BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-92-18
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: An analysis of local error control for dissipative,
               contractive and gradient dynamical systems
        TYPE:: Manuscript
      AUTHOR:: Stuart, Andrew M.
      AUTHOR:: Humphries, Antony R.
        DATE:: November 1992
       PAGES:: 52
    ABSTRACT:: 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 $\cal B$ which all trajectories
               eventually enter and remain inside. The exponentially
               contractive problems studied have a unique, globally
               attracting equilibrium point and thus they are also
               dissipative since the absorbing set $\cal B$ 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 $t \rightarrow\ \infty$. If the set
               of equilibria is bounded then the gradient systems are also
               dissipative. The aim is to find conditions under which
               numerical methods with local error control replicate these
               large-time dynamical features. The results are proved without
               recourse to asymptotic expansions for the truncation error.

               Standard embededed Runge-Kutta pairs are analysed together
               with several non-standard error control strategies. These
               non-standard strategies are easy to implement and have
               desirable properties within certain of the classes of
               problems studied. 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
               algebraically stable and explicit algebraically 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 algebraically stable embedded pairs. In
               particular, it is shown that for dissipative problems the
               discrete dynamical system has an absorbing set ${\cal
               B}_{\tau}$ and is hence dissipative. For exponentially
               contractive problems the radius of ${\cal B}_{\tau}$ is
               proved to be proportional to a positive power of $\tau$. For
               gradient systems the numerical solution enters and remains in
               a small ball about one of the equilibria and the radius of
               the ball $\rightarrow$ 0 as $\tau\ \rightarrow$ 0. 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 whilst for
               error per step strategies the conditions are initial data
               dependent. Thus error per unit step strategies are
               considerably more robust.
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-92-18