We consider the numerical approximation of linear damped wave systems
by Galerkin approximations in space and appropriate time-stepping schemes. Based on
a dissipation estimate for a modified energy, we prove exponential decay of the physical
energy on the continuous level provided that the damping is effective everywhere in the
domain. The methods of proof allow us to analyze also a class of Galerkin approximations
based on a mixed variational formulation of the problem. Uniform exponential stabil-
ity can be guaranteed for these approximations under a general compatibility condition
on the discretization spaces. As a particular example, we discuss the discretization by
mixed finite element methods for which we obtain convergence and uniform error esti-
mates under minimal regularity assumptions. We also prove unconditional and uniform
exponential stability for the time discretization by certain one-step methods. The valid-
ity of the theoretical results as well as the necessity of some of the conditions required
for our analysis are demonstrated in numerical tests
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.
We consider the identification of a nonlinear friction law in a one-dimensional
damped wave equation from additional boundary measurements. Well-posedness of the
governing semilinear hyperbolic system is established via semigroup theory and con-
traction arguments. We then investigte the inverse problem of recovering the unknown
nonlinear damping law from additional boundary measurements of the pressure drop
along the pipe. This coefficient inverse problem is shown to be ill-posed and a varia-
tional regularization method is considered for its stable solution. We prove existence of
minimizers for the Tikhonov functional and discuss the convergence of the regularized so-
lutions under an approximate source condition. The meaning of this condition and some
arguments for its validity are discussed in detail and numerical results are presented for
illustration of the theoretical findings
We consider the numerical approximation of compressible flow in a pipe net-
work. Appropriate coupling conditions are formulated that allow us to derive a variational
characterization of solutions and to prove global balance laws for the conservation of mass
and energy on the whole network. This variational principle, which is the basis of our fur-
ther investigations, is amenable to a conforming Galerkin approximation by mixed finite
elements. The resulting semi-discrete problems are well-posed and automatically inherit the
global conservation laws for mass and energy from the continuous level. We also consider the
subsequent discretization in time by a problem adapted implicit time stepping scheme which
leads to conservation of mass and a slight dissipation of energy of the full discretization.
The well-posedness of the fully discrete scheme is established and a fixed-point iteration is
proposed for the solution of the nonlinear systems arising in every single time step. Some
computational results are presented for illustration of our theoretical findings and for demon-
stration of the robustness and accuracy of the new method