Consistent SUPG-method for transient transport problems: Stability and convergence

We consider the time/space discretization of the transient advection equation. Discretization in space is performed by the streamline upwind Petrov-Galerkin method and in time we use an A-stable finite difference operator. The formulation is strongly consistent in the sense that the time derivative is included in the stabilization term. Uniform stability of the general formulation is proved under a regularity condition on data, or a moderate inverse CFL-condition that allows for optimal choices of the discretization parameters. Both the backward Euler method (BDF1), the Crank-Nicolson scheme and the second-order backward differentiation formula (BDF2) enter the framework and quasi-optimal convergence is proved for these schemes.