### Aspects of Numerical Time Integration (Summer Semester 2014)

• Lecturer:
• Classes: Lecture (0161300), Problem class (0161310)
• Weekly hours: 2+2
Schedule
Lecture: Friday 11:30-13:00 Z 1
Problem class: Tuesday 15:45-17:15 K 2
Lecturers
Lecturer JProf. Dr. Katharina Schratz
Office hours: By Appointment
Room Kollegiengebäude Mathematik (20.30)
Email:
Lecturer, Problem classes Dr. Patrick Krämer
Office hours: by appointment
Room 3.025 Kollegiengebäude Mathematik (20.30)
Email: patrick.kraemer3@kit.edu

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.