The Numerical Solution of Compressible Flows in Time Dependent Domains Vaclav Kucera This work is considered with the numerical solution of inviscid compressible fluid flow in moving domains. Specifically, we assume the boundary part of the domain (impermeable walls) are time dependent . We consider the Euler equations, which describe the movement of inviscid compressible fluids. We present two formulations of the Euler equations in the ALE (Arbitrary Lagrangian-Eulerian) form. These two formulations are discretized in space by the discontinuous Galerkin method. We apply a semi-implicit linearization with respect to time to obtain a numerical scheme requiring the solution of only one linear system on each time level. We apply the method to two basic situations: flow in a channel with moving walls and flow around a moving (vibrating) profile.