TY - INPR A1 - Alldredge, Graham A1 - Frank, Martin A1 - Giesselmann, Jan T1 - On the convergence of the regularized entropy-based moment method for kinetic equations N2 - The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal. Y1 - 2023 U6 - https://doi.org/https://doi.org/10.5802/smai-jcm.93 VL - 9 ER - TY - INPR A1 - Gugat, Martin A1 - Giesselmann, Jan T1 - An Observer for pipeline flow with hydrogen blending in gas networks: exponential synchronization N2 - We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the L2-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights. Y1 - 2023 ER - TY - INPR A1 - Egger, Herbert A1 - Giesselmann, Jan T1 - Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization N2 - We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem. KW - gas transport KW - doubly nonlinear parabolic problems KW - relative entropy estimates KW - exponential stability KW - structure preserving discretization Y1 - 2023 ER - TY - INPR A1 - Giesselmann, Jan A1 - Krupa, Sam T1 - Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems N2 - In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments. KW - Conservation laws KW - entropy conditions KW - entropy solutions KW - shocks, KW - a posteriori error estimates Y1 - 2023 ER - TY - INPR A1 - Giesselmann, Jan A1 - Kolbe, Niklas T1 - A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model N2 - We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ. KW - Keller-Segel KW - chemotaxis; KW - nonlinear diffusion KW - finite volume scheme KW - a posteriori error analysis Y1 - 2023 ER - TY - INPR A1 - Giesselmann, Jan A1 - Kwon, Kiwoong T1 - A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model N2 - We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results. KW - Keller-Segel KW - chemotaxis KW - nonlinear diffusion KW - discontinuous Galerkin scheme KW - a posteriori error analysis Y1 - 2023 ER - TY - INPR A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Breiten, Tobias A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Hiller, Benjamin A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview (Extended English Version) N2 - With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. Y1 - 2023 ER - TY - JOUR A1 - Giesselmann, Jan A1 - Gugat, Martin A1 - Kunkel, Teresa T1 - Observer-based data assimilation for barotropic gas transport using distributed measurements N2 - We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks. Y1 - 2023 U6 - https://doi.org/10.4310/CMS.240918203214 VL - 22 SP - 2271 EP - 2309 PB - Communications in Mathematical Sciences ER - TY - JOUR A1 - Martin, Gugat A1 - Giesselmann, Jan A1 - Kunkel, Teresa T1 - Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks N2 - The flow of gas through networks of pipes can be modeled by coupling hyperbolic systems of partial differential equations that describe the flow through the pipes that form the edges of the graph of the network by algebraic node conditions that model the flow through the vertices of the graph. In the network, measurements of the state are available at certain points in space.Based upon these nodal observations, the complete system state can be approximated using an observer system. In this paper we present a nodal observer, and prove that the state of the observer system converges to the original state exponentially fast. Numerical experiments confirm the theoretical findings. Y1 - 2021 U6 - https://doi.org/10.1093/imamci/dnab029 CY - IMA Journal of Mathematical Control and Information ER - TY - INPR A1 - Giesselmann, Jan A1 - Egger, Herbert T1 - Stability and asymptotic analysis for instationary gas transport via relative energy estimates N2 - We consider the transport of gas in long pipes and pipeline networks for which the dynamics are dominated by friction at the pipe walls. The governing equations can be formulated as an abstract dissipative Hamiltonian system which allows us to derive perturbation bounds by means of relative energy estimates. As particular consequences, we obtain stability with respect to initial conditions and model parameters and quantitative estimates in the high friction limit. Our results are established in detail for the flow in a single pipe and through the energy-based modelling they naturally generalize also to pipe networks. KW - gas transport on networks KW - asymptotic limits KW - hyperbolic balance laws KW - relative energy estimates KW - singular perturbations Y1 - 2020 ER - TY - INPR A1 - Sarna, Neeraj A1 - Giesselmann, Jan A1 - Benner, Peter T1 - Data-Driven Snapshot Calibration via Monotonic Feature Matching N2 - Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast $m$-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the $m$-width does not only depend on $m$ but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method. Y1 - 2020 ER - TY - JOUR A1 - Gugat, Martin A1 - Giesselmann, Jan T1 - Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks N2 - The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the $H^1$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon observability inequalities for the $L^2$ and the $H^1$-norm. Y1 - 2020 U6 - https://doi.org/10.1051/cocv/2021061 CY - ESAIM:COCV ER - TY - INPR A1 - Egger, Herbert A1 - Giesselmann, Jan A1 - Philippi, Nora A1 - Kunkel, Teresa T1 - An asymptotic-preserving discretization scheme for gas transport in pipe networks N2 - We consider the simulation of barotropic flow of gas in long pipes and pipe networks. Based on a Hamiltonian reformulation of the governing system, a fully discrete approximation scheme is proposed using mixed finite elements in space and an implicit Euler method in time. Assuming the existence of a smooth subsonic solution bounded away from vacuum, a full convergence analysis is presented based on relative energy estimates. Particular attention is paid to establishing error bounds that are uniform in the friction parameter. As a consequence, the method and results also cover the parabolic problem arising in the asymptotic large friction limit. The error estimates are derived in detail for a single pipe, but using appropriate coupling conditions and the particular structure of the problem and its discretization, the main results directly generalize to pipe networks. Numerical tests are presented for illustration. Y1 - 2021 ER - TY - INPR A1 - Giesselmann, Jan A1 - Kunkel, Teresa T1 - Identification of minimal number of measurements allowing synchronization of a nodal observer for the wave equation N2 - We study a state estimation problem for a 2x2 linear hyperbolic system on networks with eigenvalues with opposite signs. The system can be seen as a simplified model for gas flow through gas networks. For this system we construct an observer system based on nodal measurements and investigate the convergence of the state of the observer system towards the original system state. We assume that measurements are available at the boundary nodes of the network and identify the minimal number of additional measurements in the network that are needed to guarantee synchronization of the observer state towards the original system state. It turns out that for tree-shaped networks boundary measurements suffice to guarantee exponential synchronization, while for networks that contain cycles synchronization can be guaranteed if and only if at least one measurement point is added in each cycle. This is shown for a system without source term and for a system with linear friction term. Y1 - 2024 ER - TY - JOUR A1 - Giesselmann, Jan A1 - Karsai, Attila A1 - Tscherpel, Tabea T1 - Energy-consistent Petrov-Galerkin time discretization of port-Hamiltonian systems N2 - For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a Hamiltonian function, which represents an energy in the system and is conserved or dissipated along solutions. For infinite-dimensional systems this structure is preserved under suitable Galerkin discretization in space. The numerical scheme is energy-consistent in the sense that the Hamiltonian of the approximate solutions at time grid points behaves accordingly. This structure preservation property is achieved by specific design of a continuous Petrov-Galerkin (cPG) method in time. It coincides with standard cPG methods in special cases, in which the latter are energy-consistent. Examples of port-Hamiltonian ODEs and PDEs are presented to visualize the framework. In numerical experiments the energy consistency is verified and the convergence behavior is investigated. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2404.12480 ER - TY - INPR A1 - Giesselmann, Jan A1 - Kwon, Kiwoong A1 - Lee, Min-Gi T1 - Relative entropy technique in terms of position and momentum and its application to Euler-Poisson system N2 - This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of N particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension d≥2, the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing L2 state differences before the formation of δ-shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as δ-shocks, vacuums, and shock discontinuities in the weak solutions. Y1 - 2024 ER - TY - INPR A1 - Birke, Gunnar A1 - Engwer, Christian A1 - Giesselmann, Jan A1 - May, Sandra T1 - Error analysis of a first-order DoD cut cell method for 2D unsteady advection N2 - In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the L∞-in-time L2-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results. KW - cut cell KW - discontinuous Galerkin method KW - DoD Stabilization KW - a priori error estimate KW - unsteady advection Y1 - 2024 ER - TY - INPR A1 - Berrens, Arne A1 - Giesselmann, Jan T1 - A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport N2 - We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end we derive an abstract stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumption that coincide with those made in weak uniqueness results for this system. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme approximating the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error. KW - cross-diffusion KW - ion transport KW - finite-volume approximation KW - a posteriori error estimates KW - diffusion equation Y1 - 2025 ER - TY - INPR A1 - Giesselmann, Jan A1 - Ranocha, Hendrik T1 - Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions N2 - We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings. Y1 - 2025 ER - TY - INPR A1 - Brunk, Aaron A1 - Giesselmann, Jan A1 - Tscherpel, Tabea T1 - A posteriori existence of strong solutions to the Navier-Stokes equations in 3D N2 - Global existence of strong solutions to the three-dimensional incompressible Navier--Stokes equations remains an open problem. A posteriori existence results offer a way to rigorously verify the existence of strong solutions by ruling out blow-up on a certain time interval, using only numerical solutions. In this work we present such a result for the Navier--Stokes equations subject to periodic boundary conditions, which makes use of a version of the celebrated blow-up criterion in the critical space $L^\infty(L^3)$ by Iskauriaza, Serëgin and Shverak (2003). Our approach is based on a conditional stability estimate in $L^2$ and $L^3$. The a posteriori criterion that, if satisfied, verifies existence of strong solutions, involves only negative Sobolev norms of the residual. We apply the criterion to numerical approximations computed with mixed finite elements and an implicit Euler time discretisation. A posteriori error estimates allow us to derive a fully computable criterion without imposing any extra assumptions on the solution. While limited to short time intervals, with sufficient computational resources in principle the criterion might allow for a verification over longer time intervals than what can be achieved by theoretical means. KW - Navier-Stokes KW - blow-up KW - a posteriori estimates KW - critical space KW - reconstruction Y1 - ER - TY - INPR A1 - Boerner, Pascal A1 - Giesselmann, Jan A1 - Kumar, Varun M. A1 - Pfetsch, Marc E. A1 - Thiele, Michael A1 - Tscherpel, Tabea T1 - Gas Mixtures on Networks: Modeling, Simulation and Optimization N2 - This chapter addresses mathematical models for isothermal mixtures of hydrogen and natural gas, motivated by the need for reliable simulation tools in future low-carbon energy systems. We analyze several classes of mixture models and investigate their convergence properties in the regime of strong interaction between constituents, covering stationary and instationary single-pipe settings as well as network flows. Since mixture models critically depend on the choice of pressure law, we compare the industry-standard GERG equation of state with simplified alternatives that preserve convex energies and reduce computational costs. For network applications, we discuss consistent coupling conditions across model classes, explore optimization of steady flows using the algebraic Weymouth formulation, and provide numerical evidence for its applicability in relevant operating regimes. The study reveals when simplified models are justified and outlines key open challenges for the modeling of gas mixtures. Y1 - ER -