Summation by parts operators for finite difference approximations of second derivatives

Finite difference operators approximating second derivatives and satisfying a summation by parts rule have been derived for the fourth, sixth and eighth order case by using the symbolic mathematics software Maple. The operators are based on the same norms as the corresponding approximations of the first derivative, which makes the construction of stable approximations to general parabolic problems straightforward. The error analysis shows that the second derivative approximation can be closed at the boundaries with an approximation two orders less accurate than the internal scheme, and still preserve the internal accuracy.