Splitting methods are a well-established tool for numerically integrating time-dependent partial differential equations. These methods split the vector field into disjoint components, which are integrated separately using an appropriate time step. The individual flows are then combined to obtain the desired numerical approximation.
The merits and pitfalls of splitting methods will be discussed on the basis of several examples. It is shown that splitting methods can have superior geometric properties (such as preservation of positivity and favorable long-term behavior) compared to standard time integration schemes. Moreover, typical stability constraints in standard discretizations can frequently be overcome. An additional advantage of using splitting methods is that pre-existing code is often available for implementation.
On the other hand, the use of splitting methods also requires a certain amount of care. The presence of (non-trivial) boundary conditions can cause strong order reduction, resulting in computational inefficiency. Additionally, dealing with non-smooth initial data poses a significant challenge. To overcome these issues, the integrators must be adapted accordingly as explained in the talk.
Alexander Ostermann is Professor of Numerical Analysis and Scientific Computing at the University of Innsbruck, where he is also the Dean of the Faculty of Mathematics, Computer Science and Physics.