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 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
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.
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.
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.
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.
Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization
(2024)
We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem.
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.
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
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.