A relaxation scheme for the equations of isentropic gas dynamics on a network with jump transmission conditions

This paper addresses the numerical modeling of isentropic gas dynamics on one-dimensional networks, focusing on the Euler equations with a novel class of transmission conditions at network junctions, termed the Jump Transmission Condition (JTC). Unlike traditional models that enforce continuity of density at junctions, the (JTC) allows for a density jump proportional to the flux, reflecting phenomena such as bottlenecks in biological or traffic networks. The authors propose a relaxation scheme based on the vector BGK approach, which ensures mass conservation and enforces the (JTC) without requiring the explicit solution of the Riemann problem at the junction. The scheme is analyzed for its mathematical properties, including entropy dissipation and positivity, and is compared with classical solvers based on the explicit solution of the Riemann problem. Numerical experiments on simple and complex networks demonstrate the accuracy, robustness, and flexibility of the proposed method, especially in handling both subsonic and supersonic regimes and in extending to general networks.