Moritz Schneider

Moritz Schneider, M.Sc.

Dolivostraße 15
64293 Darmstadt

Office: S4|10 114

+49 6151 16-23172
+49 6151 16-23164


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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 part
NUMDIFF 15, Halle (Saale)
May 15, 2018 Superconvergent IMEX Peer methods
Numerics Seminar, TU Darmstadt
September 11, 2017 Superconvergent IMEX Peer methods
SciCADE 2017, Bath
October 27, 2016 The Contour Method and its applications
Numerics Seminar, TU Darmstadt