An Accuracy Progressive Sixth-Order Finite-Difference Scheme

Abstract

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 for numerical calculation is proposed. The major features of the AP scheme are three-point, high-order accuracy, and accuracy progressive. The lower-order scheme acts as a ‘‘source’’ term in the higher-order scheme. This treatment keeps three-point schemes with high accuracy. The analytical error estimation shows the sixth-order accuracy that the AP scheme can reach. The Fourier analysis of errors indicates the accuracy improvement from lower-order to higher-order AP schemes. The Princeton Ocean Model (POM) implemented for the Japan/East Sea (JES) is used to evaluate the AP scheme. Consider a horizontally homogeneous and stably stratified JES with realistic topography. Without any forcing, initially motionless ocean will keep motionless forever; that is to say, there is a known solution (V 5 0). Any nonzero model velocity 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 unforced flow in the JES. The three-point sixthorder AP scheme is shown to have error reductions by factors of 10‐20 compared to the second-order difference scheme. Due to their three-point grid structure, the AP schemes can be easily applied to current ocean and atmospheric models.