Convergence results for linear multistep methods for quasi-singular perturbed problem systems

Stiff behavior occurs in a variety of ODE systems relevant in applications. The notion of stiffness is a phenomenological one, and a stability and error analysis of numerical methods has been based either on simple models or particular problem structures. In particular, stiff initial value problems in standard singular perturbation form are well understood. However, problems of this type exhibit a very simple phase space geometry, namely the stiff eigendirections also behave stiff in another sense, i.e., they are almost parallel. This motivates us to consider a more general nonlinear class of stiff ODE systems depending on a small parameter. In particular, we investigate the convergence properties of the implicit Euler scheme applied to problems of this type, and linear multistep schemes will further be investigated. 1 The class of quasi-singular perturbed problems The idea of extending stability theory by considering a more general class of problems (not only of singular perturbed form) is not new, but has already been done by Auzinger, Schranz-Kirlinger and Frank (see [1]). In this paper, a new ansatz, basing on a generalised form of singular pertubation, is introduced. Definition 1 We call an ODE sytem a singular perturbed (SPP) problem iff it takes the form ż1 = F (z1, z2), (1) ż2 = 1 ǫ F (z1, z2), (2) where the real part of the spectrum of the Jacobian matrix Dz2F ♯ is smaller than some −b0 for b0 > 0. There is a well developped theory about the dynamics of SPP, basing on the works of Fenichel [2]. In [3] Nipp has shown that under some reasonable assumptions, SPP have a strongly attractive smooth invariant manifold with moderate flow. Definition 2 A system ẏ = G(y) is called quasi-singular perturbed (QSPP) iff there exists a transformation y = Φ(z) with ‖DΦ‖ = O(1) and ‖D(Φ)‖ = O(1) which transforms the system to singular perturbed form.