### PhD and Diploma thesis and publications

- Splitting methods for nonlinear Dirac equations with thirring type interaction in the nonrelativistic limit regime, CRC 1173 Preprint, November 2018, joint work with JProf. Dr. Katharina Schratz (KIT, Karlsruhe) and Dr. Xiaofei Zhao (Wuhan University, China)
- Numerical Integrators for Maxwell-Klein-Gordon and Maxwell-Dirac Systems in Highly to Slowly Oscillatory Regimes, PhD thesis, 2017, supervised by JProf. Dr. Katharina Schratz
- Efficient time integration of the Maxwell-Klein-Gordon equation in the non-relativistic limit regime, CRC 1173 Preprint, July 2016, joint work with JProf. Dr. Katharina Schratz
- The Method of Multiple Scales for nonlinear Klein-Gordon and Schrödinger Equations Diploma thesis, December 2013, supervised by Prof. Dr. Marlis Hochbruck and JProf. Dr. Katharina Schratz

### Teaching

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.

## SS 2015:

# Exercises to the lecture *Aspects of Numerical Time Integration*

The exercises will take place in room **3.069** in the Kollegiengebäude Mathematik 20.30 **tuesday 15:45**. First exercise on the 14.4.2015.

## 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 want to 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 laplacian . - 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 01 explicit Euler method for an ODE, Order Plots

Exercise Sheet 02 Lie and Strang splitting for an ODE

Exercise Sheet 03 Finite Differences / Störmer-Verlet for a linear wave equation

Exercise Sheet 04 *(corrected version)* Spectral Methods in Matlab (see also L.Trefethen - Spectral Methods in Matlab (2000) or in my diploma thesis, chapter 4.1 )

Exercise Sheet 05 Spatial order and efficiency of spectral methods and finite differences for a time-dependent problem

Exercise Sheet 06 Temporal and spatial order of a Strang splitting method with the space discretization by spectral methods applied to the NLS

Exercise Sheet 07 Conservation of norm and energy / NLS in 2D

## WS 2014/15:

# Exercises to the lecture *Splitting Methods*

The exercises will take place in room **1C-03** in building Allianzgebäude 5.20 **wednesday 15:45**. First exercise on the 22.10.2014.

## Important Note: Please install *MATLAB* on your Laptop and bring it with you to the problem class if possible

In the exercises to the lecture *Splitting Methods* we want to learn how to use splitting methods as efficient numerical time integrators. The aim of splitting methods is to brake down a complicated, costly problem into simpler subproblems which very often can be solved very efficiently.

For example solving the nonlinear Schrödinger equation

(*NLS*)

with a Runge-Kutta method is very costly.

On the other hand, breaking down the *NLS* into the two subproblems

(*S1*) and (*S2*)

allows to construct an efficient time integrator:

We can solve the subproblems (*S1*) and (*S2*) even exactly in time and combine their solutions in order to obtain an approximation to .

In the problem class

- we will firstly have a short introduction to the numerical software
*MATLAB*, which we will use for the**practical handling of numerical time integration methods**for ODEs and later on also for PDEs. - we will also
**deepen the theoretical understanding of splitting methods and their applications** - of course if there are
**questions concerning the lecture**we try to**resolve**these questions together.

## Exercise Sheets

Exercise Sheet 1 explicit/implicit Euler method, exact solution of an ODE

Exercise Sheet 2 Lie splitting example and order of Strang splitting

Exercise Sheet 3 adjoint of a method

Exercise Sheet 4 global error of Strang splitting, auxiliary results for the BCH formula

Exercise Sheet 5 BCH formula and Lie derivative

Exercise Sheet 6 Lemma by Gröbner

Exercise Sheet 7 Third order splitting method

Exercise Sheet 8 Symplectic Euler method for a Hamiltonian system (harmonic oscillator)

Exercise Sheet 9 2-Body Kepler problem

Exercise Sheet 10 Some theory on symplectic mappings

Exercise Sheet 11 The implicit midpoint rule

## SS 2014:

# Proseminar: *Approximation von Funktionen*

Please go to the german site.

# Exercises to the lecture *Aspects of Numerical Time Integration*

The exercises will take place in room **K 2** (Kronenstraße 32) **tuesday 15:45**. First exercise on the 15.4.2014.

## 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 want to 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 laplacian . - 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 Explicit/Implicit Euler Method

Exercise Sheet 2 Order Plots and Splitting Methods for ODEs

Exercise Sheet 3 Space discretization with finite differences and spectral methods

Exercise Sheet 4 Animated plots and transport equation

Exercise Sheet 5 Linear Schrödinger equation with potential

Exercise Sheet 6 Nonlinear Schrödinger equation

Exercise Sheet 7 Norm and Energy conservation of Lie and Strang solutions of the NLS

Exercise Sheet 8 Numerical order of Lie and Strang solutions of the NLS

Exercise Sheet 9 Regularity of numerical solutions of the NLS & NLS in 2D