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The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations $5 presented. Different integration methods are tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, homogeneous gasphase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications of this result to the development of fast integration techniques for combustion kinetic rate equations are discussed. Introduction The ordinary differential equations (ODE'S) describing complex chemical reactions are characterized by widely different time constants. Although the differential equations are stable, standard numerical techniques such as the explicit Runge-Kutta and Adams methods are prohibitively expensive to use because of the severe steplength restriction imposed by the requirements for numerical stability. equation Such systems of differential re commonly referred to as "stiff" systems. ?-P The problem of stiffness has been recognized for some time, e.g.,6 and several techniques have been developed for stiff ODE'S. At the pre e t time, the packaged codes EPISODE' and LSODE3,g represent the most extensively documented, tested and used routines for stiff ODE's. Amon s era1 codes examined in recent detailed studies,PO-(ix LSOOE was found to be the fastest for solvina chemical kinetic rate equations. combustion device modelers that LSODE is not fast enough for economical calculat' ns of multidimenHowever, it is4recognized by sional reacting flow problems. 18 The numerical solution of combustion kinetic rate equations is complicated by the existence of a narrow region ("heat release" zone) where the species concentrations and temperature change rapidly, as illustrated in Fig. 1 for a typical batch reaction combustion problem. In the heat release regime, especially in the early part, many of the species and the temperature have positive time constants -an indication that the governing ODE'S are unstable. Since small steplengths are required for solving unstable O D E ' S , the use of methods designed for stiff problems -designated herein as "stiff methods" -may be inefficient. During *NRC-NASA Research Associate and Assistant Professor of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, Michigan 48109. Member, AIAA. Copyri*hl 0 Amtrlcm ln6111uli of Aeron~ulirr m d A s l r ~ n ~ ~ l i n . Inr. 1985. All rlshlr rrxned. early heat release explicit "nonstiff methods" -i.e., methods suitable for nonstiff problems -may be adequate. However, implicit methods are more accurate than explicit methods, which are therefore used only a predictors in predictorcorrector algorithm^.^^ The corrector equations are iterated until convergence is obtained. not clear what corrector iteration technique is optimal in h nonstiff regime. t Slmple or functional 1i9p7 and J a c ~ b i N e w t o n ~ ~ > ~ ~ iteration techniques have been used because they avoid the expense associated with forming and inverting Jacobian matrices, which is required by NewtonRaphson iteration. lengths can be used with Newton-Raphson iteration. For unstable ODE's this advantage may not be of much help and it is therefore not apparent which technique is the most efficient. During late heat release and equilibration the governing ODE's are stable so that Newton-Raphson iteration is the optimal convergence method. In these regimes, especially during equilibration, the different species approach the equilibrium state at different rates and the OOE's are stiff -i.e., classical numerical techniques will require prohibitive amounts of computer time in these regimes. Here, stiff methods are better suited to solving the problem. I n developing an efficient algorithm to solve combustion kinetic rate equations, it is important to recognize and accommodate the widely different characteristics of the three regimes (induction, heat release and equilibration) encountered in a typical combustion problem. the problem changes character, occurs in other areas and schemes have been proposed for auto tic switching between stiff and nonstiff methods. It is However, much larger stepSuch a situation where

Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods