The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form y' = g'(x) + \\y -g(x)\.The stability and accuracy of numerical approximations to the solution v = g(x), obtained using implicit one-step integration methods, are studied. An S-stability property is introduced for this problem, generalizing the concept of /4-stability. A set of stiffly accurate onestep methods is identified and the concept of stiff order is defined in the limit Re( -X) -» oe. These additional properties are enumerated for several classes of ^-stable one-step methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initial-value problems obtained using such methods. A family of methods based on a compromise between accuracy and stability considerations is recommended for use on practical problems.

We have developed a generalized mathematical model for the autoignition of hydrocarbons under the conditions of high pressure and temperature achieved in a rapid-compression machine. The model is able to simulate the essential phenomena of the two-stage autoignition of alkanes under these conditions; these are a well-defined cool flame that is often quenched rapidly and completely before the onset of a sharp ignition. It also pre­dicts correctly the transition to single-stage autoignition at even higher temperatures and the variation with temperature of the characteristic induction periods. The model is based on a degenerate-branched-chain mechanism. We show that it must contain as necessary features two termination processes, one linear and the other quadratic in radial concen­tration, and two routes for the formation of branching agent, one of which involves intermediate products of oxidation. The model also predicts, without any adjustment of the kinetic parameters, the essential pheno­mena of cool-flame and ignition behaviour that are observed at low pressures.

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A detailed mathematical model of the non-isothermal oxidation of acetaldehyde has been found to give a realistic simulation of (i) single and multiple cool flames, their limits, amplitudes and induction periods; (ii) two-stage ignition; and (iii) the negative temperature coefficient for the maximum rate of slow combustion. A simplified form of the model, valid over a limited range of conditions, has been subjected to mathematical analysis to provide interpretations of the effects simulated by the detailed model. It is concluded that cool flames are thermokinetic effects often, but not exclusively, of an oscillatory nature, and that a satisfactory account of cool-flame phenomena must necessarily take reactant consumption into account.