Finite difference approximations of the second derivative in space appearing in, parabolic, incompletely parabolic systems of, and second order hyperbolic, partial differential equations are 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 boundary without reducing the global order of accuracy.
This result is generalised to initial-boundary value problems with an mth order principal part. Then, the boundary accuracy can be lowered m orders.
Further, it is shown that summation-by-parts operators with approximating second derivatives are pointwise bounded. Linear and nonlinear computations corroborates the theoretical results.
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