(1) The 
-coordinate, pressure gradient error depends on the choice of
difference schemes. By choosing an optimal scheme, we may reduce the error in
a great deal without increasing the horizontal resolution. Analytical analysis
shows that the truncation error ratio between the fourth-order scheme and the
second-order scheme is proportional to 
, and the truncation error
ratio between the sixth-order scheme and the second-order scheme is
proportional to 
The compact scheme may reduce near 30% error
for the fourth-order difference and more than 50% error for the sixth order difference.
(2) The Semi-Spectral Primitive Equation Model (SPEM) Version 3.9 is used to demonstrate the benefit of using the sixth-order scheme. A series of calculations of unforced flow in the vicinity of an isolated seamount are performed. Over a wide range of parameter space as well as a great parametric domain of numerical stability, the sixth-order scheme has error reductions by factors of 5 comparing to the fourth-order difference scheme, and by factors of 50 comparing to the second-order difference scheme. Furthermore, the compact scheme reduces error by 20-30% for the fourth-order difference and more than 50-55% for the sixth-order difference scheme.
(3) Using the sixth-order scheme does not require much more CPU time. Taking SPEM3.9 as an example, the CPU time for the sixth-order scheme is almost the same as for the fourth-order scheme, and 10% more than for the second-order scheme, and the CPU time for the compact scheme (both fourth- and sixth-order) is only 5% more than the ordinary scheme.
(4) Since the fourth-order different scheme has error reductions by factors of 10 comparing to the second-order difference scheme, there is no real advantage to going to a higher order scheme if the bottom topography is not too complicated. The need for a lot of accuracy will go up with increasingly complex bottom topography on small scales, so one might expect that future demand of the accuracy will increase as models strive for more realism.