Damped wave systems on networks: exponential stability and uniform approximations
- 71 Downloads
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.
Mathematics Subject Classification35L05 35L50 65L20 65M60
The authors are grateful for financial support by the German Research Foundation (DFG) via Grants IRTG 1529 and TRR 154 subproject C04, and by the “Excellence Initiative” of the German Federal and State Governments via the Graduate School of Computational Engineering GSC 233 at Technische Universität Darmstadt.
- 3.Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations Volume 25 of Studies in Mathematics and Its Applications. North-Holland, Amsterdam (1991)Google Scholar
- 4.Banks, H.T., Ito, K., Wang, C.: Exponentially stable approximations of weakly damped wave equations. In: Estimation and Control of Distributed Parameter Systems, Volume 100 of International Series of Numerical Mathematics, pp. 6–33. Birkhäuser, Basel (1991)Google Scholar
- 8.Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: Mixed Finite Elements, Compatibility Conditions, and Applications Volume 1939 of Lecture Notes in Mathematics. Springer, Berlin (2008)Google Scholar
- 14.Cramer, D., Latushkin, Y.: Gearhart–prüss theorem in stability for wave equations: a survey. In: Lecture Note in Pure and Applied Mathematics, pp. 105–119 (2003)Google Scholar
- 15.Dáger, R., Zuazua, E.: Wave propagation, observation and control in \(1-d\) flexible multi-structures Volume 50 of Mathématiques and Applications (Berlin) [Mathematics and Applications]. Springer, Berlin (2006)Google Scholar
- 17.Egger, H., Kugler, T.: Uniform exponential stability of Galerkin approximations for damped wave systems. arXiv:1511.08341 (2015)
- 25.Günther, M., Feldmann, W., ter Maten, J.: Modelling and discretization of circuit problems. In: Ciarlet, P.G. (ed.) Handbook of Numerical Analysis. vol. XIII, pp. 523–659. Elsevier, Amsterdam (2005)Google Scholar