The Essential Stability of Local Error Control for Dynamical Systems

Abstract

Although most adaptive software for initial value problems is designed with an accuracy requirement—control of the local error—it is frequently observed that stability is imparted by the adaptation. This relationship between local error control and numerical stability is given a firm theoretical underpinning. The dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations: dissipative, contractive, and gradient systems. Dissipative dynamical systems are characterised by having a bounded absorbing set $\mathcal{B}$ which all trajectories eventually enter and remain inside. The exponentially contractive problems studied have a unique, globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set $\mathcal{B}$ may be chosen to be a ball of arbitrarily small radius around the equilibrium point. The gradient systems studied are those for which the set of equilibria comprises isolated points and all trajectories are bounded so that each trajectory converges to an equilibrium point as $t \to \infty $. If the set of equilibria is bounded then the gradient systems are also dissipative. Conditions under which numerical methods with local error control replicate these large-time dynamical features are described. The results are proved without recourse to asymptotic expansions for the truncation error. Standard embedded Runge–Kutta pairs are analysed together with several nonstandard error control strategies. Both error per step and error per unit step strategies are considered. Certain embedded pairs are identified for which the sequence generated can be viewed as coming from a small perturbation of an algebraically stable scheme, with the size of the perturbation proportional to the tolerance $\tau $. Such embedded pairs are defined to be essentially algebraically stable and explicit essentially stable pairs are identified. Conditions on the tolerance $\tau $ are identified under which appropriate discrete analogues of the properties of the underlying differential equation may be proved for certain essentially stable embedded pairs. In particular, it is shown that for dissipative problems the discrete dynamical system has an absorbing set $\mathcal{B}_\tau $ and is hence dissipative. For exponentially contractive problems the radius of $\mathcal{B}_\tau $ is proved to be proportional to $\tau $. For gradient systems the numerical solution enters and remains in a small ball about one of the equilibria and the radius of the ball is proportional to $\tau $. Thus the local error control mechanisms confer desirable global properties on the numerical solution. It is shown that for error per unit step strategies the conditions on the tolerance $\tau $ are independent of initial data while for error per step strategies the conditions are initial-data dependent. Thus error per unit step strategies are considerably more robust.