EsCUT: Entropy-stable high-order CUT-cell discontinuous Galerkin methods

Project information

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Principal Investigators:

Project staff:

  • Gunnar Birke (University of Münster)
  • Louis Petri (University of Mainz)

The aim of this project is the development of new stabilisation techniques to obtain robust and efficient entropy-stable numerical methods for non-linear hyperbolic conservation laws on cut-cell meshes for demanding simulations of under-resolved flows in complex geometries. These new high-resolution, structure-preserving methods will be designed for novel solution concepts such as dissipative weak solutions. Nonlinear hyperbolic balance laws play a crucial role in many important applications such as aircraft design and environmental/climate research. Two major challenges for high- resolution numerical simulation methods for these applications are complex geometries and under-resolved features, necessarily present in turbulent flows. To solve these issues, two types of stabilisation are required: (i) cut-cell stabilisations to cope with time step restrictions of small cut cells (ii) stabilisation of the baseline scheme for under-resolved flows and turbulence. To avoid overly dissipative schemes and pave the way to a better understanding of turbulent flows in complex geometries, we will develop novel entropy-stable high-resolution schemes with robust cut-cell stabilisations for discontinuous Galerkin methods. The basic idea is to reformulate cut-cell stabilisations in a first step to make them approachable by an entropy analysis. Based thereon, we will analyse the stability properties in detail and develop novel approaches guaranteeing entropy stability properties. The final deliverable of this project are entropy-stable DG methods on cut-cell meshes that can be used efficiently with explicit time integration methods under reasonable time step constraints for the multi-dimensional compressible Euler equations. In an upcoming extension of this project, we plan to extend these novel algorithms to the compressible Navier-Stokes equations and apply them to the simulation of turbulent behaviour in complex geometries, in particular in the high Reynolds number regime.

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 

Marco Artiano, Hendrik Ranocha, Saurav Samantaray: "Asymptotic-Preserving and Well-Balanced Linearly Implicit IMEX Schemes for the Anelastic Limit of the Isentropic Euler Equations with Gravity", arxiv:2604.11573 (2026)
 
Arpit Babbar, Qifan Chen, Hendrik Ranocha: "Compact Runge-Kutta flux reconstruction methods with entropy and/or kinetic energy preserving fluxes", arxiv:2604.02125, (2026)
 
Wittenstein, Collin and Marks, Vincent and Ricchiuto, Mario and Ranocha, Hendrik: "GPU-Accelerated Energy-Conserving Methods for the Hyperbolized Serre-Green-Naghdi Equations in 2D" arXiv:2601.02540 (2026)
 

Louis Petri, Sigrun Ortleb, Gunnar Birke, Christian Engwer, Hendrik Ranocha: "Domain-of-dependence-stabilized cut-cell discretizations of linear kinetic models with summation-by-parts properties", arxiv.2601.05817 (2026)

Daniel Doehring, Jesse Chan, Hendrik Ranocha, Michael Schlottke-Lakemper, Manuel Torrilhon, Gregor Gassner: Volume Term Adaptivity for Discontinuous Galerkin Schemes, arxiv:2603.24189 (2026)
 
Marco Artiano, Hendrik Ranocha: On Affordable High-Order Entropy-Conservative/Stable and Well-Balanced Methods for Nonconservative Hyperbolic Systems, arxiv2603.18978 (2026)
 
Marco Artiano, Arpit Babbar, Michael Schlottke-Lakemper, Gregor Gassner, Hendrik Ranocha: Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes, arXiv:2603.16290 (2026)

2025

Hendrik Ranocha, Mario Ricchiuto: "Structure-Preserving Approximations of the Serre-Green-Naghdi Equations in Standard and Hyperbolic Form", Numerical Methods for Partial Differential Equations, Volume 41, Issue 4, e70016, https://doi.org/10.1002/num.70016 (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)

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

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

Lampert, Joshua and Wittenstein, Collin and Ranocha, Hendrik: "DispersiveShallowWater.jl: A Julia library of structure-preserving numerical methods for dispersive wave equations" Journal of Open Source Software, 10(116), 9361, https://doi.org/10.21105/joss.09361 (2025)

Marco Artiano, Oswald Knoth, Peter Spichtinger, Hendrik Ranocha: "Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows", arxiv:2509.10311 (2025)

Louis Petri, Gunnar Birke, Christian Engwer, Hendrik Ranocha: "The domain-of-dependence stabilization for cut-cell meshes is fully discretely stable", arXiv:2508.05372 (2025)

A. Babbar, V. Churavy, M. Schlottke-Lakemper, H. Ranocha: "Automatic differentiation for Lax-Wendroff-type discretizations", arXiv:2506.11719 (2025)

H. Ranocha, D. Ketcheson: "Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger equations" arxiv:2512.16352 (2025)
 
A. Babbar, H. Ranocha: "Compact Runge-Kutta flux reconstruction methods for non-conservative hyperbolic equations" arxiv:2512.08611 (2025)
 
Sebastian Bleecke, Abhijit Biswas, David I. Ketcheson, Hendrik Ranocha, Jochen Schutz: "Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation", arXiv:2511.10044 (2025)
 
Hendrik Ranocha, David I. Ketcheson: High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation and its hyperbolization", 2025, arXiv:2510.14335
 
Ferdinand Thein, Hendrik Ranocha: "Computing Radially-Symmetric Solutions of the Ultra-Relativistic Euler Equations with Entropy-Stable Discontinuous Galerkin Methods", arXiv:2508.21427 (2025)
 
Wasilij Barsukow, Christian Klingenberg, Lisa Lechner, Jan Nordström, Sigrun Ortleb, Hendrik Ranocha: "Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators", arXiv:2507.11068 (2025)
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