In this lecture we will firstly deepen our knowledge on *splitting methods*.

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.

In this lecture we will discuss (amongst others)

- Splitting methods for inhomogeneous evolution equations, their order reduction when „non-trivial“ boundary conditions are present, as well as the derivation of correction methods in order to overcome the „curse of boundary conditions“

- High-order splitting methods for analytic semigroups

- Splitting methods for the non-linear Schrödinger equation with polynomial nonlinearity

- Modified energy of splitting schemes applied to linear Schrödinger equations with potential

Furthermore we will investigate efficient numerical time-integrators for *highly-oscillatory problems* such as

- Averaging methods
- Gautschi-type methods
- Fourier expansion techniques

**Exercises**

In the exercises we will deepen some theoretical results and carry out practical implementations. For further informations see the exercise description on the following homepage:

Homepage Patrick Krämer

**
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. Attendance of the lecture „Splitting methods“ (winter term 2013/14) is helpful.

References: Will be given in the lecture.