Interpolation methods characteristics

Pseudo-Newton-Raphson methods have traditionally been the preferred algorithms with ab initio wave function. The interpolation methods tend to have a somewhat poor convergence characteristic, requiring many function and gradient evaluations, and have consequently primarily been used in connection with semi-empirical and force field methods. 

Approximation of characteristic lines by piecewise smooth functions Forces are often described in terms of data obtained from measurements and given in tables. Approximating or interpolating these data introduces discontinuities into the problem depending on the particular interpolation method. 

D based methods were developed a few years later still aiming at solving peak shift issues but with a tendency to focus more on NMR [22,26,27]. Indeed, they often converge faster, especially because they do not include the costly interpolation step, characteristic of the C/E methods, and because of the significantly smaller number of feasible correction functions within the search space. Furthermore, they are more conservative in terms of... 

Rotor/run conditions SW 55 Ti rotor at 55000 rpm for approximately 6 hours. These recommendations form the core of any procedure. SpinPro usually considers more factors in the rotor selection process than does the expert. In determining the run speed, SpinPro considers every possible reason to reduce the run speed. If there are none, the rotor is run at full speed. When there are reasons (e.g., when using salt gradients, bottles, differential pelleting, or discontinuous runs), the run speed may have to be reduced dramatically, from 80,000 rpm to 40,000 rpm, for example. There are many cases of rotors being run too slow for the application or too fast for safety. Accurate determination of the run time is a complex problem based on the gradient characteristics, calculations, interpolations from numerical tables, and experience. SpinPro employs all of these methods in order to infer run times for many special cases. 

The dynamic model calculations are done in a two phase process. In the first phase, the solids stream dynamic equations are integrated for the reactor while keeping gas stream conditions constant. When necessary, intermediate gas stream values are obtained by interpolation between storage nodes. These calculations proceed down the reactor for the distance and time method of lines, and back up the reactor for the method of characteristics. In the second phase, the gas stream steady state equations are integrated from the bottom to the top of the reactor while keeping the solids stream conditions constant. Intermediate solids stream values are obtained by interpolation between storage nodes. 

When deriving these expressions, it was assumed that velocity at all the cell faces is positive. In other cases, suitable modifications to include appropriate upstream nodes (in place of 0ww and 0ss) should be made. It can be seen that the continuity equation indicates that the last term inside the bracket of Eq. (6.19) will always be zero for constant density flows. The behavior of numerical methods depends on the source term linearization employed and interpolation practices. Before these practices are discussed, a brief discussion of the desired characteristics of discretization methods will be useful. The most important properties of the discretization method are ... 

---