Abstract. High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-byparts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area.

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.