  We explore in this article the possibilities and limitations of the so-called
energy method for analyzing the stability of finite difference approximations
to the transport equation with extrapolation numerical boundary conditions at
the outflow boundary. We first show that for the most simple schemes, namely
the explicit schemes with a three point stencil, the energy method can be
applied for proving stability estimates when the scheme is implemented with
either the first or second order extrapolation boundary condition. We then
examine the case of five point stencils and give several examples of schemes
and second order extrapolation numerical boundary conditions for which the
energy method produces stability estimates. However, we also show that for the
standard first or second order translatory extrapolation boundary conditions,
the energy method cannot be applied for proving stability of the classical
fourth order scheme originally proposed by Strang. This gives a clear
limitation of the energy method with respect to the more general approach based
on the normal mode decomposition.
