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Splitting methods in Geometric Integration
Robert McLachlan
Massey University, New Zealand
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The traditional effort of numerical analysis and computational mathematics
has been that of rendering physical phenomena into algorithms that produce
sufficiently precise, affordable and robust numerical approximations.
While not disregarding precision, affordability and robustness, GI is
concerned also with producing numerical approximation preserving the
qualitative attributes of the solution to the extent it is possible.
Examples of GI algorithms for differential equations include
symplectic, Lie group, volume-preserving, energy-preserving, reversible,
and symmetry-preserving integrators. All these classes of systems can be
treated in a unified way by
(i) Considering classifications of dyamical systems, for example by their
group of diffeomorphisms (all the above examples form groups);
(ii) Splitting the given vector field $X$ into a sum $\sum_i X_i$, where
the flow of each piece $X_i$ lies in the same group as the flow of $X$ but
is easy to calculate exactly; and
(iii) Composing the flows of the $X_i$ so as to form an integrator for
$X$ of the desired order.
The resulting methods are generally simple, fast, explicit, and (because
of their geometric properties) give qualitatively good results, especially
for very long integration times. Hence it is not surprising that they are
used (and have been re-invented) in many fields such as celestial mechanics,
accelerator physics, quantum mechanics, molecular dynamics, and fluid dynamics.
A longer introdution (7 pages) is available
here (figures).
The lectures will be based on the recent review article "Splitting Methods"
by Robert McLachlan and Reinout Quispel, to appear in Acta Numerica 2002.
The article is available here
and will be handed out at the school.
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