BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-92-16
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: Model problems in numerical stability theory for initial
               value problems
        TYPE:: Manuscript
      AUTHOR:: Stuart, Andrew M.
      AUTHOR:: Humphries, Antony R.
        DATE:: November 1992
       PAGES:: 42
    ABSTRACT:: In the past numerical stability theory for initial value
               problems in ordinary differential equations has been
               dominated by the study of problems with essentially trivial
               dynamics. Whilst this has resulted in a coherent and
               self-contained body of knowledge, it has not thoroughly
               addressed the problems of real interest in applications.
               Recently there have been a number of studies of numerical
               stability for wider classes of problems admitting more
               complicated dynamics. This on-going work is unified and
               possible directions for future work are outlined. In
               particular, striking similarities between this new developing
               stability theory and the classical non-linear stability
               theory are emphasised.

               The classical theories of $A$, $B$, and algebraic stability
               for Runge-Kutta methods are briefly reviewed, and it is
               emphasised that the classes of equations to which these
               theories apply - linear decay and contractive problems - only
               admit trivial dynamics. Four other categories of equations -
               gradient, dissipative, conservative and Hamiltonian systems -
               are considered. Relationships and differences between the
               possible dynamics in each category, which range from multiple
               competing equilibria to fully chaotic solutions, are
               highlighted and it is stressed that the wide range of
               possible behaviour allows a large variety of applications.
               Runge-Kutta schemes which preserve the dynamical structure of
               the underlying problem are sought, and indications of a
               strong relationship between the developing stability theory
               for these new categories and the classical existing stability
               theory for the older problems are given. Algebraic stability,
               in particular, is seen to play a central role. The effects of
               error control are considered, and multi-step methods are
               discussed briefly. Finally, various open problems are
               described.
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-92-16