G. Kirlinger:

"Linear multistep methods applied to stiff inntial value problems - a survey";

Mathematical and Computer Modelling,40(2004), 11-12; 1181 - 1192.

The numerical approximation of solutions of

differential equations has been and continues to be one of the principal concerns of numerical analysis. Linear multistep methods and in particular backward differentiation formulas (BDFs) are frequently used for the numerical integration of stiff intial value problems. Such stiff problems appear in a variety of applications.

While the intuitive meaning of stiffness is clear to all specialists,

much controversy has been about its correct mathematical definition. We present a historical development of the concept of stiffness. A survey of convergence results for special classes of stiff problems based on these different concepts of stiffness is given, e.g. for

linear, stiff systems, problems in singular perturbation form,

non-autonomous stiff systems, and rather general nonlinear stiff

problems. Different approaches to prove convergence of linear

multistep methods applied to stiff initial value problems are

introduced. It is further indicated that the corresponding proofs for

singular perturbation problems are compatible with a nonlinear

transformation and thus convergence of a quite general class of

nonlinear problems seems to be covered.

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