@TechReport{ it:2003-012,
author = {Ken Mattsson and Jan Nordstr{\"o}m},
title = {Finite Difference Approximations of Second Derivatives on
Summation by Parts Form},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2003},
number = {2003-012},
month = feb,
note = {Updated 2003-03-21},
abstract = {Finite difference approximations of second derivatives,
which satisfy a summation by parts rule have been derived
for the 4th, 6th and 8th order case, by using the symbolic
mathematics software Maple. The operators are based on the
same norms as the corresponding approximations of the first
derivate, which make the construction of stable
approximations to general parabolic problems
straightforward. The error analysis show 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.
However, if an energy estimate cannot be obtained, the
overall convergence rate is reduced by one order. }
}