Title: Modified Equations and Shadow Hamiltonians

Speaker:  Robert Skeel, Purdue University

Generally, the most fruitful way to analyze the accuracy of a discrete
approximation to a dynamical system is to express the effect of
discretization as a modification to the right-hand side vector field.
Such a modified, or ``shadow,'' vector field can be expressed uniquely
as a formal expansion in powers of the step size, which evidentally
converges only for linear problems.  Nonetheless, a truncated
expansion very accurately represents many features of the discrete
dynamics.  For a Hamiltonian system, the shadow vector field is that
of a Hamiltonian system if and only if the integrator is symplectic.
The Hamiltonian is a conserved quantity, and the near existence of a
shadow Hamiltonian is of considerable interest.  It is possible to
construct highly accurate shadow Hamiltonian approximations using
information readily available from the numerical integration.
Properties of this construction are briefly described.  Also reported
are remarkable results obtained from constructions with accuracy order
as high as 24 and applied to systems as complex as the molecular
dynamics of water.  These experiments shed light on theoretical
properties of shadow Hamiltonians and also give practical information
about the accuracy of a simulation.