2018-10-10
10:15 at HCP F 43.4

Splitting methods: basics, analysis and applications

Alexander Ostermann

Numerical Analysis Group, University of Innsbruck, Austria

Splitting methods are a well-established tool for the numerical integration of time-dependent partial differential equations. The basic idea behind these methods is to split the vector field into disjoint parts, carry out integration steps for each vector field, and combine the single flows in the right way to obtain the sought after numerical approximation. Merits and pitfalls of splitting methods will be discussed with the help of various examples. Among those are reaction-diffusion equations, the Vlasov-Poisson equation (a kinetic model in plasma physics), the Korteweg-de Vries (KdV) and the Kadomtsev–Petviashvili (KP) equations. It is shown that splitting methods can have superior geometric properties (such as preservation of positivity and favourable long-term behaviour) as compared to standard time integration schemes. Moreover, it is often possible to overcome a CFL condition present in standard discretizations. Another benefit of splitting methods is the fact that they can be implemented by resorting to existing methods and codes for simpler problems, and they often admit parallelism in a straightforward way. On the other hand, the application of splitting methods requires also some care. The presence of (non-trivial) boundary conditions can lead to a strong order reduction and consequently to computational inefficiency. Moreover, stability is always an issue with splitting methods, in particular in non-Hilbert space norms and for nonlinear problems.