Mathematical modelling of one-way pressure-activated valves for gas

We model the gas flow through two pipes characterized by a maximal capacity and joined by a one-way, pressure-activated valve that can be partially open. Along the pipes, the gas flow is described by the one-dimensional isothermal Euler system. We provide a general framework to model the valve through suitable coupling conditions encoded by specially designed coupling Riemann solvers, which relate the passing flow to the pressure difference across the valve. Within this framework, we introduce two coupling Riemann solvers that respectively maximize and minimize the valve opening. We analyze their mathematical properties, including coherence and $\\mathrm{L}^1_{\\mathrm{loc}}$-continuity with respect to the initial data and valve parameters. A key feature of the model is the lack of coherence for some solvers, a phenomenon related to valve chattering. We present numerical experiments based on the Glimm scheme, which illustrate the impact of the valve parameters on chattering and highlight the numerical instabilities related to incoherent couplings.