Damped wave systems on networks: Exponential stability and uniform approximations
Provides a theoretical foundation for stability of damped wave propagation in pipe networks, relevant for engineering applications like gas or water distribution.
This paper proves exponential stability and convergence to equilibrium for damped wave systems on networks, generalizing results from single pipes to network topologies. Numerical tests with mixed finite elements confirm the theoretical decay rate bounds.
We consider a damped linear hyperbolic system modelling 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 proof of the exponential stability estimate is based on a variational formulation of the problem, some graph theoretic results, and appropriate energy estimates. The main arguments are rather generic and can be applied also for the analysis of Galerkin approximations. Uniform exponential stability can be guaranteed for the resulting semi-discretizations under mild compatibility conditions on the approximation spaces. 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.