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Thu, 26 Jul 2018 13:32:30 +0200Thu, 26 Jul 2018 13:32:30 +0200Mixed-Integer Programming Techniques for the Connected Max-k-Cut Problem
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/239
We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved.Christopher Hojny; Imke Joormann; Hendrik Lüthen; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/239Thu, 26 Jul 2018 13:32:30 +0200Strictly and Γ-Robust Counterparts of Electricity Market Models: Perfect Competition and Nash-Cournot Equilibria
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/238
This paper mainly studies two topics: linear complementarity problems (LCPs) for modeling electricity market equilibria and optimization under uncertainty. While there have been quite some attempts to deal with uncertain LCPs in a stochastic - i.e., distributional - sense, robust LCPs have only gained attention very recently. In this paper, we consider both perfectly competitive and Nash-Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition this particularly means that the two classical welfare theorems also hold in both considered robust cases. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of Γ-robustifications of Nash-Cournot models. Thus, an analogue of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.Anja Kramer; Vanessa Krebs; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/238Fri, 13 Jul 2018 16:21:28 +0200The Impact of Potential-Based Physics Models on Pricing in Energy Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/237
Pricing of access to energy networks is an important issue in liberalized energy sectors because of the natural monopoly character of the underlying transport infrastructures. We introduce a general pricing framework for potential-based energy flows in arbitrarily structured transport networks. In different specifications of our general pricing model we discuss first- and second-best pricing results and compare different pricing outcomes of potential-free and potential-based energy flow models. Our results show that considering nonlinear laws of physics leads to significantly different pricing results on networks and that these differences can only be seen in sufficiently complex, e.g., cyclic, networks as they can be found in real-world situations.Lars Schewe; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/237Wed, 06 Jun 2018 12:59:51 +0200Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/236
We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.Martin Gugat; Lionel Rosier; Vincent Perrollazarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/236Tue, 05 Jun 2018 07:37:11 +0200The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/235
It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function - in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute global optimal solutions in finite time and we also provide a worst-case iteration bound. However, first numerical experiences reveal that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.Martin Schmidt; Mathias Sirvent; Winnifried Wollnerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/235Thu, 26 Apr 2018 13:07:18 +0200Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/234
We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal design of zonal pricing electricity markets. This together with nonlinearities due to economic modeling yields extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof, whereas the second is a tailored generalized Benders decomposition. We prove for both methods that they yield global optimal solutions. Finally, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation.Thomas Kleinert; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/234Wed, 07 Feb 2018 14:27:16 +0100Robust Optimal Discrete Arc Sizing for Tree-Shaped Potential Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/233
We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing's optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern Germany.Martin Robinius; Lars Schewe; Martin Schmidt; Detlef Stolten; Johannes Thürauf; Lara Welderpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/233Thu, 01 Feb 2018 15:28:47 +0100A Bayesian approach for parameter identification in gas networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/231
The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the
friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.Soheil Hajian; Michael Hintermüller; Claudia Schillings; Nikolai Strogiespreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/231Mon, 29 Jan 2018 14:11:31 +0100Optimal Model Switching for Gas Flow in Pipe Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/232
We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.Fabian Rüffler; Volker Mehrmann; Falk Hantearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/232Mon, 29 Jan 2018 14:11:31 +0100DYNAMIC BOUNDARY CONTROL GAMES WITH NETWORKS OF STRINGS
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/229
Consider a star-shaped network
of strings. Each string is governed by the wave equation.
At each boundary node of the network there is
a player that performs Dirichlet boundary control action
and in this way influences the system state.
At the central node, the states are coupled
by algebraic conditions in such a way that the energy is conserved.
We consider the corresponding antagonistic game
where each player minimizes a certain quadratic objective function
that is given by the sum of a control cost and
a tracking term for the final state.
We prove that under suitable assumptions
a unique Nash equilibrium exists
and give an explicit representation
of the equilibrium strategies.Martin Gugat; Sonja Steffensenarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/229Fri, 19 Jan 2018 11:16:58 +0100Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/227
Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.M. Hassan Farshbaf-Shaker; Rene Henrion; Dietmar Hömbergarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/227Tue, 09 Jan 2018 11:32:43 +0100Subdifferential characterization of probability functions under Gaussian distribution
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/226
Probability functions figure prominently in optimization problems
of engineering. They may be nonsmooth even if all input data are smooth.
This fact motivates the consideration of subdifferentials for such typically just
continuous functions. The aim of this paper is to provide subdifferential formulae
of such functions in the case of Gaussian distributions for possibly
infinite-dimensional decision variables and nonsmooth (locally Lipschitzian)
input data. These formulae are based on the spheric-radial decomposition of
Gaussian random vectors on the one hand and on a cone of directions of
moderate growth on the other. By successively adding additional hypotheses,
conditions are satisfied under which the probability function is locally Lipschitzian
or even differentiable.Abderrahim Hantoute; Rene Henrion; Pedro Perez-Arosarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/226Tue, 09 Jan 2018 11:32:42 +0100On the identification of the friction coefficient in a semilinear system for gas transport through a network
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/225
An identification problem for the friction parameter in a semilinear system of balance laws, describing the transport of gas through a passive network of pipelines, is considered. The existence of broad solutions to the state system is proven and sensitivity results for the corresponding solution operator are obtained. The existence of solutions to the output least squares formulation of the identification problem, based on noisy measurements over time at fixed spatial positions is established. Finally, numerical experiments validate the theoretical findings.Michael Hintermüller; Nikolai Strogiespreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/225Tue, 28 Nov 2017 10:57:52 +0100Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/224
The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.Volker Mehrmann; Martin Schmidt; Jeroen J. Stolwijkpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/224Mon, 27 Nov 2017 12:56:12 +0100Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/223
In this paper we extend the implicit-explicit (IMEX) methods of Peer type
recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203–215, 2017]
to a broader class of two-step methods that allow the construction of super-
convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of ex-
plicit methods to efficiently solve systems of ordinary differential equations with
both stiff and non-stiff parts included in the source term. To construct super-
convergent IMEX-Peer methods with favourable stability properties, we derive
necessary and sufficient conditions on the coefficient matrices and apply an
extrapolation approach based on already computed stage values. Optimised
super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are
given as result of a search algorithm carefully designed to balance the size of
the stability regions and the extrapolation errors. Numerical experiments and
a comparison to other IMEX-Peer methods are included.Moritz Schneider; Jens Lang; Willem Hundsdorferpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/223Sun, 26 Nov 2017 20:46:40 +0100Uniqueness of Market Equilibria on Networks with Transport Costs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/222
We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated.Vanessa Krebs; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/222Wed, 22 Nov 2017 11:30:24 +0100Maximizing the storage capacity of gas networks: a global MINLP approach
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/221
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.Robert Burlacu; Herbert Egger; Martin Groß; Alexander Martin; Marc Pfetsch; Lars Schewe; Mathias Sirvent; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/221Tue, 21 Nov 2017 15:12:15 +0100Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/220
This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a first-best benchmark. We provide conditions under which the solution of the simultaneous game is also the solution of the corresponding MNCP. However, equilibria cannot be determined by the MNCP as the transmission system operator's (TSO’s) first-order conditions are insufficient, which goes back to nonconvexities of the gas flow model. This also implies that the welfare maximization problem may have multiple solutions that sometimes do not even coincide with any of the market equilibria. Our analysis shows that, even in the absence of strategic firms, market interaction fails to implement desirable outcomes from a welfare perspective due to the TSO’s incentive structure. We conclude that the technical environment calls for a market design that commits the TSO to a welfare objective through regulation and propose a design where the market solution corresponds to a welfare maximum and vice versa.Veronika Grimm; Julia Grübel; Lars Schewe; Martin Schmidt; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/220Tue, 21 Nov 2017 13:42:30 +0100Optimal Boundary Control of Hyperbolic Balance Laws with State Constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/219
In this paper we analyze the optimal control of initial-boundary value problems
for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Hereby,
we suppose that the initial and the boundary data switch between different C¹-functions at certain
switching points, where the C¹ -functions and the switching points are considered as the control. For
a class of cost functionals, we prove first order necessary optimality conditions for the corresponding
optimal control problem with state contraints. Furthermore, we use a Moreau-Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality
conditions for the regularized problems and finally prove convergence to the solution of the optimal
control problem with state constraints.Johann Michael Schmitt; Stefan Ulbrichpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/219Tue, 21 Nov 2017 13:19:55 +0100Improving branching for disjunctive polyhedral models using approximate convex decompositions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/218
Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior.Benjamin Hiller; Tom Waltherreporthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/218Tue, 21 Nov 2017 13:08:05 +0100Numerical optimal control of instationary gas transport with control and state constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/214
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.Herbert Egger; Thomas Kugler; Winnifried Wollnerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/214Mon, 20 Nov 2017 19:59:45 +0100Joint model of probabilistic/robust (probust) constraints with application to gas network optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/215
Optimization tasks under uncertain conditions abound in many
real-life applications. Whereas solution approaches for probabilistic constraints
are often developed in case the uncertainties can be assumed to follow a
certain probability distribution, robust approaches are usually used in case
solutions are sought that are feasible for all realizations of uncertainties within
some pre-defined uncertainty set. As many applications contain different types
of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints as its appears in
optimization problems under uncertainty. Focussing on complex uncertain gas
network optimization problems, we show the relevance of this class of problems
for the task of maximizing free booked capacities in an algebraic model for a
stationary gas network. We furthermore present approaches for their solution.
Finally, we study the problem of controlling a transient system that is governed
by the wave equation. The task consists in determining controls such that a
certain robustness measure remains below some given upper bound, with high
probability.Dennis Adelhütte; Denis Aßmann; Tatiana Gonzàlez Grandòn; Martin Gugat; Holger Heitsch; Frauke Liers; René Henrion; Sabrina Nitsche; Rüdiger Schultz; Michael Stingl; David Wintergerstpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/215Mon, 20 Nov 2017 19:59:45 +0100Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/213
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.Herbert Egger; Anke Böttcherpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/213Mon, 20 Nov 2017 19:03:43 +0100Super-convergence and post-processing for mixed finite element approximations of the wave equation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/212
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.Herbert Egger; Bogdan Radupreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/212Mon, 20 Nov 2017 19:03:42 +0100A mixed variational discretization for non-isothermal compressible flow in pipelines
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/211
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.Herbert Eggerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/211Mon, 20 Nov 2017 18:58:29 +0100