Adaptive integration of delay differential equation

M. Zennaro, A. Bellen

Dipartimento di Scienze Matematiche, Università di Trieste, I-34100 Trieste, Italy

zennaro@univ.trieste.it

Contributed talk

We consider the numerical solution of nonstiff delay differential equations (DDEs) of the form

$$\left\{ \begin{array}{ll} y'(t)=f\Big(t,y(t),y\big(t-\tau(... ..., \\ y(t)=\phi(t), \ \ \ t\leq t_{0}, \end{array} \right.$$

by means of variable stepsize continuous Runge-Kutta methods of discrete (nodal) order $p$ and uniform order $q\leq p$. Our aim is to ensure, as far as possible, the proportionality of the maximum global error to a given tolerance $TOL$ and to minimize the computational cost. In this context, $TOL$ is meant as a tolerance per unit step. The approach is explained in our forthcoming book [1, Chapter 7]. It is wellknown that, for ordinary differential equations (ODEs), the stepsize control mechanism for a (discrete) advancing method of order $p$ is mostly based on the use of a second method, the error-estimating method, of different order $p'$, usually $p'=p+1$ or $p'=p-1$. The extension to DDE solvers is not straighforward, and its mathematical justification is far from being obvious. However, the conclusions we arrive at are consistent with the results already available in the literature. In particular, we shall stress the following two points:

• If the (discrete) error-estimating method has order $p'=p+1$, then the uniform order of the continuous advancing method should be $q=p$. On the contrary, if $p'=p-1$, then $q=p-1$ is sufficient.
• Both the discrete and the uniform local errors should be controlled, rather than the sole discrete local error as is done in ODE codes. More precisely, the given tolerance $TOL$ should bound the discrete local error per unit step and, at the same time, the absolute uniform local error. To this aim, one can use a continuous error-estimating method of uniform order $q'$ other than $q$, say $q'=q-1$. However, under certain hypotheses on the problem and on the method, one might be satisfied with controlling the discrete local error only.

We shall conclude with some numerical illustrations using the classical explicit 4-stage RK method of order 4 with suitable continuous extension of uniform order 4.