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.

中文翻译：

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管网中气体输送的一种渐近保持离散化格式

我们考虑模拟长管道和管网中气体的正压流动。基于控制系统的哈密顿方程，提出了一种使用空间混合有限元和时间隐式欧拉方法的完全离散逼近方案。假设存在一个远离真空的平滑亚音速解，基于相对能量估计提出了一个完整的收敛分析。特别注意建立在摩擦参数中一致的误差界限。因此，该方法和结果也涵盖了在渐近大摩擦极限中出现的抛物线问题。误差估计是针对单个管道详细得出的，但使用适当的耦合条件和问题的特定结构及其离散化，主要结果直接推广到管网。数值测试用于说明。