arXiv subjects

Jan Giesselmann

Publications and source records attributed to Jan Giesselmann.

At least 19 recordsLinked to original sources

arXiv2026

Justification and structure- and asymptotic-preserving discretizations of a hyperbolized Cahn-Hilliard equation

We study a hyperbolic approximation ("hyperbolization") of the Cahn-Hilliard (CH) equation, originally proposed by Dhaouadi, Dumbser, and Gavrilyuk (2025, DOI: 10.1098/rspa.2024.0606) and study its convergence towards the CH model in a relaxation limit both via formal asymptotic expansions and, for a slightly modified approximation, via the relative energy framework. Moreover, we develop energy-stable semidiscretizations of the CH equation and of this hyperbolization using upwind summation-by-parts operators in space. Subsequently, we combine them with (additive) implicit-explicit (IMEX) Runge-Kutta methods based on a convex-concave splitting. We show that the resulting method is asymptotic preserving, i.e., it converges in the limit of the relaxation parameter to a stable discretization of the original CH equation. The choice of the necessary parameters is guided by the a priori error estimate based on the relative energy framework.

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arXiv2026

Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions

We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency.

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arXiv2026

A Posteriori Error Analysis of Runge-Kutta Discontinuous Galerkin Schemes with SIAC Post-Processing for Nonlinear Convection-Diffusion Systems

We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.

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arXiv2026

Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions

We consider a structure-preserving finite-volume scheme for the Euler-Korteweg (EK) and Navier-Stokes-Korteweg (NSK) equations. We prove that its numerical solutions converge to energy-variational solutions of EK or NSK under mesh refinement. Energy-variational solutions constitute a novel solution concept that has recently been introduced for hyperbolic conservation laws, including the EK system, and which we extend to the NSK model. Our proof is based on establishing uniform estimates following from the properties of the structure-preserving scheme, and using the stability of the energy-variational formulation under weak convergence in the natural energy spaces.

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arXiv2026

On the consistency of the Domain of Dependence cut cell stabilization

So called cartesian cut cell meshes provide efficient ways to generate meshes but do require tailored numerical methods to not suffer from stabilization issues, especially in the hyperbolic regime where the application of explicit time stepping schemes is common. In this scenario, due to potentially arbitrarily small cut cells, an infeasible restriction is imposed on the time step size. The Domain of Dependence (DoD) stabilization allows for a time step size based on the underlying Cartesian mesh. Being an extension of a discontinuous Galerkin (DG) method, one would expect similar accuracy properties as in the pure DG case. While numerical results do support this expectation, on the analytical level this has only been investigated thoroughly for $k=0$. Error analysis typically hinges on a consistency result. In this contribution we prove such a result for the DoD stabilization given an arbitrary polynomial degree and an exact solution of sufficient regularity. This in turn could open the way towards a more refined analysis of the method even in the high-order case.

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arXiv2026

Convergence Analysis of a Fully Discrete Observer For Data Assimilation of the Barotropic Euler Equations

We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations.

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arXiv2026

Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System

The Navier-Stokes-Cahn-Hilliard (NSCH) system governs the diffuse-interface dynamics of two incompressible and immiscible fluids. We consider a relaxation approximation of the NSCH system that is composed by a system of first-order hyperbolic balance laws and second-order elliptic operators. We prove first that the solutions of an initial boundary value problem for the approximation recover the limiting NSCH system for vanishing relaxation parameters. To cope with the singular limit we exploit the fact that the approximate solutions dissipate an almost quadratic energy, and employ the relative entropy-framework. In the second part of the work we provide numerical evidence for the analytical results, even in flow regimes not covered by the assumptions needed for the theoretical results. Using a novel marker-and-cell conservative finite-difference approach for both the approximation and the limit system, we are able to compute physically relevant interfacial flow problems including Ostwald ripening and high-velocity flow.

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arXiv2026

A Structure Preserving Finite Volume Scheme for the Navier-Stokes-Korteweg Equations

We present a semi-discrete finite volume scheme for the local NavierStokes-Korteweg and Euler-Korteweg systems. Our scheme is applicable for equidistant Cartesian meshes in one and two space dimensions. In contrast to other works, which employ, for example, hyperbolic approximations of the equations or auxiliary-variable approaches leading to extended systems, our scheme operates directly on the original system. We prove that it conserves mass and momentum and is energy stable. Numerical experiments complement our theoretical findings, showing that the scheme is convergent of order one if employed with explicit or implicit time discretisation.

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arXiv2025

A posteriori analysis for nonlinear convection-diffusion systems

This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

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arXiv2025

A posteriori existence of strong solutions to the Navier-Stokes equations in 3D

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, Ser\"egin 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.

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arXiv2025

A posteriori existence for the Keller-Segel model via a finite volume - finite element scheme

We derive two forms of conditional a posteriori error estimates for a finite volume scheme approximating the parabolic-elliptic Keller-Segel system. The estimates control the error in the $L^\infty(0,T, L^2(\Omega))$- and $L^2(0,T;H^1(\Omega))$-norm and exhibit linear convergence in the mesh size, as observed in numerical experiments. Crucially, we show that, as long as the condition of the error estimate is satisfied, a weak solution exists. This means, as long as the numerical solution has good properties, we can rigorously infer existence of an exact solution.

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arXiv2025

Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions

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.

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arXiv2025

A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport

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 a 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 assumptions that coincide with those made in weak uniqueness results for this system. Additionally, when electrical forces are present, we assume that the solvent concentration is uniformly bounded from below. 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 that approximates the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error.

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arXiv2025

Extending relative entropy for Korteweg-Type models with non-monotone pressure:large friction limit and weak-strong uniqueness

In this paper we study weak-strong uniqueness and singular relaxation limits for the Euler--Korteweg and Navier--Stokes--Korteweg systems with non monotone pressure. Both weak-strong uniqueness and the relaxation limit are investigated using relative entropy technique. We make use of the enlarged formulation of the model in terms of the drift velocity introduced in [6], generalizing in this way results proved in [17] for the Euler-Korteweg model, by allowing more general capillarity functions, and the result contained in [8] for the monotone pressure case.

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arXiv2024

Error analysis of a first-order DoD cut cell method for 2D unsteady advection

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 $L^\infty$-in-time $L^2$-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results.

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arXiv2024

Identification of minimal number of measurements allowing synchronization of a nodal observer for the wave equation

We study a state estimation problem for a $2\times 2$ 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.

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arXiv2024

Energy-consistent Petrov-Galerkin time discretization of port-Hamiltonian systems

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.

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arXiv2024

Robust a posteriori error control for the Allen-Cahn equation with variable mobility

In this work, we derive a $\gamma$-robust a posteriori error estimator for finite element approximations of the Allen-Cahn equation with variable non-degenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.

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