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Research Group 3: Scientific Computing

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.039

Address
Hausadresse:
Zimmer 3.039
Englerstr. 2
Kollegiengebäude Mathematik (20.30)
76131 Karlsruhe

Postadresse:
Karlsruher Institut für Technologie (KIT)
Fakultät für Mathematik
Institut für Angewandte und Numerische Mathematik
Arbeitsgruppe 3: Wissenschaftliches Rechnen
Englerstr. 2
Kollegiengebäude Mathematik (20.30)
D-76131 Karlsruhe

Office hours:
Mon-Thu 9-12 Uhr

Tel.: +49 721 608 42062

Fax.: +49 721 608 43197

Splitting Methods (Winter Semester 2016/17)

Lecturer: JProf. Dr. Katharina Schratz
Classes: Lecture (0106200), Problem class (0106300)
Weekly hours: 2+2


Schedule
Lecture: Tuesday 15:45-17:15 SR 3.61
Problem class: Thursday 15:45-17:15 SR 2.67
Lecturers
Lecturer JProf. Dr. Katharina Schratz
Office hours: By Appointment
Room Kollegiengebäude Mathematik (20.30)
Email:
Problem classes Dr. Patrick Krämer
Office hours: by appointment
Room 3.025 Kollegiengebäude Mathematik (20.30)
Email: patrick.kraemer3@kit.edu

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.

Exercises

Exercise Sheet 1
Exercise Sheet 2
Exercise Sheet 3
Exercise Sheet 4
Exercise Sheet 5
Exercise Sheet 6
Exercise Sheet 7
Exercise Sheet 8
Exercise Sheet 9
Exercise Sheet 10