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 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.
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 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 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.
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.
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
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
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.
