Gear integration algorithm

Note PSpice does not support the Gear integration option. It instead relies on a modified trapezoidal-Gear integration algorithm for transient timestep operation. 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

Interpretive running for testing, or compiled FORTRAN for optimum speed. Eight integration algorithms, including improved Gear/Hindmarsh methods. 

The mass balance differential equations for the gas phase and the surface species were solved simultaneously as a function of time and discrete position, using a variable step integration algorithm (Gear) [8]. 

Figure 14.4 Gear and implicit Euler numerical integration algorithms.

The set of PDEs (10.39)-(10.46), with obvious initial and boundary conditions was solved numerically according to the method of lines, based on axial discretization with backward finite differences and on time integration by Gear s algorithm. 

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. 

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. 

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