Damped wave systems on networks: exponential stability and uniform approximations
Submission Status: | under review |
---|
- 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 realiza-
tion 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.