Team
Principal Investigators:
- Jan Giesselmann,
Technical University of Darmstadt - Sebastian Krumscheid,
Karlsruhe Institute of Technology
Project staff:
- Kiwoong Kwon
Abstract
This project addresses the numerical approximation of statistical solutions of the barotropic Navier-Stokes equations, one of the fundamental equations in fluid mechanics. Statistical solutions are a novel solution concept for compressible Navier-Stokes equations that is motivated by turbulence modeling and is thought to address issues with well-posedness that persist for deterministic solution concepts. Statistical solutions can be understood as time-parametrized probability measures on function spaces induced by a random initial datum. Hence, a statistical solution can be approximated by an empirical measure obtained from samples from the initial distribution that are evolved with a numerical scheme for the deterministic, barotropic Navier-Stokes equations. In the convection-dominated case that we focus on, a typical numerical scheme would be of Runge-Kutta discontinuous Galerkin type. We aim to provide reliable, efficient, and robust a posteriori error estimators for these schemes, i.e., upper error bounds for errors caused by discretization in space-time and stochastic space that are computable from the numerical solution. We will combine the relative entropy stability framework with suitable reconstructions of the numerical solution to establish these error bounds. Furthermore, we plan to employ our a posteriori error estimator to construct adaptive, highly efficient multi-level Monte Carlo schemes for approximating quantities of interest pertinent to statistical solutions.
Publications:
2026
Jan Giesselmann, Hendrik Ranocha: "Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions", SMAI J. Comput. Math (2026)
Gunnar Birke, Christian Engwer, Jan Giesselmann, Sandra May: Error analysis of a first-order DoD cut cell method for 2D unsteady advection, J Sci Comput 106, 1 (2026). https://doi.org/10.1007/s10915-025-03091-w
Jan Giesselmann, Jens Keim, Fabio Leotta, Christian Rohde: "Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System", arxiv.2601.18463 (2026)
Jan Giesselmann, Philipp Öffner, Robert Sauerborn: "A Structure Preserving Finite Volume Scheme for the Navier-Stokes-Korteweg Equations" arxiv:2601.08498 (2026)
2025
Giada Cianfarani Carnevale, Jan Giesselmann: Extending relative entropy for Korteweg-Type models with non-monotone pressure:large friction limit and weak-strong uniqueness, Comm. Math. Sci., Volume 23, no. 7, 1983-1998, https://dx.doi.org/10.4310/CMS.250802033320 (2025)
Aaron Brunk, Jan Giesselmann, Maria Lukacova-Medvidova: Robust a posteriori error control for the Allen-Cahn equation with variable mobility, SIAM J. Num. Anal., Volume 63, no. 4, 1540-1560, https://doi.org/10.1137/24M1646406 (2025)
Aaron Brunk, Jan Giesselmann, Tabea Tscherpel: "A posteriori existence of strong solutions to the Navier-Stokes equations in 3D", 2025, arXiv:2509.25105
Arne Berrens, Jan Giesselmann: "A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport", https://arxiv.org/abs/2502.08306 (2025)
2024
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
Fabio Leotta, Jan Giesselmann: A priori error estimates of Runge-Kutta discontinuous Galerkin schemes to smooth solutions of fractional conservation laws, ESAIM: M2AN, 58 (4), 1301–1315, 2024. https://doi.org/10.1051/m2an/2024043