Linear multistep methods applied to stiff initial value problems—A survey

Abstract

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 formulae (BDFs) are frequently used for the numerical integration of stiff initial value problems. Such stiff problems appear in a variety of applications. While the intuitive meaning of stiffness is clear to all specialists, there has been much controversy 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, nonautonomous stiff systems, and rather general nonlinear stiff problems. Different approaches proving 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.