TABLE OF CONTENTS In 1 we define singularly perturbed ODE, i.e. ODE where the highest order derivative has a coefficient depending on a small parameter ?, and analyse the solution behaviour when ? approaches zero. In 2 we introduce the method of matched asymptotic expansions, which gives approximations for the solution in terms of (truncated) power series in this parameter ?. In 3 it is shown that treating such problems numerically may impose a severe restriction on the step size for certain methods including all explicit ones). This so-called stiffness problem necessitates the introduction of new numerical stability notions, such as A-stability, as is done in 5, after a more precise scrutiny of the increment per step has been carried out in 4. The most important class of LMM for use in stiffness problems, viz. of backward difference formulae, is described in 6, along with some practical implementation aspects, including the step size control.
Consider a scalar higher order linear ODE. If we let the coefficient of the highest order term go to zero we obtain an equation of one order less; the latter equation also needs one initial condition less than the former. Now if this coefficient is very small but nonzero, we may expect solutions of this equation to resemble (in some way) appropriate solutions of the equation without the highest order term; but the limit case must have a singular behaviour in the sense that the solution of this problem...
