Damped wave systems on networks: exponential stability and uniform approximations

We consider a damped linear hyperbolic system modeling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a unique steady state is proven in the absence of driving forces. Under mild assumptions on the network topology and the model parameters, we show exponential stability and convergence to equilibrium. This generalizes related results for single pipes and multi-dimensional domains to the network context. Our proofs are based on a variational formulation of the problem, some graph theoretic results, and appropriate energy estimates. These arguments are rather generic and allow us to consider also Galerkin approximations and to prove the uniform exponential stability of the resulting semi-discretizations under mild compatibility conditions on the approximation spaces. A subsequent time discretization by implicit Runge-Kutta methods then allows to obtain fully discrete schemes with uniform exponential decay behavior. A particular realization by mixed finite elements is discussed and the theoretical results are illustrated by numerical tests in which also bounds for the decay rate are investigated.