  Stability estimates for Streamline Upwind Petrov-Galerkin (SUPG) finite
element method with different time integration schemes for the solution of a
scalar transient convection-diffusion-reaction equation in a time-dependent
domain are derived. The deformation of the domain is handled with the arbitrary
Lagrangian-Eulerian (ALE) approach. In particular, the non-conservative form of
the ALE scheme is considered. The implicit Euler, the Crank-Nicolson, and the
backward-difference~(BDF-2) methods are used for temporal discretization. It is
shown that the stability of the semi-discrete (continuous in time) ALE-SUPG
equation is independent of the mesh velocity, whereas the stability of the
fully discrete problem is only conditionally stable. The theoretical
considerations are illustrated by a numerical example. Further, the dependence
of numerical solution on the choice of stabilization parameter $\delta_k$ is
also presented.
