefined. Although high-order accurate backward and forward difference formulas are available, their use often forces simple matrix structures into numerical forms that are not suitable for efficient inversion. 

To solve Eqs. 26 and 27, the initial values at n = 0 and n= must be known. Additionally, storing the velocities and stresses from both the current and the previous time step increases memory requirements. Approximating the spatial derivatives with the central difference formula and the temporal derivatives with the forward difference formula may seem a convenient alternative. However, such a combination leads to a FD scheme that is unconditionally unstable, i.e., it will not converge regardless of the value of the Courant number C. The stability of a numerical scheme is often analyzed using the von Neumann method, which is based on Fourier decomposition of the numerical solution. 

The forward difference formula in Eq. 18 represents the first-order approximation to the first spatial derivative, while the central difference formula (25) represents the second-order approximation. Both schemes can be derived by replacing the functional values at /(zq Az) with a Taylor expansion ... 

At grid point j, we use a central-difference formula to approximate the local value of the second derivative of the velocity. 

Here, the values of the first derivatives are evaluated at the mid-points between the grid locations. We then use yet other central-difference formulas for these mid-point values,... 

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