Martin Arnold
A note on the uniform perturbation index
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 52, 33-46 (1998)
65L05 Initial value problems
Abstract: For a given differential-algebraic equation (DAE) the perturbation index
gives a measure for the sensitivity of a solution w.r.t. small perturbations.
If we consider, however, {\em classes} of DAEs (e.g. all DAEs that arise
as semi-discretizations of a given partial DAE by the method of lines)
then the error bound in the definition of the perturbation index may become
arbitrarily large even if the perturbation index does not exceed 1.
We illustrate this fact by 2 examples and define as alternative the {\em uniform} perturbation index
that gives simultaneously error bounds for {\em all} DAEs of a given
class. We prove that in one example each individual DAE has perturbation
index 1 but the uniform perturbation index is 2. Another example illustrates
that the class of all finite difference semi-discretizations may even
have {\em no} uniform perturbation index if the given partial DAE has
perturbation index 2.
Keywords: ifferential-algebraic equations, perturbation index, Baumgarte stabilization, partial DAEs, method of lines