Research & Publications
Modeling and Analysis of Nonlinear Closures in Hierarchical Moment Equations Using Discontinuous Galerkin Methods
Jan Habscheid (Supervisor: Eda Yilmaz & Manuel Torrilhon) Master Thesis, RWTH Aachen University (03/2026)
To be published!
Rarefied Gases Moment Methods Extended Gramian Closure Discontinuous Galerkin
▶ Show Abstract
The classical continuum mechanical equations of Navier-Stokes-Fourier (NSF) accurately describe gases with a high collision frequency, characterized by a small Knudsen number (Kn). However, the NSF equations fail for gases with a high Knudsen number, which corresponds to a low collision frequency. Solving Boltzmann’s equation directly is a highly accurate approach, but it is computationally very demanding due to its high dimensionality. Another method is to approximate the Boltzmann equation by directly solving for macroscopic properties, such as density, velocity, and temperature, which can be described by moments. Therefore, the so-called moment system is solved, consisting of an infinitely large system of hierarchically coupled partial differential equations (PDEs), with the moments as unknowns. Solving this system is impossible, as each moment depends on the next higher moment, which in turn depends on an even higher moment. Therefore, the moment system is truncated after \(M\) equations, requiring a moment closure for the \((M+1)\)-th moment. This thesis discusses the extended Gramian closure, a recently developed moment closure based on orthogonal polynomials. The extended Gramian closure has attractive structure-preserving properties. It overcomes shortcomings of classical closures, such as the lack of global hyperbolicity in Grad’s closure and the computational effort required by the maximum entropy closure. This work introduces the theoretical framework of the extended Gramian closure and suggests an updated closure under the condition that the equations are truncated where \(M\) is odd. Rigorous mathematical proofs for the structure-preserving properties of this updated closure are discussed. While the closure is constructed in one dimension, a projection method for applying it to a three-dimensional velocity space, based on matrix rotations, is introduced. Further, the extended Gramian closure is numerically approximated with the discontinuous Galerkin method, and its accuracy is analyzed and compared with that of the Gramian closure and Grad’s closure. To this end, various benchmark problems, including the shock tube, shock structure, Landau damping, and two-stream instability, are analyzed.
A finite element solver for a thermodynamically consistent electrolyte model
Jan Habscheid, Satyvir Singh, Lambert Theisen, Stefanie Braun, Manuel Torrilhon
Computer Physics Communications (02/2026)
Electrochemistry Electrical Double Layer Thermodynamics Electrolyte Models FEniCS Finite Element Method
▶ Show Abstract
In this study, we present a finite element solver for a thermodynamically consistent electrolyte model that accurately captures multicomponent ionic transport by incorporating key physical phenomena such as steric effects, solvation, and pressure coupling. The model is rooted in the principles of non-equilibrium thermodynamics and strictly enforces mass conservation, charge neutrality, and entropy production. It extends beyond classical frameworks like the Nernst–Planck system by employing modified partial mass balances, the electrostatic Poisson equation, and a momentum balance expressed in terms of electrostatic potential, atomic fractions, and pressure, thereby enhancing numerical stability and physical consistency. Implemented using the FEniCSx platform, the solver efficiently handles one- and two-dimensional problems with varied boundary conditions and demonstrates excellent convergence behavior and robustness. Validation against benchmark problems confirms its improved physical fidelity, particularly in regimes characterized by high ionic concentrations and strong electrochemical gradients. Simulation results reveal critical electrolyte phenomena, including electric double layer formation, rectification behavior, and the effects of solvation number, Debye length, and compressibility. The solver’s modular variational formulation facilitates its extension to complex electrochemical systems involving multiple ionic species with asymmetric valences. We publicly provide the documented and validated solver framework.
Numerical Treatment of a Thermodynamically Consistent Electrolyte Model
Jan Habscheid (Supervisor: Lambert Theisen & Manuel Torrilhon) Bachelor Thesis, RWTH Aachen University (09/2024)
Electrochemistry Electrical Double Layer Thermodynamics Electrolyte Models FEniCSx Finite Element Method
▶ Show Abstract
Batteries play a crucial role in the energy transition. The production of green energy depends on external factors. Storing energy in batteries is necessary to access green energy at any time. Better optimized batteries are essential for the future. Lifetime, loading time, and energy loss are just some aspects that must be improved to prepare for a greener future. Numerical simulations are crucial to understanding and optimizing batteries’ behavior. Those simulations enable researchers to test many different materials without considerable additional expenses to, for example, find the best combination of anions and cations. The classical Nernst-Planck model for the ion transport in an electrolyte fails to predict the correct concentration in the boundaries of the electrolyte. This work will present and analyze a thermodynamically consistent electrolyte model with dimensionless units under isothermal conditions. A simplified version of the system for the one-dimensional equilibrium of an ideal mixture and the incompressible limit will be considered. The numerical implementation of the model with the open-source software FEniCSx will be discussed. Furthermore, the influence of different boundary conditions, material parameters, solvation, and compressibility on the electric potential, pressure, and ion concentration will be investigated, and the model will be compared with the classical Nernst-Planck model. Examples of the double layer capacity and electrolytic diode will be considered.