# Moritz Schneider, M.Sc.

Work
Dolivostraße 15

64293
Darmstadt

Office: S4|10 114

work +49 6151 16-23172

fax +49 6151 16-23164

moschneider@mathematik.tu-...

# Research

## Superconvergent implicit-explicit (IMEX) Peer Methods

*with Jens Lang (Darmstadt) and Rüdiger Weiner (Halle)*

The spatial discretization of certain time-dependent PDEs (e.g. advection-reaction-diffusion systems) yields large systems of ODEs where the right-hand side admits a splitting into a stiff and non-stiff part. We construct time integrators that combine the favorable stability properties of implicit methods and the low computational costs of explicit schemes. In order to guarantee consistency and, thus, convergence, the implicit and explicit integrator must fit together. A natural way to construct these implicit-explicit (IMEX) Peer methods is to start with an appropriate implicit scheme and extrapolate it in a suitable manner.

Peer methods have the advantage that all stage values have the same order and, hence, order reduction for stiff systems is avoided. Further, there remain enough free parameters such that additional properties can be guaranteed. This includes optimal zero-stability, A-stability of the implicit part and, in particular, superconvergence. We focus on the construction of new superconvergent IMEX schemes for different numbers of stages. In addition, Peer methods can be easily adapted to the setting of variable step sizes. The realization of superconvergent methods for variable step sizes is the subject of ongoing research.

# Publications

M. Schneider, J. Lang, R. Weiner**Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes**

arXiv:1902.01161 arXiv-file

M. Schneider, J. Lang, W. Hundsdorfer**Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part**

J. Comput. Physics, Vol. 367, pp. 121-133, 2018; doi:10.1016/j.jcp.2018.04.006; arXiv-file

# Teaching

winter term 2018/19 |
Mathematics III for ET (assistant)Dr. Kersten Schmidt |

summer term 2018 |
Numerical Linear Algebra (assistant)Prof. Dr. Jens Lang Numerics of hyperbolic differential equations (assistant)Prof. Dr. Jens Lang |

winter term 2017/18 |
Introduction to numerical mathematics (assistant)Prof. Dr. Jens Lang |

summer term 2017 |
Mathematics II for Computer Science (assistant)Prof. Dr. Thomas Streicher |

winter term 2016/17 |
Mathematics I for Computer Science (assistant)Prof. Dr. Thomas Streicher |

summer term 2016 |
Numerical Linear Algebra (tutor)Dr. Alf Gerisch Introduction to mathematical modelling (tutor)Prof. Dr. Jens Lang |

winter term 2015/16 |
Introduction to numerical mathematics (tutor)Prof. Dr. Herbert Egger |

summer term 2015 |
Mathematics IV for ET / III for CS (tutor)Prof. Dr. Stefan Ulbrich |

winter term 2014/15 |
Mathematics III for ET (tutor)Prof. Dr. Hans-Dieter Alber |

summer term 2014 |
Mathematics II for ET (tutor)Prof. Dr. Hans-Dieter Alber |

winter term 2013/14 |
Mathematics I for ET (tutor)Prof. Dr. Hans-Dieter Alber |

summer term 2013 |
Linear Algebra for Physics (tutor)Prof. Dr. Matthias Schneider |

# Talks

September 6, 2018 |
Superconvergent IMEX Peer methods with A-stable implicit partNUMDIFF 15, Halle (Saale) |

May 15, 2018 |
Superconvergent IMEX Peer methodsNumerics Seminar, TU Darmstadt |

September 11, 2017 |
Superconvergent IMEX Peer methodsSciCADE 2017, Bath |

October 27, 2016 |
The Contour Method and its applicationsNumerics Seminar, TU Darmstadt |