*Exercise sheets below -- please scroll down.*

### Description of the lecture:

Due to their computational advantage splitting methods are nowadays omnipresent in scientific computing. They pursue the intention to break down a complicated problem

into a series of simpler subproblems. In the context of time integration a

common idea is to split up the right-hand side and to decompose the given

evolution equation into a sequence of subproblems, which in many situations can be solved far more efficiently or even exactly. The exact solution of the full-problem is then approximated by the composition of the flows associated to the simpler subproblems.

Firstly we will investigate the error behavior of splitting methods

for ordinary differential equations. The analysis will be based on the Baker-Campbell-Hausdorff formula and the calculus of Lie derivatives. We will in particular discuss splitting methods applied to Hamiltonian systems of ODEs and analyze in how far geometric properties (such as the energy of the system) are conserved within this numerical approach. We will then analyze splitting approaches for certain partial differential equations, such as linear Schrödinger equations, Schrödinger equations with a polynomial nonlinearity, as well as the so-called dimension splitting for parabolic evolution equations.

In the exercises we will deepen some theoretical results and carry out practical implementations.

Prerequisites: One should be familiar with basic concepts of the numerical time integration of ODEs and PDEs and functional analysis. A basic knowledge of the theory of semigroups is helpful.

References: Will be given in the lecture.