Gear Algorithms

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

With these reaction rate constants, differential reaction rate equations can be constructed for the individual reaction steps of the scheme shown in Figure 10.3-12. Integration of these differential rate equations by the Gear algorithm [15] allows the calculation of the concentration of the various species contained in Figure 10.3-12 over time. This is. shown in Figure 10.3-14. 

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-HaU, Englewood Cliffs, NJ, 1971 Gear Algorithm, QCPE Program No. QCMP022. 

The Gear Algorithm [15], based on the Adams formulas, adjusts both the order and mesh size to produce the desired local truncation error. BuUrsch and Sloer method [16, 22] is capable of producing accurate solutions using step sizes that arc much smaller than conventional methods. Packaged Fortran subroutines for both methods are available. 

The preceding equations form a set of algebraic and ordinary differential equations which were integrated numerically using the Gear algorithm (21) because of their nonlinearity and stiffness. The computation time on the CRAY X-MP supercomputer for a typical case in this paper was about 5 min. Further details on the numerical implementation of the algorithm are provided in (Richards, J. R. et al. J. ApdI. Polv. Sci.. in press). 

A systematic stepwise method for numerical integration of a rate expression [indeed, of any differential equation y = f(x,y) with an initial value y(Xo) = Vo] to determine the time evolution of the rate process. See also Numerical Computer Methods Numerical Integration Stiffness Gear Algorithm... 

NUMERICAL COMPUTER METHODS NUMERICAL INTEGRATION STIFFNESS GEAR ALGORITHM... 

In the simulation, the density of the silicon structure is chosen as that of c-Si (2.33 g/cm ), since the density of a-Si without voids is close to that of c-Si [17]. The atoms are initially arranged as the diamond structure with periodic boundary conditions. They move according to the intermolecular forces based on the potential function, Eq. (3), and these movements can be described by the classical momentum equations. The momentum equations are integrated by the Gear algorithm with a time step of 0.002 ps and the average temperature of the structure is kept constant by the momentum scaling method. 

EROS handles concurrent reactions with a kinetic modeling approach, where the fastest reaction has the highest probability to occur in a mixture. The data for the kinetic model are derived from relative or sometimes absolute reaction rate constants. Rates of different reaction paths are obtained by evaluation mechanisms included in the rule base that lead to partial differential equations for the reaction rate. Three methods are available that cover the integration of the differential equations the GEAR algorithm, the Runge-Kutta method, and the Runge-Kutta-Merson method [120,121], The estimation of a reaction rate is not always possible. In this case, probabilities for the different reaction pathways are calculated based on probabilities for individual reaction steps. 

The stability regions are reported in Figure 2.8, where (j = hX. 2.14.7.2 Gear Algorithms... 

The Gear algorithms are stable for stiff problems, whereas the Adams-Moulton are unstable with orders larger than 2. 

Figure 2.9 Stability regions for the family of Gear algorithms.

In the simulations performed in our lab and discussed in this chapter, the fifth-order Nordsieck-Gear algorithm was used. In this algorithm, Taylor series expansions of the existing positions and time derivatives of each atom are used to predict the next positions and time derivatives for that atom. Using a particular value for the timestep St) m the expansion, each atom is moved some predicted distance and the predicted fifth-order time derivatives (velocity, acceleration, etc.) are also updated. The force on each atom is calculated from the predicted positions and the difference between the predicted acceleration and that obtained in the force calculation (from Fi = ntjOj) is used to correct the position and derivatives. 

The simulation of protein dynamics has been reviewed authoritatively by McCammon and Kaiplus, who have participated fully in this development. A detailed description has been given of the potential functions used and of the application of the Gear algorithm to integrating the Newtonian equations of motion to permit the calculation of the movement of atoms over the potential energy... 

This algorithm is extremely simple and it behaves very nicely as can be seen from a comparison with Gear-algorithm for a dynamics simulation of a macromolecule (ref. 6). Fig. 3 shows the rms error in the total energy of the system (which should be constant), compared with the rms fluctuation in kinetic... 

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