An asymptotic-preserving scheme for isentropic flow in pipe networks

We considered the simulation of isentropic flow in pipelines and pipe networks. Standard operating conditions in pipe networks suggested an emphasis to simulate low Mach and high friction regimes—however, the system was stiff in these regimes and conventional explicit approximation techniques proved quite costly and often impractical. To combat these inefficiencies, we developed a novel asymptotic-preserving scheme that was uniformly consistent and stable for all Mach regimes. The proposed method for a single pipeline followed the flux splitting suggested in Haack et al., in which the flux was separated into stiff and non-stiff portions then discretized in time using an implicit-explicit approach. The non-stiff part was advanced in time by an explicit hyperbolic solver; we opted 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 reduced this implicit stage to a discretization of a linear elliptic equation. To extend to full pipe networks, the scheme on a single pipeline was paired with coupling conditions defined at pipe-to-pipe intersections to ensure a mathematically well-posed problem. We showed that the coupling conditions remained well-posed at the low Mach/high friction limit—which, when used to define the ghost cells of each pipeline, resulted in a method that was accurate across these intersections in all regimes. The proposed method was tested on several numerical examples and produced accurate, non-oscillatory results with run times independent of the Mach number.