Numerical calculations convergence equations

To transform the potential smoothly and to avoid large perturbations, Aj has been increased in small steps from 0 to 1. To evaluate numerically AF(AES) with equations (43) and (44), we have used in each case the mapping potential Hm for which the most populated S subset is the one centred at AES. Following this procedure, each calculation converges very fast. 

These expressions are substituted in the summation equation to generate Equation 2.50. The value obtained for a is used to update the mixed K-values, and all the computations are repeated until the convergence is achieved. The numerical calculations at convergence are shown below ... 

The solution in Equation 8.26 is inconvenient for several reasons. Each term in the series contains two coefficients w and A ) which require numerical calculation. In the case of a linear wall reaction, these quantities depend on the wall kinetic parameter, and this relationship is recently obtained in a simple and explicit manner by Lopes et al. [40]. In addition, whenever this slowly convergent series is used to describe the inlet region, a large number of terms may be required so that a satisfactory result is obtained. The efficient evaluation of the terms in Graetz series has been the object of many studies. Housiadas et al. [51] presented a comparative analysis between several methods to estimate these terms, remarking the numerical issues associated with the rigorous calculation of these quantities. However, this was done for uniform wall concentration (Dirichlet boundary condition), excluding the important case of finite reaction rates. 

Doing this speeds up the numerical calculation, which otherwise is slow because of the slow thermal response of the wall. From here the Gauss Siedel iteration method can be employed to solve the system of algebraic equations with a reasonable convergence criterion, for example. 

In order to calculate a , numerical differentiation in equation (10.40a) was applied because it is very difficult to express analytically p)" as a function of E from equation (10.38). This numerical approach, however, requires a much higher accuracy for the total HF energy than in other applications. Therefore, the convergence threshold for the density matrix elements was set at 10" -10 . Having determined at different field strengths, equation (10.40a) can be approximated by the numerical difference quotient for a ... 

Equations A3.11.114(b) and A3.11.115(b) are in a fonn that is convenient to use for potential scattering problems. One needs only to detemiine the phase shift 5 for each i, then substitute into these equations to detemiine the cross sections. Note that in the limit of large i, finiist vanish so that the infinite sum over partial waves iwill converge. For most potentials of interest to chemical physics, the calculation of finiist be done numerically. 

The hnal step in the design calculations for a laminar flow reactor is determination of mixing-cup averages based on Equation (8.4). The trapezoidal rule is recommended for this numerical integration because it is easy to implement and because it converges O(Ar ) in keeping with the rest of the calculations. 

---