Numerical stability, resolvent conditions and delay differential equations Original Research Article

This paper deals with the problem of estimating a priori the error growth in discretizations of linear initial value problems. A review is presented of various recent stability estimates which are valid under the Kreiss resolvent condition or under a strengthened version thereof. Moreover, a conjecture is formulated to the effect that errors cannot grow at a faster rate than sβ, where β < 1 and s denotes the order of the matrices under consideration. Also a weaker version of the Kreiss resolvent condition is discussed. Under that condition, a stability estimate is proved which grows linearly with the order of the matrices under consideration. The paper concludes by presenting an application of the Kreiss resolvent condition in the error growth analysis for discretizations of delay differential equations.