Technische Universität Darmstadt
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We consider a compressible mixture model for binary fluids with two velocities in one space dimension in the low-Mach, high-friction regime. Building on the framework in [Egger, Giesselmann 2023], we establish stability with respect to the data. Furthermore, we analyse a high-order, energy-consistent numerical discretisation. In particular, we prove asymptotic-preserving error estimates for a first-order time discretisation combined with a spatial discretisation of arbitrary order, under suitable regularity assumptions. The scheme combines a modified continuous Petrov-Galerkin discretisation in time, introduced in [Giesselmann, Karsai, Tscherpel 2025] for a general class of nonlinear port-Hamiltonian systems, with a mixed finite element discretisation in space, both of arbitrary order. By developing and using an approximation operator adapted to the nonlinear structure of the model, we obtain optimal-order error estimates in the spatial discretisation parameter. Our results also apply to a one-component fluid with friction, extending existing schemes and error estimates to higher-order spatial discretisations.
We derive an adjoint gradient representation for the objective functional of an optimal control problem of the unique entropy solution to the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation or balance laws.
The GRP is an initial value problem where the initial state is piecewise C^1 with exactly one discontinuity. In a recent work, differentiability properties of the solution operator of the GRP were derived. In particular, the differentiability of a class of tracking-type functionals was shown. We build on these results to derive an adjoint gradient representation for such functionals.
The adjoint problem is a system of linear transport equations with a discontinuous coefficient and discontinuous terminal state. We prove the existence and uniqueness of a piecewise Lipschitz continuous solution to the adjoint problem which satisfies suitable interior boundary conditions along all shock curves. The adjoint state generally contains infinitely many discontinuities but its total variation remains uniformly bounded in time.
We develop efficient algorithms for a fundamental network design problem arising in potential-based flow models, which are central to many energy transport networks (e.g., hydrogen and electricity). In contrast to classical network flow problems, the nonlinearities inherent in potential-based networks introduce significant new challenges. We address these challenges through intricate reductions to classical combinatorial optimization problems, such as (constrained) shortest path problems, enabling the application of well-established algorithmic techniques to compute exact and approximate solutions efficiently. Finally, we complement these algorithmic results with matching complexity results concerning the hardness and non-approximability of the considered problem variants.
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.
This paper deals with the mixing of gases in stationary networks. We first derive a model and pressure law for mixing, for which there is empirical evidence for its accuracy. The model is based on the change of the speed of sound in gas mixtures.
We then consider stationary gas networks. The existence result for solutions of single gas flow on networks is extended to mixtures. Further, we establish an easy to check criterion for uniqueness of solutions on the network. This results in a uniqueness proof of solutions on all networks with a mixture of natural gas and low hydrogen percentages. We then develop a solution algorithm that alternates between the solution of a single gas problem and update of the mixture ratios. In a computational study, different model variants and their impact on performance are compared.
Moreover, the increased complexity of solving stationary gas transport problems with mixing is evaluated.
This document aims to provide a concise and clear introduction to the topic of gas flow modeling. We present several models for gas flow, organized into hierarchies based on complexity. We discuss in detail the modeling of individual components such as valves and compressors. Network model classes based on purely algebraic relations and energy-based port-Hamiltonian models are included, along with a brief overview of basic numerical methods for hyperbolic balance laws and port-Hamiltonian systems.
We do not claim completeness and refer in many places to the existing literature.
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.
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.
We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels.
It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones.
We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls.
We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable (C^1) states left and right of it. The control consists of the C^1 parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon T>0 to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small.
Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to C^0. In physical coordinates, this implies that the shock curves in C^1 and the states between the shocks in the topology of C^0 depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals.
