Approximations that result from the natural matching of two stable dissipative difference schemes across a coordinate line are shown to be stable. The basic idea is to reformulate the matching scheme consistent to an equivalent initial boundary value problem and to verify the algebraic conditions for stability of such systems. An interesting comparison to the above result is the case of redefinition of a scheme at a single point. In particular, we show that some unstable perturbations do not upset the stability of the Lax–Wendroff scheme.