Integrals algorithms that approximate

We consider the algorithms that approximate the integral I as follows ... 

In Table 1 the CPU time required by the two methods (LFV and SISM) for 1000 MD integration steps computed on an HP 735 workstation are compared for the same model system, a box of 50 water molecules, respectively. The computation cost per integration step is approximately the same for both methods so that th< syieed up of the SISM over the LFV algorithm is deter-... 

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Since that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Since radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the integration of continuity equations involving radical and molecular species requires special integration algorithms (25). An approximate method known as pseudo steady-state approximation has been used in chemical kinetics for many years (26,27). The errors associated with various approximations in predicting the product distribution have been given (28). 

To avoid the need for special procedures and modification of the integration algorithm, delays may be modeled by using rational approximations, e.g., Fade functions or multiple first-order lags in series. Experimentation suggests that 10 series lags is adequate for most applications, so this is used as a default. The approximation should be checked by comparison with a more detailed model where it is believed to be particularly significant. 

Although analog circuits that approximate 7(0 have been proposed (23), the function is usually evaluated by a numerical integration technique on a computer. Several different algorithms have been proposed for the evaluation (24, 25). The i-t data are usually divided into N equally spaced time intervals between / = 0 and t = tf, indexed by j then 7(0 becomes I k t), where At = tfIN and k varies between 0 and A, representing / = 0 and... 

It is possible to reduce the size of the time step such that the system configuration better approximates the constrained system configuration. However, this defies the practical reason for introducing constraints in the first place, namely, the ability to use a larger time step for increased computational efficiency. If the integration algorithm has an error in the coordinates on the order of the time step then in the worst case of holonomic constraints lin-... 

Inserting Eq. [49] into Eq. [50] shows that the forces of constraint are accurate to the same order as the integration algorithm adopted. Hence, with the Verlet algorithm the computation of the approximate forces of constraint at t reduces to the computation of the (y). Equivalently, by inserting Eq. [48] into Eq. [47], one obtains... 

All of the steps in the two algorithms are implemented using an array of computer packages that are becoming more integrated. Note that steps 4 and 5 deserve special attention, as they are the basis for the approximate models generated for the dynamic C R analysis. 

All the integration algorithms assume that the positions, velocities, and accelerations can be approximated by a Taylor series expansion ... 

Finally, practical computation with Fourier series is easy because there are fast algorithms to approximate them on digital computers. One starts with the observation that the integrals in equation (2) can be approximated by sums ... 

---