P-stability properties of Runge-Kutta methods for delay differential equations

SummaryRecently the author and others have studied a class of Runge-Kutta methods for DDEs. These methods make use of certain constrained meshes, which allow to get optimal order results. In this paper we study their asymptotic stability properties, using the test equation $$\begin{gathered} y'(t) = ay(t) + by(t - \tau ),t > 0 \hfill \\ y(t) = g(t)for - \tau \leqq t \leqq 0 \hfill \\ \end{gathered} $$ wherea,b∈ℂ, σ>0, andg(t) is continuous and complex valued. It is known thaty(t)→0 ast→+∞ if |b|<-Re(a). In particular, we show that any Astable one-step collocation method for ODEs inherits the same property when it is applied to DDEs with a constrained mesh (i.e. it is P-stable).

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