How to reduce the computational error is a key issue in numerical modeling
and simulation. The higher the order of the difference scheme, the less the
truncation error and the more complicated the computation. For compromise,
a simple, three-point accuracy progressive (AP) finite-difference scheme f
or numerical calculation is proposed. The major features of the AP scheme a
re three-point, high-order accuracy, and accuracy progressive. The lower-or
der scheme acts as a "source'' term in the higher-order scheme. This treatm
ent keeps three-point schemes with high accuracy. The analytical error esti
mation shows the sixth-order accuracy that the AP scheme can reach. The Fou
rier analysis of errors indicates the accuracy improvement from lower-order
to higher-order AP schemes. The Princeton Ocean Model (POM) implemented fo
r the Japan/East Sea (JES) is used to evaluate the AP scheme. Consider a ho
rizontally homogeneous and stably stratified JES with realistic topography.
Without any forcing, initially motionless ocean will keep motionless forev
er; that is to say, there is a known solution (V = 0). Any nonzero model ve
locity can be treated as an error. The stability and accuracy are compared
with those of the second-order scheme in a series of calculations of unforc
ed flow in the JES. The three-point sixth-order AP scheme is shown to have
error reductions by factors of 10-20 compared to the second-order differenc
e scheme. Due to their three-point grid structure, the AP schemes can be ea
sily applied to current ocean and atmospheric models.