The Unconditional Instability of Inflow-Dependent Boundary Conditions in Difference Approximations

It is well known that for a mixed initial-boundary hyberbolic system to be well-defined it is necessary to impose additional boundary conditions only on the inflow eigenspace of the problem. We prove the discrete analogue of the above concerning difference approximations to such a system; that is, imposing numerical boundary conditions which are at least zeroth-order accurate with an inflow part of the interior equations leads to unconditional instability.