Team
Principal Investigators:
- Michael Herty,
RWTH Aachen - Mária Lukáčová,
Johannes Gutenberg-Universität Mainz
Project staff:
- Shaoshuai Chu (Aachen)
- Changsheng Yu (Mainz)
Associated staff:
- Yizhou Zhou (RWTH Aachen)
- Simon Schneider (University of Mainz)
Abstract
Mathematical models arising in science and engineering inherit several sources of uncertainties, such as model parameters, initial and boundary conditions. In order to predict reliable results, deterministic models are insufficient and more sophisticated methods are needed to analyse the influence of uncertainties on numerical solutions. Progress in analytical results and corresponding design of numerical schemes for elliptic and parabolic equations, have, however, not yet been fully expanded towards the hyperbolic problems. A main obstacle is posed by the nonlinear transport that causes the loss of regularity of a solution that propagates also in the random space and the possible loss of hyperbolicity in intrusive Galerkin methods.
Uncertainty quantification of the Euler equations is intrinsically connected to statistical hydrodynamics.
The idea of considering random or statistical solutions of compressible fluid flow models is natural in order to describe turbulent fluid behaviour. Our aim in this proposal is to deepen the understanding of random/statistical solutions of compressible Euler equations both from the theoretical as well as numerical point of view.
The proposed project aims to achieve three goals: Firstly, we introduce and analyse a new concept of ensemble-averaged solutions, the random dissipative solutions. Their existence will be proved via convergence of suitable uncertainty quantification methods, based on polynomial chaos expansion. To this end, we work with inherently stochastic compactness arguments. In order to quantify errors of numerical approximations the random relative energy inequality will be derived.
Applying a set-valued compactness framework, K-convergence, we approximate turbulent Reynolds stress and energy dissipation. Secondly, using the formulation of random dissipative solutions we propose and analyse novel numerical schemes.
Here, moment approximations will be used to derive effective equations for the evolution of statistical moments.
Thirdly, we study the low Mach number limit of the random weakly-compressible Euler equations by means of asymptotic preserving numerical schemes. Consequently, we address multiscale phenomena in hyperbolic problems, investigate a delicate interplay between randomness and hyperbolic transport and contribute to overarching goals of SPP 2410 to design entropy stable and structure preserving numerical schemes.
Publications
2026
E. Feireisl, , M. Lukacova-Medvidova, H. Mizerova, C. Yu; "Monte Carlo method and the random isentropic Euler system", Stoch. Partial Differ. Equ. Anal. Comput, https://doi.org/10.1007/s40072-026-00417-z (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. 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
Herty, Michael and Thein, Ferdinand: Boundary feedback control for hyperbolic systems, https://doi.org/10.1051/cocv/2024062, (2024)
Niklas Kolbe, Michael Herty, Siegfried Müller. Numerical schemes for coupled systems of nonconservative hyperbolic equations, SIAM J. Numer. Anal., 62(5):2143-2171, DOI:10.1137/23M1615176
Michael Herty, Niklas Kolbe, Siegfried Mueller: A central scheme for two coupled hyperbolic systems, Commun. Appl. Math. Comput. 6, 2093–2118, DOI:10.1007/s42967-023-00306-5
2023
Michael Herty, Niklas Kolbe, Siegfried Mueller, Central schemes for networked scalar conservation laws, Networks and Heterogeneous Media, 18(1), 310--340, 2023, DOI:10.3934/nhm.2023012