## C04

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#### Keywords

- First-Discretize-Then-Optimize (1)
- Galerkin projection (1)
- Mixed-Integer Nonlinear Programming (1)
- Power-to-Gas (1)
- Storage Capacity Maximization (1)
- Transient Gas Transport Optimization (1)
- asymptotic limits (1)
- gas transport on networks (1)
- hyperbolic balance laws (1)
- partial differential-algebraic equations (1)

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.

Stability and asymptotic analysis for instationary gas transport via relative energy estimates
(2020)

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.

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.

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.

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.

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.

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.

Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries
(2017)

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.

Super-convergence and post-processing for mixed finite element approximations of the wave equation
(2017)

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.

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.