By requiring the minimum truncation error for the sixth-order difference
scheme we obtain the sixth-order compact scheme
The ''left'' boundary point is computed by
The ''right'' boundary point has the similar formulation.
The last terms in (4) and (12) suggest that the compact scheme can provide
more accuracy. The error ratio between the sixth-order compact and ordinary
schemes is
which means a more than 50% reduction of the truncation error when we use the
compact scheme for the 6th-order accuracy. We use the pressure gradient error
reduction in the 
-coordinate system as an example to illustrate the
benefit of using high-order compact difference schemes.