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

Abstract

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.

An asymptotic-preserving discretization scheme for gas transport in pipe networks

“…As predicted by our theoretical results, we observe linear convergence for both density and mass flux uniform for all parameters ε and, in particular, also in the parabolic limit ε = 0. Further note that the errors, and actually also the solutions, are very similar for all values of ε ≤ 0.01, which clearly indicates the asymptotic convergence of solutions with ε ց 0, which was proven in [4] for the continuous problem. 6.2.…”

“…with dissipation functional D(ρ, w) = e∈E ℓ e 0 a e γ e ρ e |w e | 3 ≥ 0, which again follows directly from the particular form of the variational formulation. Based on relative energy estimates, the stability of solutions to (37)-( 41) with respect to perturbations in the initial conditions and the problem parameters has been analysed in [4]. Here we use a similar reasoning to extend our discretization scheme and error estimates to gas networks.…”

“…structure becomes apparent by an appropriate reformulation of the equations and a variational characterization of its solutions, which immediately leads to an energy dissipation inequality; see [4,5] and Section 2 fro details. In the weak formulation, the coupling and boundary conditions are incorporated variationally, which allows for a structure-preserving discretization by Galerkin projection.…”

“…The consideration of shocks or discontinuities, as in [14], is beyond the scope of this work. Our analysis is based on discrete stability of the proposed scheme, which is established via relative energy estimates and uses similar arguments as in [4], where the asymptotic limit ε ց 0 of (1)- (2) was investigated on the continuous level. Relative energy or entropy estimates are a well known tool for the analysis of quasi-linear partial differential equations; see [15] for an overview on its use in parabolic equations and [16] for applications in hyperbolic balance laws.…”