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Institutes

Institute for Applied and Numerical Mathematics

Research Group 3: Scientific Computing

Seminar (Highly oscillatory problems)

Seminar (Highly oscillatory problems)

Seminar (Highly oscillatory problems) (Winter Semester 2017/18)

Lecturer: JProf. Dr. Katharina Schratz
Classes: Seminar (0126400)
Weekly hours: 2

The topics of the seminar will be discussed in a first meeting on Monday, 16.10.2017, at 15:45 in room 2.058 (Math Building 20.30).

We recommend this seminar to Master students and also late Bachelor students who have already attended lectures on the (numerical) analysis of differential equations.
Programming skills in MATLAB are recommended.

It will take place as a block seminar on one or two days in January 2018.

Schedule
Seminar:	Monday 16.10.2017 15:45 First Meeting en-block class	SR 2.058 (Building 20.30)
	Block Seminar in January 2018 en-block class

Lecturers
Lecturer	JProf. Dr. Katharina Schratz
	Office hours: By Appointment
	Room Kollegiengebäude Mathematik (20.30)
	Email:
Lecturer	Dr. Patrick Krämer
	Office hours: by appointment
	Room 3.025 Kollegiengebäude Mathematik (20.30)
	Email: patrick.kraemer3@kit.edu

Motivation

Consider a (linear) harmonic oscillator (a pendulum consisting of a spring with a mass)
$\quad y''(t)=-\omega^2 y(t),\qquad y(0)=-1,\quad y'(0)=0,\quad \omega>0$
with exact solution $y(t)=-\cos(\omega t)$ (deviation of the mass from the equilibrium at time $t\in[0,T]$ ).
Small values of $\omega$ thereby belong to nonstiff springs and slow oscillations, whereas large $\omega$ belong to stiff springs and rapid oscillations. For large $\omega$ the latter example describes a highly oscillatory problem (see link to a simulation movie below).

$Spring Pendulum$

Harmonic oscillator with

a nonstiff spring with $\omega=1$ (green, left)
an intermediate spring with $\omega=10$ (yellow, middle) and
a stiff spring with $\omega=20$ (red, right)

For a simulation movie of the oscillations click HERE

Seminar Content

Motivated by this linear example, in this seminar we treat linear and nonlinear highly oscillatory problems. Due to the rapid oscillations, standard time integration schemes, such as explicit Euler, Störmer-Verlet and related Runge-Kutta type methods, only allow very small time steps in order to guarantee stability of the exact solution. This leads to huge computational costs.

The aim of this seminar is to discuss the construction and analysis of methods which can cope with the rapid oscillations and thus allow to overcome these numerical challenges.

References

E. Hairer, C. Lubich, G. Wanner: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. 2nd Edition. (Springer 2006)