Finally, solving for the second derivative we find
This result is another of those which are widely used in computational physics simulations. Note that one needs
three data values to obtain this second derivative. The centered character of the approximation has the
benefit that it does not build in any directional bias. Of course, should one need a second derivative at one
of the end points one would have to find a set of interior values which would also have the third derivative
terms cancelling. We leave such a construction as an exercise for the reader.