Project C5: Adaptive hybrid multiscale simulations of soft matter fluids
We develop and analyse efficient, hybrid multiscale methods that bridge the continuum-particle gap by combining a discontinuous Galerkin method for the macroscopic model with molecular dynamics. In the second funding period we have focused on the description of non-Newtonian fluids, particularly polymer melts, in simple and complex geometries as well as on theoretical convergence analysis of numerical schemes taking multiscale effects into account.
Building on these results we will study the flow behaviour of polymer mixtures and develop methods to separate polymers with similar molecular masses based on differences in rheological properties (WP1). Applying a probabilistic concept of solutions, we will extend our convergence analysis to these non-Newtonian systems (WP3) and investigate effects of uncertainty with machine learning techniques (WP4). In the final funding period, we would also like to expand the scope of our project to a novel, exciting class of quasi-particles, which on a coarse-grained level can be described in terms of soft matter physics. In close collaboration with experimental colleagues at the JGU Mainz we will probe and study rheological properties of magnetic skyrmions (WP2). In summary, the main goals of the final project period are two-fold: i) investigate the applicability of our hybrid multiscale method to new exciting systems (polymer mixtures, magnetic quasi-particles) and theoretical analysis (convergence and error analysis, uncertainty quantification).
Publications
Temperature-driven turbulence in compressible fluid flows
Eduard Feireisl, Mária Lukáčová-Medvid'ová, Bangwei She, Yuhuan Yuan
Arxiv preprint
Link: https://doi.org/10.48550/arXiv.2603.28158
Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system
Eduard Feireisl, Mária Lukáčová-Medvid'ová, Bangwei She, Yuhuan Yuan
Arxiv preprint
Link: https://doi.org/10.48550/arXiv.2603.20758
Oscillatory approximations and maximum entropy principle for the Euler system of gas dynamics
Eduard Feireisl, Mária Lukáčová-Medvid'ová, Changsheng Yu
Arxiv preprint
Link: https://doi.org/10.48550/arXiv.2505.02070
A note on relations between convexity and concavity of thermodynamic functions
Mária Lukáčová-Medvid'ová, Ferdinand Thein, Gerald Warnecke, Yuhuan Yuan
Arxiv preprint
Link: https://doi.org/10.48550/arXiv.2510.24440
Estimatable variation neural networks and their application to scalar hyperbolic conservation laws
Mária Lukáčová-Medvid'ová, Simon Schneider
Arxiv preprint
Link: https://doi.org/10.48550/arXiv.2409.08909
Maximal dissipation and well-posedness of the Euler system of gas dynamics
E. Feireisl, A. Jüngel, M. Lukáčová-Medvid'ová
Archive for rational mechanics and analysis 250, 53 (2026)
Link: https://doi.org/10.1007/s00205-026-02205-4
Convergence of a hyperbolic thermodynamically compatible finite volume scheme for the Euler equations
Michael Dumbser, Mária Lukáčová-Medvid'ová, Andrea Thomann
Numerische Mathematik 158, 715-747 (2026)
Link: https://doi.org/10.1007/s00211-025-01522-2
Hybrid multiscale method for polymer melts: Analysis and simulations
Ranajay Datta, Mária Lukáčová-Medvid'ová, Andreas Schömer, Peter Virnau
ZAMM 106, e70438 (2026)
Link: https://doi.org/10.1002/zamm.70438
Solving random hyperbolic conservation laws using linear programming
Shaoshuai Chi, Michael Herty, Mária Lukáčová-Medvid'ová, Yizhou Zhou
SIAM J. Sci. Comp. 48, A1184-A1205 (2026)
Link: https://doi.org/10.1137/25M1731654
Real-Time Modeling of Skyrmion Dynamics in Arbitrary 2D Spatially Dependent Pinning Potential Landscapes
S.F. Fröhlich, T. Sparmann, M.A. Brems, J. Rothörl, F. Kammerbauer, K. Raab, K. Sachin, M. Kläui, P. Virnau
Comm. Phys. 9, 243 (2026)
Link: https://doi.org/10.1038/s42005-026-02766-6
Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems
Kuzmin, D., Lukáčová-Medvid'ová, M., Öffner, P.
J. Num. Math. 34, 1-20 (2026)
Link: https://doi.org/10.1515/jnma-2024-0123
Influence of topology on rheological properties of polymer ring melts
Ranajay Datta, Peter Virnau
The Journal of Chemical Physics 162, 094902 (2025)
Link: https://doi.org/10.1063/5.0250414
Realizing Quantitative Quasiparticle Modeling of Skyrmion Dynamics in Arbitrary Potentials
M.A. Brems, T. Sparmann, S.M. Fröhlich, L.-C. Dany, J. Rothörl, F. Kammerbauer, E.M. Jefremovas, O. Farago, M. Kläui, P. Virnau
Physical Review Letters 134, 046701 (2025)
Link: https://doi.org/10.1103/PhysRevLett.134.046701
Convergence Analysis of the Monte Carlo Method for the Random Navier–Stokes–Fourier System
M. Lukáčová-Medvid'ová, B. She, Y. Yuan
SIAM J. Numer. Anal. 63, no. 3, 1254–1280 (2025)
Link: https://doi.org/10.1137/23M1563232
Penalty method for the Navier–Stokes–Fourier system with Dirichlet boundary conditions: convergence and error estimates
M. Lukáčová-Medvid'ová, B. She, Y. Yuan
Numer. Math. 157, no. 3, 1079–1132 (2025)
Link: https://doi.org/10.1007/s00211-025-01468-5
What is a limit of structure-preserving numerical methods for compressible flows? Numerical mathematics and advanced applications
M. Lukáčová-Medvid'ová, B. She, Y. Yuan
ENUMATH 2023. Vol. 1, 17–33, Lect. Notes Comput. Sci. Eng., 153, Springer, Cham (2025)
Link: https://doi.org/10.1007/978-3-031-86173-4_2
Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system
M. Lukáčová-Medvid'ová, B. She, Y. Yuan
IMA J. Numer. Anal. 45, no. 2, 1054–1101 (2025)
Link: https://doi.org/10.1093/imanum/drae024
Conditional regularity for the compressible Navier-Stokes equations with potential temperature transport
M. Lukáčová-Medvid'ová, A. Schömer
J. Differential Equations 423, 1–40 (2025)
Link: https://doi.org/10.1016/j.jde.2024.12.028
An implicit-explicit solver for a two-fluid single-temperature model
Lukáčová-Medvid'ová, M., Peshkov, I., and Thomann, A.
Journal of Computational Physics, 498, 112696, (2024)
Link: https://doi.org/10.1016/j.jcp.2023.112696
New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties
Chertock, A., Herty, M., Iskhakov, A. S., Janajra, S., Kurganov, A., and Lukáčová-Medvid’ová, M.
Communications on Applied Mathematics and Computation, 1-34, (2024)
Link: https://doi.org/10.1007/s42967-024-00392-z
Convergence of Numerical Methods for the Navier–Stokes–Fourier System Driven by Uncertain Initial/Boundary Data
Feireisl, E., Lukáčová-Medvid’ová, M., She, B., and Yuan, Y.
Foundations of Computational Mathematics, 1-53, (2024)
Link: https://doi.org/10.1007/s10208-024-09666-7
Skyrmion flow in periodically modulated channels
Klaus Raab, Maurice Schmitt, Maarten A. Brems, Jan Rothörl, Fabian Kammerbauer, Sachin Krishnia, Mathias Kläui, Peter Virnau
Physical Review E 110, L042601 (2024)
Link: https://doi.org/10.1103/PhysRevE.110.L042601
Growth and aging in a few phase-separating active matter system
Florian Dittrich, Jiarul Midya, Peter Virnau, Subir K. Das
Physical Review E 108, 024609 (2023)
Link: https://doi.org/10.1103/PhysRevE.108.024609
300-Times-Increased Diffusive Skyrmion Dynamics and Effective Pinning Reduction by Periodic Field Excitation
R. Gruber, M.A. Brems, J. Rothörl, T. Sparmann, M. Schmitt, I. Kononenko, F. Kammerbauer, M.-A. Syskaki, O. Farago, P. Virnau, M. Kläui
Advanced Materials 35, 2208922 (2023)
Link: https://doi.org/10.1002/adma.202208922
Local characteristic decomposition based central-upwind scheme
Alina Chertock, Shaoshuao Chu, Michael Herty, Alexander Kurganov, Mária Lukáčová-Medvid'ová
J. Comp. Phys. 473, 111718
Link: https://doi.org/10.1016/j.jcp.2022.111718
Constructing coarse-grained skyrmion potentials from experimental data with Iterative Boltzmann Inversion
Y. Ge, J. Rothörl, M.A. Brems, N. Kerber, R. Gruber, T. Dohi, M. Kläui, P. Virnau
Comm. Phys. 6, 30 (2023)
Link: https://doi.org/10.1038/s42005-023-01145-9
Viscosity of flexible and semiflexible ring melts: Molecular origins and flow-induced segregation
Ranajay Datta, Fabian Berressem, Friederike Schmid, Arash Nikoubashman, Peter Virnau
Macromolecules 56, 7247-7255 (2023)
Link: https://doi.org/10.1021/acs.macromol.3c01046
Error estimates of a finite volume method for the compressible Navier–Stokes–Fourier system
Danica Basarić, Hana Mizerová, Bangwei She, Yuhuan Yuan
Math.Comp. 92, 2543–2574 (2023)
Link: https://doi.org/10.1090/mcom/3852
Compressible Navier-Stokes equations with potential temperature transport: stability of the strong solution and nu-merical error estimates
M. Lukáčová-Medvid'ová, A. Schömer
Journal of Mathematical Fluid Mechanics 25, 1 (2023)
Link: https://doi.org/10.1007/s00021-022-00733-z
Convergence of a stochastic collocation finite volume method for the compressible Navier–Stokes system.
E. Feireisl, M. Lukáčová-Medviďová
The Annals of Applied Probability, 33(6A), 4936-4963 (2023)
Link: https://doi.org/10.1214/23-AAP1937
On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions
Abgrall, R., Lukáčová-Medvid’ová, M. and Öffner, P.
Mathematical Models and Methods in Applied Sciences, 33(01), 139-173, (2023)
Link: https://doi.org/10.1142/S0218202523500057
Convergence of discontinuous Galerkin schemes for the Euler equations via dissipative weak solutions
Lukáčová-Medvid’ová, M. and Öffner, P.
Applied Mathematics and Computation, 436, 127508, (2023)
Link: https://doi.org/10.1016/j.amc.2022.127508
Statistical solutions for the Navier–Stokes–Fourier system
Feireisl, E., and Lukáčová-Medvid’ová, M.
Stochastics and Partial Differential Equations: Analysis and Computations, 1-25, (2023)
Link: https://doi.org/10.1007/s40072-023-00298-6
Error estimates of a finite volume method for the compressible Navier–Stokes–Fourier system
Danica Basarić, Mária Lukáčová-Medviďová, Hana Mizerová, Bangwei She and Yuhuan Yuan
Math. Comp. 92 (2023), 2543-2574, (2023)
Link: https://doi.org/10.1090/mcom/3852
An all Mach number finite volume method for isentropic two-phase flow
Lukáčová-Medvid’ová, M., Puppo, G. and Thomann, A.
Journal of Numerical Mathematics, 31(3), 175-204, (2023)
Link: https://doi.org/10.1515/jnma-2022-0015
Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations
V. Kucera, M. Lukáčová-Medvid'ová, S. Nölle, J. Schütz
Numerische Mathematik 150, 79-103 (2022)
Link: https://doi.org/10.1007/s00211-021-01240-5
Approximating viscosity solutions of the Euler system
E. Feireisl, M. Lukáčová-Medvid'ová, S. Scheiner, B. Whe
Mathematics of Computation 91, 2129-2164 (2022)
Link: https://doi.org/10.1090/mcom/3738
Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System
Mária Lukáčová-Medvid'ová, Bangwei She, Yuhuan Yuan
J. Scientific Computing 91, 71 (2022)
Link: https://doi.org/10.1007/s10915-022-01843-6
On a Hybrid Continuum-Kinetic Model for Complex Fluids
A. Chertock, P. Degond, G. Dimarco, and M. Lukáčová-Medvid'ová
Partial Differ. Equ. Appl. 3, Paper No. 63, 28 pp. (2022)
Link: https://doi.org/10.1007/s42985-022-00198-9
Convergence and Error Analysis for Compressible Fluid Flows with Random Data: Monte Carlo Method
E. Feireisl, M. Lukáčová-Medvid'ová, B. She, B., Y. Yuan
Math. Mod. Meth. Appl. Sci. 32 (14), 2887-2925 (2022)
Link: https://doi.org/10.1142/S0218202522500671
Penalization method for the Navier-Stokes-Fourier system
Danica Basarić, Eduard Feireisl, Mária Lukáčová-Medviďová, Hana Mizerová, Yuhuan Yuan
ESAIM: Mathematical Modelling and Numerical Analysis, (2022)
Link: https://doi.org/10.1051/m2an/2022063
Existence of Dissipative Solutions to the Compressible Navier-Stokes System with Potential Temperature Transport
M. Lukáčová-Medvid'ová, A. Schömer
Journal of Mathematical Fluid Mechanics 24, 82 (2022)
Link: https://doi.org/10.1007/s00021-022-00713-3
A multi-scale method for complex flows of non-Newtonian fluids
F. Tedeschi, G.G. Giusteri, L. Yelash, M. Lukáčová-Medvid’ova
Mathematics in Engineering 4, 6 (2022)
Link: https://doi.org/10.3934/mine.2022050
Circuits and excitations to enable Brownian token-based computing with skyrmions
Maarten A. Brems, Mathias Kläui, Peter Virnau
Applied Physics Letters 119, 132405 (2021)
Link: https://doi.org/10.1063/5.0063584
Shear thinning in oligomer melts - molecular origins and applications
R. Datta, L. Yelash, F. Schmid, F. Kummer, M. Oberlack, M. Lukáčová-Medvid'ová, P. Virnau
Polymers 13 (16), 2806 (2021)
Link: https://doi.org/10.3390/polym13162806
Computing oscillatory solutions of the Euler system via K-convergence
Eduard Feireisl, Mária Lukáčová–Medvid’ová, Bangwei She, Yue Wang
Mathematical Models and Methods in Applied Sciences 31 (03), 537-576 (2021)
Link: https://doi.org/10.1142/S0218202521500123
Commensurability between Element Symmetry and the Number of Skyrmions Governing Skyrmion Diffusion in Confined Geometries
Chengkun Song, Nico Kerber, Jan Rothörl, Yuqing Ge, Klaus Raab, Boris Seng, Maarten A. Brems, Florian Dittrich, Robert M. Reeve, Jianbo Wang, Qingfang Liu, Peter Virnau, Mathias Kläui
Advanced Functional Materials 31 (19), 2010739 (2021)
Link: https://doi.org/10.1002/adfm.202010739
Book: Numerical methods for compressible fluid flows
E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerova, B. She
Springer, Modeling, Simulation and Applications, Vol. 20 (2021)
Link: https://doi.org/10.1007/978-3-030-73788-7
Skyrmion Lattice Phases in Thin Film Multilayer
Jakub Zázvorka, Florian Dittrich, Yuqing Ge, Nico Kerber, Klaus Raab, Thomas Winkler, Kai Litzius, Martin Veis, Peter Virnau, Mathias Kläui
Advanced Functional Materials 30 (46), 2004037 (2020)
Link: https://doi.org/10.1002/adfm.202004037
Convergence of finite volume schemes for the Euler equations via dissipative measure--valued solutions
E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová
Foundations of Computational Mathematics 20, 923-966 (2020)
Link: https://doi.org/10.1007/s10208-019-09433-z
A finite volume scheme for the Euler system inspired by the two velocities approach
E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerova
Numerical Mathematics 144 (89-132), (2020)
Link: https://doi.org/10.1007/s00211-019-01078-y
K-convergence of finite volume solutions of the Euler equations
Mária Lukáčová-Medvid'ová
Finite volues for complex applications IX, 323, 25-37 (2020)
Link: https://doi.org/10.1007/978-3-030-43651-3\_2
K-convergence as a new tool in numerical analysis
E.Feireisl, M. Lukáčová-Medvid'ová, H. Mizerova
IMA Journal of Numerical Analysis 40, 2227–2255 (2020)
Link: https://doi.org/10.1093/imanum/drz045
On the convergence of a finite volume method for the Navier–Stokes–Fourier system
E.Feireisl, M. Lukáčová-Medvid'ová, H. Mizerova, B. She
IMA Journal of Numerical Analysis, (2020)
Link: https://doi.org/10.1093/imanum/draa060
Thermal skyrmion diffusion used in a reshuffler device
Jakub Zázvorka, Florian Jakobs, Daniel Heinze, Niklas Keil, Sascha Kromin, Samridh Jaiswal, Kai Litzius, Gerhard Jakob, Peter Virnau, Daniele Pinna, Karin Everschor-Sitte, Levente Rózsa, Andreas Donges, Ulrich Nowak, Mathias Kläui
Nature Nanotechnology 14 (7), 658-661 (2019)
Link: https://doi.org/10.1038/s41565-019-0436-8
Convergence of a finite volume scheme for the compressible Navier-Stokes system
E.Feireisl, M. Lukáčová-Medvid'ová, H. Mizerova
ESAIM: Mathematical Modelling and Numerical Analysis. 53, 1957–1979 (2019)
Link: https://doi.org/10.1051/m2an/2019043
An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions
A. Chertock, A. Kurganov, M. Lukáčová-Medvid'ová, S. Nur Oezcan
Kinetic and Related Models 12 (1), 195–216 (2019)
Link: https://doi.org/10.3934/krm.2019009
Convergence of a mixed finite element finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions
E. Feireisl, M. Lukáčová-Medvid'ová
Foundations of Computational Mathematics 18 , 703–730 (2018)
Link: https://doi.org/10.1007/s10208-017-9351-2
Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime
E. Feireisl, M. Lukáčová-Medvid'ová, S. Necasova, A. Novotny, B. She
SIAM Multiscale Modeling and Simulation 16 (1), 150–183 (2018)
Link: https://doi.org/10.1137/16M1094233
Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model
Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvid'ová, Nikalaos Sfakianakis
Discrete and continuous dynamics systems 23, 4397-4431 (2018)
Link: https://doi.org/10.3934/dcdsb.2018169
Free-energy barriers for crystal nucleation from fluid phases
Peter Koss, Antonia Statt, Peter Virnau, Kurt Binder
Physical Review E 96, 042609 (2017)
Link: https://doi.org/10.1103/PhysRevE.96.042609
Molecular dynamics simulations in hybrid particle-continuum schemes: Pitfalls and caveats
S. Stalter, L. Yelash, N. Emamy, A. Statt, M. Hanke, M. Lukáčová-Medvid’ová, P. Virnau
Computer Physics Communications 224, 198-208 (2017)
Link: https://doi.org/10.1016/j.cpc.2017.10.016
Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation
G. Bispen, M. Lukáčová-Medvid'ová, L. Yelash
Journal of Computational Physics 335, 222-248 (2017)
Link: https://doi.org/10.1016/j.jcp.2017.01.020
Reduced-order hybrid multiscale method combining the molecular dynamics and the discontinuous Galerkin method
N. Emamy, M. Lukácová-Medvid’ová, S. Stalter, P. Virnau, L. Yelash
VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2017, 15 pp. (2017)
Link: https://upcommons.upc.edu/server/api/core/bitstreams/d23e104d-eb34-4172-a25f-a88c2e732085/content
Analysis and numerical solution of the Peterlin viscoelastic model (PhD Thesis)
Mizerova Hana
Johannes Gutenberg-Universität (2015)
Link: https://doi.org/10.25358/openscience-3633
Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method
B. J. Block, M. Lukáčová-Medvid’ová, P. Virnau, L. Yelash
The European Physical Journal Special Topics 210 (1), 119-132 (2012)
Link: https://doi.org/10.1140/epjst/e2012-01641-0