SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Plenary Lecture

Splitting methods for autonomous and non-autonomous perturbed systems

S. Blanes

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.

Organized by         Universidad de Valladolid     IMUVA