**Note:** The Lecture and Exercise class begin in the summer semester 2017 at 4th of May 2017 in the second week of the semester.

# Aspects of Numerical Time Integration (Summer Semester 2017)

Lecturer: | JProf. Dr. Katharina Schratz |
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Classes: | Lecture (0161300), Problem class (0161310) |

Weekly hours: | 2+2 |

Lecture: | Thursday 9:45-11:15 | SR 3.69 | Begin: 4.5.2017 |
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Problem class: | Thursday 15:45-17:15 | SR 3.61 | Begin: 4.5.2017 |

Lecturer | JProf. Dr. Katharina Schratz |
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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 |

# 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.

# Exercise Class

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

If you are student at KIT you can get a free license key for *MATLAB* at the KIT software shop. Then you can download the software from mathworks.com after registration.

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

## Exercise Sheets

Exercise Sheet 1

Exercise Sheet 2

Exercise Sheet 3

Exercise Sheet 4 (updated 20.07.2017)

Exercise Sheet 5 (updated 20.07.2017)

# References

- 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)