We consider the simulation of isentropic flow in pipelines and pipe networks. Standard operating conditions in pipe networks suggest an emphasis to simulate low Mach and high friction regimes -- however, the system is stiff in these regimes and conventional explicit approximation techniques prove quite costly and often impractical. To combat these inefficiencies, we develop a novel asymptotic-preserving scheme that is uniformly consistent and stable for all Mach regimes. The proposed method for a single pipeline follows the flux splitting suggested in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955--980], in which the flux is separated into stiff and non-stiff portions then discretized in time using an implicit-explicit approach. The non-stiff part is advanced in time by an explicit hyperbolic solver; we opt for the second-order central-upwind finite volume scheme. The stiff portion is advanced in time implicitly using an approach based on Rosenbrock-type Runge-Kutta methods, which ultimately reduces this implicit stage to a discretization of a linear elliptic equation. To extend to full pipe networks, the scheme on a single pipeline is paired with coupling conditions defined at pipe-to-pipe intersections to ensure a mathematically well-posed problem. We show that the coupling conditions remain well-posed in the low Mach/high friction limit -- which, when used to define the ghost cells of each pipeline, results in a method that is accurate across these intersections in all regimes. The proposed method is tested on several numerical examples and produces accurate, non-oscillatory results with run times independent of the Mach number.

翻译：本文研究管道及管道网络中可压缩等熵流动的数值模拟问题。管道网络的标准运行工况表明，需要重点模拟低马赫数和高摩擦区域——然而在这些区域中系统呈现刚性，传统的显式近似方法计算成本高昂且往往不切实际。为克服这些效率缺陷，我们提出了一种新颖的渐近保持格式，该格式在所有马赫数区域均具有一致收敛性和稳定性。针对单管道的所提方法遵循[Haack等人，Commun. Comput. Phys., 12 (2012), pp. 955--980]中建议的通量分裂策略，即将通量分解为刚性部分与非刚性部分，随后采用隐式-显式方法进行时间离散。非刚性部分通过显式双曲求解器进行时间推进，我们选用二阶中心迎风有限体积格式。刚性部分则采用基于Rosenbrock型Runge-Kutta方法的隐式格式进行时间推进，最终将该隐式阶段简化为线性椭圆方程的离散形式。为扩展至完整管道网络，将单管道格式与管道交汇处定义的耦合条件相结合，从而保证数学上的适定性。我们证明耦合条件在低马赫数/高摩擦极限下仍保持适定性——当将其用于定义各管道的虚拟单元时，所得方法能在所有流动区域中精确处理交汇界面。通过多个算例验证，所提方法能够产生精确无振荡的结果，且计算时间与马赫数无关。