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 - JOUR A1 - Egger, Herbert A1 - Philippi, Nora T1 - A hybrid discontinuous Galerkin method for transport equations on networks JF - Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples N2 - We discuss the mathematical modeling and numerical discretization of 5 transport problems on one-dimensional networks. Suitable coupling conditions are derived that guarantee conservation of mass across network junctions and dissipation of a mathematical energy which allows us to prove existence of unique solutions. We then consider the space discretization by a hybrid discontinuous Galerkin method which provides a suitable upwind mechanism to handle the transport prob10 lem and allows to incorporate the coupling conditions in a natural manner. In addition, the method inherits mass conservation and stability of the continuous problem. Order optimal convergence rates are established and illustrated by numerical tests. Y1 - 2020 ER - TY - JOUR A1 - Egger, Herbert A1 - Philippi, Nora T1 - On the transport limit of singularly perturbed convection-diffusion problems on networks N2 - We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order O(sqrt(eps)) in the L1(L2)- norm with diffusion eps -> 0. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers. Y1 - 2020 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 - Egger, Herbert A1 - Kugler, Thomas A1 - Liljegren-Sailer, Björn A1 - Marheineke, Nicole A1 - Mehrmann, Volker T1 - On structure preserving model reduction for damped wave propagation in transport networks N2 - We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi- discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples. KW - partial differential-algebraic equations KW - port-Hamiltonian systems KW - Galerkin projection Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas T1 - Uniform exponential stability of Galerkin approximations for damped wave systems N2 - We consider the numerical approximation of linear damped wave systems by Galerkin approximations in space and appropriate time-stepping schemes. Based on a dissipation estimate for a modified energy, we prove exponential decay of the physical energy on the continuous level provided that the damping is effective everywhere in the domain. The methods of proof allow us to analyze also a class of Galerkin approximations based on a mixed variational formulation of the problem. Uniform exponential stabil- ity can be guaranteed for these approximations under a general compatibility condition on the discretization spaces. As a particular example, we discuss the discretization by mixed finite element methods for which we obtain convergence and uniform error esti- mates under minimal regularity assumptions. We also prove unconditional and uniform exponential stability for the time discretization by certain one-step methods. The valid- ity of the theoretical results as well as the necessity of some of the conditions required for our analysis are demonstrated in numerical tests Y1 - 2016 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas T1 - Damped wave systems on networks: exponential stability and uniform approximations N2 - We consider a damped linear hyperbolic system modelling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a unique steady state is proven in the absence of driving forces. Under mild assumptions on the network topology and the model parameters, we show exponential stability and convergence to equilibrium. This generalizes related results for single pipes and multi-dimensional domains to the network context. Our proof of the exponential stability estimate is based on a variational formulation of the problem, some graph theoretic results, and appropriate energy estimates. The main arguments are rather generic and can be applied also for the analysis of Galerkin approximations. Uniform exponential stability can be guaranteed for the resulting semi-discretizations under mild compatibility conditions on the approximation spaces. A particular realiza- tion by mixed finite elements is discussed and the theoretical results are illustrated by numerical tests in which also bounds for the decay rate are investigated. Y1 - 2016 ER - TY - JOUR A1 - Egger, Herbert A1 - Kugler, Thomas A1 - Strogies, Nikolai T1 - Parameter identification in a semilinear hyperbolic system JF - Inverse Problems N2 - We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and con- traction arguments. We then investigte the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a varia- tional regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized so- lutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings Y1 - 2016 VL - 33 IS - 055022 ER - TY - JOUR A1 - Egger, Herbert T1 - A robust conservative mixed finite element method for compressible flow on pipe networks N2 - We consider the numerical approximation of compressible flow in a pipe net- work. Appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions and to prove global balance laws for the conservation of mass and energy on the whole network. This variational principle, which is the basis of our fur- ther investigations, is amenable to a conforming Galerkin approximation by mixed finite elements. The resulting semi-discrete problems are well-posed and automatically inherit the global conservation laws for mass and energy from the continuous level. We also consider the subsequent discretization in time by a problem adapted implicit time stepping scheme which leads to conservation of mass and a slight dissipation of energy of the full discretization. The well-posedness of the fully discrete scheme is established and a fixed-point iteration is proposed for the solution of the nonlinear systems arising in every single time step. Some computational results are presented for illustration of our theoretical findings and for demon- stration of the robustness and accuracy of the new method Y1 - 2016 ER - TY - THES A1 - Philippi, Nora T1 - Analysis and Numerical Approximation of Transport Equations on Networks N2 - This work deals with the analysis and numerical approximation of transport problems on networks. Appropriate coupling conditions are proposed that allow to establish well-posedness of the continuous problem by semigroup theory. A discontinuous Galerkin method is proposed for the space discretization and its well-posedness and order optimal convergence rates are proven. In addition, the time discretization by the implicit Euler method is investigated. Y1 - 2020 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas A1 - Wollner, Winnifried T1 - Numerical optimal control of instationary gas transport with control and state constraints N2 - We consider the optimal control of a nonlinear hyperbolic system of balance laws on a one-dimensional network which arises in the context of gas transport in pipeline systems. State constraints, which are required for the safe operation of the system, are incorporated by a barrier method. We discuss the well-posedness of the governing system of partial differential-algebraic equations and investigate the existence of minimizers. For the numerical solution, we then consider the approximation of the state equation by mixed finite elements in space and a particular linear implicit time integration scheme that can be interpreted as a discontinuous Galerkin approximation. We establish well- posedness of this discretization scheme and prove the existence of minimizers for the corresponding discretized optimal control problem and discuss its numerical solution by a projected Gauß-Newton method. The efficient realization of the Jacobian and Hessian of the quadratic approximations that have to be minimized in every iteration of the Gauß-Newton method can be obtained via the solution of discretized sensitivity and adjoint equations. These are obtained by formal differentiation and transposition of the Galerkin methods employed for the discretization of the state equations. All approximations obtained after discretization can thus be interpreted as functions on the continuous level and, since the functional analytic setting is not changed by the Galerkin discretization, we observe mesh independence of the resulting fully discrete methods. For illustration of our theoretical results and to demonstrate the efficiency of the proposed method, we present numerical results for two test problems that model typical situations that may arise in the daily operation of gas networks. Y1 - 2017 ER - TY - JOUR A1 - Burlacu, Robert A1 - Egger, Herbert A1 - Groß, Martin A1 - Martin, Alexander A1 - Pfetsch, Marc A1 - Schewe, Lars A1 - Sirvent, Mathias A1 - Skutella, Martin T1 - Maximizing the storage capacity of gas networks: a global MINLP approach N2 - In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (PDEs). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic PDEs and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (MINLP). For the numerical solution of the MINLP, we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems (MIPs). Finally, we present two case studies that illustrate the applicability of our approach. KW - Mixed-Integer Nonlinear Programming KW - Transient Gas Transport Optimization KW - Storage Capacity Maximization KW - Power-to-Gas KW - First-Discretize-Then-Optimize Y1 - 2019 U6 - https://doi.org/10.1007/s11081-018-9414-5 VL - 20 SP - 543 EP - 573 ET - Optimization and Engineering ER - TY - INPR A1 - Egger, Herbert T1 - A class of Galerkin schemes for time-dependent radiative transfer N2 - The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to preserve basic properties like exponential stability and decay to equilibrium also on the discrete level. We present the basic a-priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure like first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis for such applications to the radiative transfer problem under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle, a finite element discretization in space, and the implicit Euler method in time. The performance of the resulting mixed PN-finite element time stepping scheme is demonstrated by computational results. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas T1 - An asymptotic preserving mixed finite element method for wave propagation in pipelines N2 - We consider a parameter dependent family of damped hyperbolic equations with interesting limit behavior: the system approaches steady states exponentially fast and for parameter to zero the solutions converge to that of a parabolic limit problem. We establish sharp estimates and elaborate their dependence on the model parameters. For the numerical approximation we then consider a mixed finite element method in space together with a Runge-Kutta method in time. Due to the variational and dissipative nature of this approximation, the limit behavior of the infinite dimensional level is inherited almost automatically by the discrete problems. The resulting numerical method thus is asymptotic preserving in the parabolic limit and uniformly exponentially stable. These results are further shown to be independent of the discretization parameters. Numerical tests are presented for a simple model problem which illustrate that the derived estimates are sharp in general. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert T1 - Enhancement of flow measurements using fluid dynamic constraints N2 - Novel experimental modalities acquire spatially resolved velocity measurements for steady state and transient flows which are of interest for engineering and biological applications. One of the drawbacks of such high resolution velocity data is their susceptibility to measurement errors. In this paper, we propose a novel filtering strategy that allows enhancement of noisy measurements to obtain reconstruction of smooth divergence free velocity and corresponding pressure fields, which together approximately comply to a prescribed flow model. The main step in our approach consists of the appropriate use of the velocity measurements in the design of a linearized flow model which can be shown to be well-posed and consistent with the true velocity and pressure fields up to measurement and modeling errors. The reconstruction procedure is formulated as a linear quadratic optimal control problem and the resulting filter has analyzable smoothing and approximation properties. We also discuss briefly the discretization of our approach by finite element methods and comment on the efficient solution of the linear optimality system by iterative solvers. The capability of the proposed method to significantly reduce data noise is demonstrated by numerical tests in which we also compare to other methods like smoothing and solenoidal filtering. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Pietschmann, Jan-Frederik A1 - Schlottbom, Matthias T1 - On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms N2 - We consider the identification of nonlinear diffusion coefficients of the form a(t,u) or a(u) in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert T1 - A mixed variational discretization for non-isothermal compressible flow in pipelines N2 - We consider the non-isothermal flow of a compressible fluid through pipes. Starting from the full set of Euler equations, we propose a variational characterization of solutions that encodes the conservation of mass, energy, and entropy in a very direct manner. This variational principle is suitable for a conforming Galerkin approximation in space which automatically inherits the basic physical conservation laws. Three different spaces are used for approximation of density, mass flux, and temperature, and we consider a mixed finite element method as one possible choice of suitable approximation spaces. We also investigate the subsequent discretization in time by a problem adapted implicit time stepping scheme for which exact conservation of mass as well as a slight dissipation of energy and increase of entropy are proven which are due to the numerical dissipation of the implicit time discretization. The main arguments of our analysis are rather general and allow us to extend the approach with minor modification to more general boundary conditions and flow models taking into account friction, viscosity, heat conduction, and heat exchange with the surrounding medium. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Radu, Bogdan T1 - Super-convergence and post-processing for mixed finite element approximations of the wave equation N2 - We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Böttcher, Anke T1 - Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries N2 - We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al. Y1 - 2017 ER -