Team
Principal Investigators:
- Christiane Helzel,
Heinrich-Heine-Universität Düsseldorf, Chair of Applied Mathematics - Mária Lukáčová,
Johannes Gutenberg-Universität Mainz
Project staff:
- Amelie Porfetye (Düsseldorf)
- Zhuyan Tang (Mainz)
Abstract
It is our strong belief that efficient, accuracy-controlled and structure-preserving schemes provide an inevitable tool for simulations of complex transport-dominated compressible flows, in particular in their high Reynolds number limit. A desirable property of numerical methods for fluid mechanical processes, in addition to obtaining high order accurate approximations on highly resolved grids, is to obtain useful approximations even on coarse grids. Active Flux methods, which are truly multi-dimensional fully discrete methods of odd order with compact stencils in space and time, are interesting candidates to achieve this goal. One goal of this project is to combine our expertise in the development of structure-preserving high-order finite volume methods to derive new third order accurate Active Flux methods for multi-dimensional hyperbolic systems that are based on a truly multi-dimensional evolution operator.
Motivated by recent progress in theoretical analysis of the Euler equations, new probabilistic solution concepts are emerging. Our second aim in this project is therefore devoted to a rigorous convergence and error analysis of the proposed Active Flux methods via dissipative measure-valued solutions and the relative energy. By means of K- and statistical-convergences we will approximate the Reynolds stress, turbulent energy dissipation as well as the mean flow quantities and their variances and explore connections to statistical turbulence models studied within SPP 2410.
In addition to rigorous numerical analysis an important outcome of our project will be an efficient parallel computational code for the Active Flux method for multi-dimensional hyperbolic conservation.
Publications:
2026
Chertock, A., Herty, M., Kurganov, A., Lukáčová-Medvid’ová, M. (2026). Challenges in Stochastic Galerkin Methods for Nonlinear Hyperbolic Systems with Uncertainty. In: Morales de Luna, T., Boscarino, S., Frolkovič, P., Müller, L.O., Escalante, C. (eds) Advances in Nonlinear Hyperbolic Partial Differential Equations. ICIAM 2023. ICIAM2023 Springer Series, vol 7. Springer, Singapore. https://doi.org/10.1007/978-981-96-9087-9_4
E. Feireisl, M. Lukacova-Medvidova, B. She, Y. Yuan: Temperature-driven turbulence in compressible fluid flows, arxiv:2603.28158
M. Anandan, K.R. Arun, A. Krishnamurty, M. Lukacova-Medvidova: "Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations", arxiv.org:2603.27421 (2026)
E. Feireisl, M. Lukacova-Medvidova, B. She, Y. Yuan: "Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system", arxiv.org:2603.20758 (2026)
M. Lukacova-Medvidova, B. She: "Lax convergence theorems and error estimates of a finite element method for the incompressible Euler system", arxiv.org:2604.00783 (2026)
A. Brunk, A. Jüngel, M. Lukacova-Medvidova: A structure-preserving numerical method for quasi-incompressible Navier--Stokes--Maxwell--Stefan systems, arXiV:2504.11892, accepted to J. Sci. Comput. 2026
A. Chertock, M. Herty, A. Ishakov, A. Ishakova, A. Kurganov, M. Lukacova-Medvidova: "Numerical study of random Kelvin-Helmholtz instability", arXiv:2511.00008 accepted to Comm. Comp. Phys. (2026)
2025
Dumbser, M., Lukáčová-Medvid’ová, M. & Thomann, A.: "Convergence of a hyperbolic thermodynamically compatible finite volume scheme for the Euler equations" Numer. Math. (2025)
M. Anandan, M. Lukáčová - Medvid'ová, S. V. Raghurama Rao: An asymptotic preserving scheme satisfying entropy stability for the barotropic Euler system, 2025, SeMA Journal
M. Anandan, M. Lukacova-Medvidova: Provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations, arXiv:2511.19679 (2025)
M. Lukacova-Medvidova, Z. Tang, Y. Yuan: "Convergence analysis for a finite volume evolution Galerkin method for multidimensional hyperbolic systems", arxiv:2511.00957, accepted to Comm. Comp. Phys. (2025)
M. Lukacova-Medvidova, S. Schneider: "Random compressible Euler flows", Proceeding HYP 2024, preprint, (2025)
Chu, S., M. Herty, M. Lukacova-Medvid’ova, and Y. Zhou: Solving random hyperbolic conservation laws using linear programming, arXiv:2501.10104, accepted to SIAM J. Sci. Comput. 2025.
2024
Mária Lukácová-Medvid’ová, Christian Rohde: Mathematical Challenges for the Theory of Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness, Jahresbericht der Deutschen Mathematiker-Vereinigung 2024, https://doi.org/10.1365/s13291-024-00290-6
Mária Lukáčová-Medvid’ová, Yuhuan Yuan: Convergence of a generalized Riemann problem scheme for the Burgers equation, Commun. Appl. Math. Comput. 6, 2215–2238 (2024). https://doi.org/10.1007/s42967-023-00338-x
Erik Chudzik, Christiane Helzel, Mária Lukáčová-Medvid’ová: Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics. J Sci Comput 99, 16 (2024). https://doi.org/10.1007/s10915-024-02462-z
Alina Chertock, Michael Herty, Arsen S. Iskhakov, Safa Janajra, Alexander Kurganov, Mária Lukáčová-Medvid’ová: New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties, Commun. Appl. Math. Comput. 6 (2024), no. 3, 2011–2044. https://doi.org/10.1007/s42967-024-00392-z
Eduard Feireisl, Mária Lukáčová-Medvid’ová, Bangwei She et al.: Convergence of Numerical Methods for the Navier–Stokes–Fourier System Driven by Uncertain Initial/Boundary Data. Found Comput Math (2024). https://doi.org/10.1007/s10208-024-09666-7
Sina Dahm, Jan Giesselmann, Christiane Helzel: Numerical discretisation of hyperbolic systems of moment equations describing sedimentation in suspensions of rod-like particles, Journal of Computational Physics, Vol. 513, p. 113162, 2024, https://doi.org/10.1016/j.jcp.2024.113162