Gear’s algorithm

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. 

A comprehensive discussion of algorithms, software and packages related to Gear s algorithm has been given by Byrne et al. [169]. 

This approach, the stage model, is simple because the equations are first-order ordinary differential ones that can be easily solved by Gear s algorithm. However, it is less accurate than the continuous model, especially when dealing with columns... 

Numerical analysis of bistability in the above system has been attempted by Kepper et al. [18] and Papsin et al. [19]. In the latter case, rate equations have been generated by considering a set of seven reactions, and the equations have been solved for seven species, assuming [lO Jo to be constant. In the former case, four rate equations are generated for four unknowns, viz., [IO ]o[l2] and [H3ASO3]. Solutions have been obtained with the use of Gear s algorithm for coupled sets of stiff differential equations. 

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 recent application of sparse-matrix techniques combined with computer optimization (vectorization) techniques has, however, improved the speed of Gear s code substantially, so that this advanced algorithm can now be used to study complex problems in multi-dimensional models (see e.g., the SMVGEAR package developed by Jacobson (1995 1998) and Jacobson and Turco, 1994). 

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