Dissipativity of multistep methods for delay dynamical systems
This paper is concerned with dissipativity properties of one--leg multistep methods for the numerical solution of initial value problems (IVPs) for delay differential equations. The IVPs under consideration have the form \[ y'(t) = f(y(t), y(t-\tau)),\;t \geq 0,\;y(t) = \varphi (t),\quad t \leq 0 \] where \( \tau\) is a positive constant, \(\varphi\) a continuous function and \(f\) satisfies the condition Re \(\langle u, f(u,v)\rangle \leq \gamma + \alpha \|u \|^2 + \beta \|v \|^2 \). Here, \(y(t)\) belongs to a complex Hilbert space with the inner product \( \langle \cdot, \cdot\rangle\) and the corresponding norm \( \|\cdot \|\) and \( \alpha, \beta \) and \(\gamma\) are real constants such that \( \alpha + \beta <0\). As remarked by the author, the last condition implies the existence of an absorbing ball around the origin and therefore the dissipativity. In this setting, the author considers one-leg multistep methods supplemented with some suitable interpolation procedures to compute the solution at the retarded argument which are extensions of the \(\theta\)-methods, stating (without proof) that dissipativity of the numerical scheme holds for A-stable methods when solving finite-dimensional IVPs, while the same property holds in the case of infinite-dimensional problems for strongly A-stable multistep methods.
scientific article; zbMATH DE number 1359987