On some stable boundary closures of finite difference schemes for the transport equation

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.