Moritz Schneider M.Sc.

Contact

work +49 6151 16-23172
fax +49 6151 16-23164

Work S4|10 114
Dolivostraße 15
64293 Darmstadt

Links

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.

Projects

Logo SFB-TRR154

SFB/TRR154: Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks

Publications

M. Schneider, J. Lang
Well-Balanced and Asymptotic Preserving IMEX-Peer Methods
In F. J. Vermolen and C. Vuik, editors. Numerical Mathematics and Advanced Applications ENUMATH 2019. Springer (2021) arXiv-file

M. Schneider, J. Lang, R. Weiner
Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes
J. Comput. Appl. Math. Vol. 387. 112501 (2021) doi:10.1016/j.cam.2019.112501; 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

summer term 2021 Introduction to numerical modeling (assistant)
Prof. Dr. Martin Kiehl
winter term 2020/21 Tutorial for Mathematics III for Mechanical Engineering (lecturer)
Prof. Dr. Christian Stinner
Introduction to numerical mathematics (assistant)
Prof. Dr. Jens Lang
summer term 2020 Numerical mathematics (mechanical engineering) (assistant)
Dr. Kersten Schmidt
winter term 2019/20 Numerics of partial differential equations (assistant)
Prof. Dr. Jens Lang
summer term 2019 Mathematics II for ET (assistant)
Dr. Kersten Schmidt
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

July 8, 2020 Superconvergent IMEX Peer methods with variable step sizes
SIAM Annual Meeting (AN20), Toronto (virtually)
October 4, 2019 Superconvergent IMEX Peer methods with variable step sizes
ENUMATH 2019, Egmond aan Zee
July 15, 2019 Superconvergent IMEX Peer methods with variable step sizes
ICIAM 2019, Valencia
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