__Splitting methods for autonomous and non-autonomous perturbed systems__

S. Blanes

**Abstract**

An important number of differential equations originated from as
diverse research areas as celestial mechanics, quantum mechanics,
Hybrid Monte Carlo, parabolic problems or some eigenvalue problems
can be considered as perturbations of problems whose solutions are
exactly solvable (or can be easily and accurately approximated).
For the numerical integration of these equations it is usually
convenient to solve separately the perturbation and the dominant
part, and then to consider appropriate compositions of their
flows. An efficient method for a given problem must take into
account the relevant aspects of the problem. For example:

- The size of the perturbation.

- The accuracy of the desired solution.

- The length of the time integration.

- Is the perturbation exactly solvable? or, can we use a low
order numerical approximation for this part?

- Do the flows admit negative time steps? and, can we use
complex coefficients having positive real part?

- Is the dominant part explicitly time dependent?

- Etc.

In this talk we present our recent works on the search of methods
for these problems (a new way to get the order conditions, the
analysis of the conditions to be solved by the coefficients in
each case, and to find the explicit coefficient for the methods).
The performance of the methods is illustrated on several numerical
examples.