On $GP$-stable adaptations of Runge-Kutta methods to

delay differential equations by one-step interpolation

Karel in 't Hout

Department of Mathematics and Statistics

Arizona State University

email: khout@math.asu.edu

This talk concerns the adaptation of Runge-Kutta methods to the numerical solution of stiff initial value problems for delay differential equations. Initial value problems of this kind arise frequently in, for example, immunology. For their effective numerical simulation, it is imperative to have stable numerical processes.

The concept of $GP$-stability forms one of the weakest stability requirements considered in the literature on numerical step-by-step methods for stiff delay equations. However, for most Runge-Kutta methods it is a long-standing open question of whether there exists a practical one-step interpolation procedure that leads to an adaptation which is $GP$-stable, i.e., all numerical solutions to the linear scalar test equation

$$U'(t) = \lambda U(t) + \mu U(t-\tau)$$

with $\Re \lambda < - \vert\mu\vert$ and $\tau >0$ vanish for any given step size. Up to now, only for certain first- and second-order $A$-stable Runge-Kutta methods it is known that there exists a useful one-step interpolation procedure (namely, linear interpolation at the gridpoints) which yields an adaptation that is $GP$-stable. In this talk, we shall present first promising results concerning $GP$-stability in the case of one-step interpolation for several (well-known) Runge-Kutta methods of orders 3 and 4