Dr. Patrick Krämer
 by appointment
 Kollegiengebäude Mathematik (20.30)
 3.025
 +49 721 608 47653
 patrick.kraemer3@kit.edu

Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik 3
Englerstraße 2
D76131 Karlsruhe
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 MaxwellKleinGordon and MaxwellDirac Systems in Highly to Slowly Oscillatory Regimes, PhD thesis, 2017, supervised by JProf. Dr. Katharina Schratz
 Efficient time integration of the MaxwellKleinGordon equation in the nonrelativistic limit regime, CRC 1173 Preprint, July 2016, joint work with JProf. Dr. Katharina Schratz
 The Method of Multiple Scales for nonlinear KleinGordon 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 pseudospectral 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 pseudospectral methods for the space approximation and splitting methods for the timeintegration
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 / StörmerVerlet 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 timedependent 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 1C03 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 RungeKutta 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 2Body 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 pseudospectral 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 pseudospectral methods for the space approximation and splitting methods for the timeintegration
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