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.
Note: Updated 2003-03-21
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