  We study the stability of the two-dimensional Lax-Wendroff scheme with a
stabilizer that approximates solutions to the transport equation. The problem
is first analyzed in the whole space in order to show that the so-called energy
method yields an optimal stability criterion for this finite difference scheme.
We then deal with the case of a half-space when the transport operator is
outgoing. At the numerical level, we enforce the Neumann extrapolation boundary
condition and show that the corresponding scheme is stable. Eventually we
analyze the case of a quarter-space when the transport operator is outgoing
with respect to both sides. We then enforce the Neumann extrapolation boundary
condition on each side of the boundary and propose an extrapolation boundary
condition at the numerical corner in order to maintain stability for the whole
numerical scheme.
