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

TL;DR

This paper introduces an asymptotic-preserving discretization scheme for simulating gas flow in pipe networks, ensuring accurate results across different flow regimes and large friction limits.

Contribution

It proposes a fully discrete scheme based on Hamiltonian reformulation, with convergence analysis and error bounds uniform in the friction parameter, applicable to pipe networks.

Findings

The scheme is asymptotic-preserving in the large friction limit.

Error bounds are uniform in the friction parameter.

Numerical tests validate the method's effectiveness.

Abstract

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.