On the order of accuracy for difference approximations of initial-boundary value problems

Finite difference approximations of the second derivative in space appearing in Finite difference approximations of the second derivative in space appearing in parabolic, incompletely parabolic and second order hyperbolic partial differential equations is considered. If the solution is pointwise bounded, we prove that finite difference approximations of those classes of equations can be closed with two orders less accuracy at the boundaries without reducing the global order of accuracy.