# C5: Adaptive hybrid multiscale simulations of soft matter fluids

The aim of this project is to develop and analyse a novel hybrid multiscale method to simulate fluids that combines the discontinuous Galerkin (dG) method and molecular dynamics (MD). Interaction between particles and continuum variables is realized dynamically, i.e., on-the-fly. Reduced order techniques are used to control the number of computationally expensive MD simulations. In the next funding period, we plan to (i) study error control by means of a-priori feedback estimates and analyse scheme convergence. (ii) Generalize the method to describe more complex physical systems such as non-Newtonian fluids and self-propelled colloidal particles, and (iii) move from benchmark geometries to real-world geometries used in microfluidics.

A multi-scale method for complex flows of non-Newtonian fluids

Mathematics in Engineering,
(2021);

Shear thinning in oligomer melts - molecular origins and applications

Polymers 13 (16),
2806
(2021);

doi:https://doi.org/10.3390/polym13162806

Computing oscillatory solutions of the Euler system via K-convergence

Mathematical Models and Methods in Applied Sciences 31 (03),
537-576
(2021);

doi:10.1142/s0218202521500123

Commensurability between Element Symmetry and the Number of Skyrmions Governing Skyrmion Diffusion in Confined Geometries

Advanced Functional Materials 31 (19),
2010739
(2021);

doi:10.1002/adfm.202010739

Numerical methods for compressible fluid flows

Springer, Modeling, Simulation and Applications , Vol. 20 (2021);

Skyrmion Lattice Phases in Thin Film Multilayer

Advanced Functional Materials 30 (46),
2004037
(2020);

doi:10.1002/adfm.202004037

Convergence of finite volume schemes for the Euler equations via dissipative measure--valued solutions

Found Comput Math 20,
923-966
(2020);

doi:10.1007/s10208-019-09433-z

A finite volume scheme for the Euler system inspired by the two velocities approach

Num. Math. 144 (89-132),
(2020);

doi:10.1007/s00211-019-01078-y

K-convergence as a new tool in numerical analysis

IMA J. Num. Anal. 40,
2227–2255
(2020);

doi:10.1093/imanum/drz045

On the convergence of a finite volume method for the Navier–Stokes–Fourier system

IMA J. Num. Anal. ,
(2020);

C5 Project

doi:10.1093/imanum/draa060

Thermal skyrmion diffusion used in a reshuffler device

Nature Nanotechnology 14 (7),
658-661
(2019);

doi:10.1038/s41565-019-0436-8

Convergence of a finite volume scheme for the compressible Navier-Stokes system

ESAIM: Math. Model. Num. 53,
1957–1979
(2019);

doi: https://doi.org/10.1051/m2an/2019043

An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions

Kinetic and Related Models 12 (1),
195–216
(2019);

URL: http://aimsciences.org//article/doi/10.3934/krm.2019009

doi:10.3934/krm.2019009

Convergence of a mixed finite element finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions

Found. Comput. Math. 18 ,
703–730
(2018);

doi: DOI: 10.1007/s10208-017-9351-2

Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime

SIAM Multiscale Model. Simul. 16 (1),
150–183
(2018);

URL: https://epubs.siam.org/doi/10.1137/16M1094233

Molecular dynamics simulations in hybrid particle-continuum schemes: Pitfalls and caveats

Computer Physics Communications,
(2017);

doi:10.1016/j.cpc.2017.10.016

Asymptotic preserving IMEX ﬁnite volume schemes for low Mach number Euler equations with gravitation

J. Comput. Phys. 335,
222-248
(2017);

doi:10.1016/j.jcp.2017.01.020

Reduced-order hybrid multiscale method combining the molecular dynamics and the discontinuous Galerkin method

VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2017,
1-15.
(2017);

URL: http://congress.cimne.com/coupled2017/frontal/default.asp

Analysis and numerical solution of the Peterlin viscoelastic model (PhD Thesis)

JGU (2015);

URL: http://ubm.opus.hbz-nrw.de/volltexte/2015/4231/

Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method

The European Physical Journal Special Topics 210 (1),
119-132
(2012);

doi:10.1140/epjst/e2012-01641-0

## Contact

- Prof. Dr. Maria Lukáčová
- Institut für Mathematik
- Universität Mainz
- Staudingerweg 9
- D-55128 Mainz
- Tel: +49 6131 39 22831
- Fax: +49 6131 39 23331
- Sekr: +49 6131 39 22270
- lukacovaRzQWG.iW@DibKnmathematik.uni-mainz.de
- http://www.mathematik.uni-mainz.de/Members/lukacova

- Dr. Peter Virnau
- Institut für Physik
- Universität Mainz
- Staudingerweg 9
- D-55128 Mainz
- Tel: +49 6131 39 20493
- Fax: +49 6131 39 20496
- virnauaEFCr@WSFpnyuni-mainz de
- https://www.komet1.physik.uni-mainz.de/people/peter-virnau