Team
Principal Investigators:
- Jan Giesselmann,
Technical University of Darmstadt - Philipp Öffner,
Clausthal University of Technology.
Project staff:
- Robert Sauerborn (Clausthal University of Technology)
Abstract
Many problems in computational fluid dynamics are described via the compressible Euler or Navier-Stokes (NS) equations. Recently, dissipative weak (DW) solutions have been introduced as a generalization to classical solution concepts. In a series of works, DW have been established as a meaningful concept from the analytical and numerical point of views. DW solutions do not have to fulfill the equations weakly but up to some defect and oscillations measures, They can be identified as limits of consistent and stable approximations and convergence towards DW solutions have been demonstrated for several structure-preserving numerical methods. Further, they are a natural extension of classical solutions since DW solutions coincide with them if either the classical solution exists, referred to as weak-strong uniqueness principle, or if they enjoy certain smoothness. A further extension to Navier-Stokes is the Navier-Stokes-Korteweg (NSK) which includes capillarity terms in the equations. Our motivation of considering NSK in our project is driven by the recent observations made by Slemrod that the rigorous passage of solutions from the mesoscopic equations (Boltzmann) to macroscopic systems, known as Hilbert sixth’s problem, will fail for the classical systems (NS and Euler) because of the appearance of van der Waals–Korteweg capillarity terms in a macroscopic description. Therefore, Korteweg systems are more suitable, for describing real fluid motions. In our project, we will extend the framework of DW solutions to NSK equations. We will define DW solutions in such a way that they form a natural extension of classical solutions, i.e. such that they satisfy a weak strong uniqueness principle, for the local and non-local NSK model. In addition, we plan to prove their global existence by demonstrating convergence of structure-preserving numerical schemes. First, structure-preserving low order schemes will be constructed and analyzed, whereas later we focus as well on the development and convergence properties of higher-order finite element (FE) schemes. Our investigations extend classical concepts and give new insights in the description of fluid- flows both theoretically and numerically, and lies in the scope of the SPP 2410 program.
Publications:
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
Janina Bender and Thomas Izgin and Philipp Öffner and Davide Torlo: "The Lax–Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws" Computers & Fluids Volume 304, 106885 (2026)
Jan Giesselmann, Philipp ̈Offner, Robert Sauerborn: "Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions", arxiv:2604.16110 (2026)
Jan Glaubitz, Armin Iske, Joshua Lampert, Philipp Öffner: "Why summation by parts is not enough" arxiv:2602.10786 (2026)
Thomas Eiter, Jan Giesselmann, Robert Lasarzik, Philipp Öffner, Robert Sauerborn: "Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions" arxiv:2603.29880 (2026)
Philipp Öffner, Per Pettersson, Andrew R. Winters: "A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension" arxiv:2604.01199 (2026)
Jan Giesselmann, Philipp Öffner, Robert Sauerborn: "Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions", arxiv:2604.16110 (2026)
Jan Giesselmann, Philipp Öffner, Robert Sauerborn: "A Structure Preserving Finite Volume Scheme for the Navier-Stokes-Korteweg Equations" arxiv:2601.08498 (2026)
Rahul Barthwal, Philipp Öffner, Christian Rohde: "Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows", arXiv:2604.16000 (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)
M. Ciallella, L. Micalizzi, V. Michel-Dansac, P. Öffner, and D. Torlo - A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations, GEM - International Journal on Geomathematics, https://link.springer.com/article/10.1007/s13137-025-00262-7 ,2025
P. Öffner, L. Petri, and D. Torlo - Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations - Applied Numerical Mathematics, https://www.sciencedirect.com/science/article/pii/S0168927424003568?via%3Dihub, 2025
H. Ranocha, A. R. Winters, M. Schlottke-Lakemper, P. Öffner, J. Glaubitz, G. J. Gassner - High-order upwind summation-by-parts methods for nonlinear conservation laws - Journal of Computational Physics 520, https://www.sciencedirect.com/science/article/pii/S0021999124007198?via%3Dihub, (2025)
D. Breit, T. C. Moyo, and P. Öffner - Discontinuous Galerkin methods for the complete stochastic Euler equations, Journal of Computational Physics 541 (2025). https://doi.org/10.1016/j.jcp.2025.114324
H. Hajduk, D. Kuzmin, P. Öffner, G. Lube - Locally Energy-Stable Finite Element Schemes for Incompressible Flow Problems: Design and Analysis for Equal-Order Interpolations, Computers & Fluids 294 (2025). https://doi.org/10.1016/j.compfluid.2025.106622
Glaubitz, J., Nordström, J. & Öffner, P. An Optimization-Based Construction Procedure for Function Space-Based Summation-by-Parts Operators on Arbitrary Grids. J Sci Comput 105, 83 (2025). https://doi.org/10.1007/s10915-025-03062-1
Kuzmin, Dmitri, Lukáčová-Medvid’ová, Mária and Öffner, Philipp. "Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems" Journal of Numerical Mathematics. (2025) https://doi.org/10.1515/jnma-2024-0123
J. Glaubitz, H. Ranocha, A. R. Winters, M. Schlottke-Lakemper, P. Öffner, G. J. Gassner - Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods, Journal of Computational Physics 529 (2025). https://doi.org/10.1016/j.jcp.2025.113841
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)
Rémi Abgrall, Philipp Öffner, Yongle Liu: "Some new properties of the PamPa scheme" arxiv:2508.17147 (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
J. Glaubitz, S.-Ch. Klein, J. Nordström, P. Öffner - Summation-by-parts operators for general function spaces: The second derivative - Journal of Computational Physics 504, 2024. https://www.sciencedirect.com/science/article/pii/S0021999124001384