Merits and pitfalls of splitting methods
Alexander Ostermann (University of Innsbruck, Austria)
For the time integration of certain types of partial differential equations splitting methods have been a popular choice since long. Important examples comprise the incompressible Navier--Stokes equations and semi-linear problems such as nonlinear Schr\"odinger equations. Meanwhile the range of application of splitting methods has been extended to many more equations and systems.
In my talk I will highlight some merits of splitting methods. First of all, when splitted in the right way, these methods have superior geometric properties (such as preservation of positivity and long term behaviour) as compared to standard time integration schemes. In certain cases, it is also possible to overcome a CFL condition present in standard discretizations. From a more practical point of view, one has to mention that splitting methods can simply be implemented by resorting to existing methods and codes for simpler problems, and they often admit parallelism in a straight-forward way.
On the other hand, the application of splitting methods also requires some care. The presence of (non-trivial) boundary conditions can lead to a strong order reduction and consequently to computational inefficiency, and stability is always an issue with splitting methods, in particular in non-Hilbert space norms and for non-linear problems. In the numerical analysis of splitting methods the regularity of the exact solution (or sometimes of the data) plays a prominent role, as will be exemplified in my talk.