Numerical stabilization with boundary controls for hyperbolic systems of balance laws

Abstract

In this dissertation, boundary stabilization of a linear hyperbolic system of balance laws is considered. Of particular interest is the numerical boundary stabilization of such systems. An analytical stability analysis of the system will be presented as a preamble. A discussion of the application of the analysis on speci c examples: telegrapher equations, isentropic Euler equations, Saint-Venant equations and Saint-Venant-Exner equations is also presented. The rst order explicit upwind scheme is applied for the spatial discretization. For the temporal discretization a splitting technique is applied. A discrete ²−Lyapunov function is employed to investigate conditions for the stability of the system. A numerical analysis is undertaken and convergence of the solution to its equilibrium is proved. Further a numerical implementation is presented. The numerical computations also demonstrate the stability of the numerical scheme with parameters chosen to satisfy the stability requirements.