Aspects of Numerical Time Integration (Summer Semester 2017)
- Lecturer: JProf. Dr. Katharina Schratz
- Classes: Lecture (0161300), Problem class (0161310)
- Weekly hours: 2+2
Note: The Lecture and Exercise class begin in the summer semester 2017 at 4th of May 2017 in the second week of the semester.
|Lecture:||Thursday 9:45-11:15||SR 3.69||Begin: 4.5.2017|
|Problem class:||Thursday 15:45-17:15||SR 3.61||Begin: 4.5.2017|
|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)|
Contents of the Lecture
In this lecture we illustrate some ideas in the convergence analysis of splitting as well as exponential integrator methods for semilinear evolution equations. As a model problem we will thereby consider the cubic Schrödinger equation. If time allows, we also show some techniques concerning the time integration of certain dispersive equations involving a derivative in the nonlinearity, e.g., the Zakharov system (a scalar model for Langmuir oscillations in a plasma). Furthermore, we give the students the opportunity to do some programming on their own in the exercises using Fourier pseudo spectral methods for the space discretization.
The exercises will take place in room 3.061 in the Kollegiengebäude Mathematik 20.30 thursday 15:45. First exercise on the 04.05.2017.
Important Note: Please install MATLAB on your Laptop and bring it with you to the Exercise if possible
In the exercises to the lecture Aspects of Numerical Time Integration our aim is to learn how to implement efficient integrators for certain partial differential equations such as the (nonlinear) Schrödinger equation.
- At first we recap how to implement simple time integrators in MATLAB, such as the explicit and implicit Euler method, for ordinary differential equations (ODE) of the form .
- Afterwards we practice the application of splitting methods for ODES
- Furthermore we learn how to implement pseudo-spectral methods, which make use of the Fast Fourier Transform to discretize spatial differential operators such as the laplace operator .
- Our aim is then to construct efficient integrators for the nonlinear Schrödinger equation which are based on pseudo-spectral methods for the space approximation and splitting methods for the time-integration
- E. Faou, Geometric Numerical Integration and Schrödinger Equations. European Math. Soc., 2012.
- C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77:2141--2153 (2008)