The GPL-Stability of Runge-Kutta Methods for Delay Differential Systems (Journal of Computational Mathematics, 2000)
This paper deals with the GPL-stability of the Implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. We focus on the stability behaviour of the Implicit Runge-Kutta (IRK) methods in the solutions of the following test systems with a delay term$$y'(t) = Ly(t) + My(t-τ), t\ge 0,$$ $$y(t)=Φ(t), t\le 0,$$where $L, M$ are $N × N$ complex matrices, $τ \gt 0$, $Φ(t)$ is a given vector function. We shall show that the IRK methods are GPL-stable if and only if it is L-stable, when we use the IRK methods to the test systems above.
Keywords: Delay differential equation; Implicit Runge-Kutta methods; GPL-stability.